Algebra and Number Theory Seminar
Questions or comments?

Posted September 10, 2003

Last modified January 27, 2004

Marco Schlichting, Universitat Essen, (Germany)

Negative K-theory of derived categories

Visit supported by Visiting Experts Program in Mathematics, Louisiana Board of Regents. LEQSF(2002-04)-ENH-TR-13

Algebra and Number Theory Seminar
Questions or comments?

Posted September 8, 2003

Last modified September 17, 2003

Helena Verrill, Mathematics Department, LSU

Examples of rigid Calabi-Yau 3-folds

Algebra and Number Theory Seminar
Questions or comments?

Posted September 11, 2003

Last modified September 17, 2003

Paul van Wamelen, Mathematics Department, LSU

Analytic Jacobians in Magma

Algebra and Number Theory Seminar
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Posted October 15, 2003

3:40 pm - 4:30 pm Room 239
Paul van Wamelen, Mathematics Department, LSU

Analytic Jacobians in Magma II

Algebra and Number Theory Seminar
Questions or comments?

Posted September 12, 2003

Last modified September 17, 2003

Charles Neal Delzell, Mathematics Department, LSU

A generalization of Polya's theorem to signomials with rational exponents

Algebra and Number Theory Seminar
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Posted November 3, 2003

3:40 pm - 4:30 pm Lockett 282
Eric Baxter, University of New Orleans

Prime time

Algebra and Number Theory Seminar
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Posted March 23, 2004

Last modified April 16, 2004

Ling Long, Iowa State University

On Atkin-Swinnerton-Dyer congruence relations

Visit supported by Visiting Experts Program in Mathematics, Louisiana

Board of Regents LEQSF(2002-04)-ENH-TR-13

Algebra and Number Theory Seminar
Questions or comments?

Posted April 26, 2004

Last modified April 27, 2004

Paulo Lima-Filho, Texas A&M

On the RO(Z/2)-graded equivariant cohomology ring of real quadrics

ABSTRACT: We provide a complete presentation of the RO(Z/2)-graded

equivariant cohomology ring of real quadrics under the action of the

Galois group. Then we exhibit its relation to classical objects in

topology and to motivic cohomology over the reals.

Visit supported by Visiting Experts Program in Mathematics, Louisiana Board of

Regents LEQSF(2002-04)-ENH-TR-13

Algebra and Number Theory Seminar
Questions or comments?

Posted September 9, 2004

Last modified September 13, 2004

Helena Verrill, Mathematics Department, LSU

Finding the Picard Fuchs differential equations of certain families of Calabi-Yau varieties

Algebra and Number Theory Seminar
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Posted September 17, 2004

3:10 pm - 4:00 pm Lockett 282
Jerome W. Hoffman, Mathematics Department, LSU

Modular forms on noncongruence subgroups and Atkin-Swinnerton_Dyer congruences

Algebra and Number Theory Seminar
Questions or comments?

Posted September 9, 2004

Last modified September 13, 2004

Marco Schlichting, Louisiana State University

Higher algebraic K-theory of forms and Karoubi's fundamental theorem

Algebra and Number Theory Seminar
Questions or comments?

Posted September 9, 2004

Last modified September 30, 2004

Ambar Sengupta, Mathematics Department, LSU

Calculus Reform, or How (super)Algebra simplifies Calculus (on manifolds)

Algebra and Number Theory Seminar
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Posted September 17, 2004

Last modified October 19, 2004

Pramod Achar, Mathematics Department, LSU

Hecke algebras and complex reflection groups

Algebra and Number Theory Seminar
Questions or comments?

Posted September 10, 2004

Last modified October 22, 2004

Robert Perlis, Mathematics Department, LSU

Disconnected thoughts on Klein's four group

Algebra and Number Theory Seminar
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Posted December 9, 2005

Locket 285
Marie-José Bertin, Université Pierre et Marie Curie, Paris

TBA

Algebra and Number Theory Seminar
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Posted March 4, 2005

2:30 pm - 3:30 pm Lockett 282
Augusto Nobile, Mathematics Department, LSU

Algorithmic equiresolution

Algebra and Number Theory Seminar
Questions or comments?

Posted March 9, 2005

Last modified March 11, 2005

Jean Bureau, Louisiana State University

The Four Conjecture

Algebra and Number Theory Seminar
Questions or comments?

Posted March 15, 2005

3:10 pm - 4:00 pm Lockett 282
Jurgen Hurrelbrink, Mathematics Department, LSU

Quadratic Forms over Fields: The Splitting Pattern Conjecture

Algebra and Number Theory Seminar
Questions or comments?

Posted March 15, 2005

3:10 am - 4:00 pm Lockett 282
Preeti Raman, Rice University

Hasse Principle for Classical groups

*Abstract*: I will discuss a conjecture due to Colliot-Thelene about Hasse
principle for algebraic groups defined over the function field of a curve
over a number field. I will also describe its relation to the
classification of hermitian forms over such fields.

Algebra and Number Theory Seminar
Questions or comments?

Posted April 4, 2005

Last modified April 7, 2005

Debra Czarneski, LSU

Zeta Functions of Finite Graphs

Algebra and Number Theory Seminar
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Posted April 11, 2005

Last modified April 14, 2005

James Madden, Mathematics Department, LSU

Ways of ordering real algebras

Algebra and Number Theory Seminar
Questions or comments?

Posted April 20, 2005

2:40 pm - 3:30 pm Lockett 282
Helena Verrill, Mathematics Department, LSU

More modular Calabi-Yau threefolds

Algebra and Number Theory Seminar
Questions or comments?

Posted September 12, 2005

3:30 pm - 4:30 pm 285 Lockett Hall
Jerome W. Hoffman, Mathematics Department, LSU

Koszul Duality

Algebra and Number Theory Seminar
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Posted September 14, 2005

2:35 pm - 3:30 pm Lockett 285
Jerome W. Hoffman, Mathematics Department, LSU

Koszul Duality II

continuation of previous algebra seminar

Algebra and Number Theory Seminar
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Posted September 21, 2005

Last modified September 28, 2005

Jerome W. Hoffman, Mathematics Department, LSU

Koszul Duality III

continuation of previous algebra seminar

Algebra and Number Theory Seminar
Questions or comments?

Posted October 4, 2005

3:30 pm Locket 285
Marco Schlichting, Louisiana State University

Algebraic K-theory of singular varieties and a conjecture of Weibel

Algebra and Number Theory Seminar
Questions or comments?

Posted October 4, 2005

3:40 pm Locket 285
Helena Verrill, Mathematics Department, LSU

Modular forms and Ramanujan's series for 1/pi

Algebra and Number Theory Seminar
Questions or comments?

Posted October 14, 2005

3:40 pm Locket 285
Pramod Achar, Mathematics Department, LSU

Koszul duality in representation theory

Algebra and Number Theory Seminar
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Posted October 15, 2005

Last modified November 2, 2005

Jorge Morales, Mathematics Department, LSU

Quaternion orders, ternary quadratic forms and hyperelliptic curves

Algebra and Number Theory Seminar
Questions or comments?

Posted November 9, 2005

3:40 pm Locket 285
Jorge Morales, Mathematics Department, LSU

Quaternion orders, ternary quadratic forms and hyperelliptic curves, part II

Algebra and Number Theory Seminar
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Posted November 9, 2005

3:40 pm Locket 285
Pramod Achar, Mathematics Department, LSU

How I learned to stop worrying and love stacks

Algebra and Number Theory Seminar
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Posted November 4, 2005

Last modified November 9, 2005

Planning meeting to decide graduate courses in algebra for next year

Algebra and Number Theory Seminar
Questions or comments?

Posted December 2, 2005

3:30 pm Locket 285
Pramod Achar, Mathematics Department, LSU

Stacks II

Before the talk, we will have a 15 minute discussion of graduate courses for next year. Graduate students welcome.

Algebra and Number Theory Seminar
Questions or comments?

Posted December 9, 2005

Last modified December 18, 2005

Marie-José Bertin, Université Pierre et Marie Curie, Paris

Lehmer's problem and Mahler measure

Algebra and Number Theory Seminar
Questions or comments?

Posted January 23, 2006

2:30 pm Locket 285
Marco Schlichting, Louisiana State University

Stabilized Witt groups, Ranicki's lower L groups, and blow ups

Algebra and Number Theory Seminar
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Posted February 1, 2006

2:30 pm Locket 276
James Madden, Mathematics Department, LSU

Orderings of commutative rings with nilpotents

Algebra and Number Theory Seminar
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Posted January 31, 2006

Last modified February 14, 2006

Edith Adan-Bante, University of Southern Mississippi Gulf Coast

On Conjugacy Classes and Finite Groups

Algebra and Number Theory Seminar
Questions or comments?

Posted March 2, 2006

2:30 pm Locket 276
Juan Marco Cervino, University of Goettingen

The Minkowski-Siegel formula for quadratic bundles on curves

Algebra and Number Theory Seminar
Questions or comments?

Posted March 22, 2006

2:40 pm Locket 276
Jerome W. Hoffman, Mathematics Department, LSU

Koszul duality for multigraded algebras

Algebra and Number Theory Seminar
Questions or comments?

Posted March 27, 2006

2:40 pm Locket 276
Marco Schlichting, Louisiana State University

Koszul duality and the derived category of coherent sheaves on a quadric (after Kapranov)

Algebra and Number Theory Seminar
Questions or comments?

Posted April 5, 2006

Last modified April 6, 2006

Jeonghun Kim, Mathematics Department, LSU
LSU graduate student of Robert Perlis

Arf equivalence of quadratic fields

Note this is an hour later than the usual algebra seminar time.

Algebra and Number Theory Seminar
Questions or comments?

Posted April 5, 2006

Last modified April 6, 2006

Jeonghun Kim, Mathematics Department, LSU
LSU graduate student of Robert Perlis

Arf equivalence of quadratic fields, Part II

Part I, given the previous week, is related to this talk, but not essential for understanding part II.

Algebra and Number Theory Seminar
Questions or comments?

Posted March 13, 2006

Last modified April 5, 2006

Edith Adan-Bante, University of Southern Mississippi Gulf Coast

On Characters and Finite Groups

Algebra and Number Theory Seminar
Questions or comments?

Posted September 5, 2006

3:40 pm Lockett 282
Seva Joukhovitski, Mathematics Department, LSU

Splitting varieties and Bloch-Kato Conjecture

Algebra and Number Theory Seminar
Questions or comments?

Posted September 12, 2006

3:40 pm Lockett 282
Seva Joukhovitski, Mathematics Department, LSU

Splitting varieties and Bloch-Kato Conjecture II

Algebra and Number Theory Seminar
Questions or comments?

Posted February 6, 2007

Last modified February 7, 2007

Max Karoubi, University of Paris 7

K-theory and characteristic classes in number theory

ABSTRACT: Let A be an arbitrary ring. We introduce a Dennis trace map mod n, from K_1(A;Z/n) to the Hochschild homology group with coefficients HH_1(A;Z/n). If A is the ring of integers in a number field, explicit elements of K_1(A,Z/n) are constructed and the values of their Dennis trace mod n are computed. If F is a quadratic field, we obtain this way non trivial elements of the ideal class group of A. If F is a cyclotomic field, this trace is closely related to Kummer logarithmic derivatives; this trace leads to an unexpected relationship between the first case of Fermat's last theorem, K-theory and the number of roots of Mirimanoff polynomials. This is joint work with Thierry Lambre, see ArXiv math.NT/0006237 for more details.

Algebra and Number Theory Seminar
Questions or comments?

Posted April 24, 2007

Last modified April 27, 2007

Jens Hornbostel, University of Regensburg, Germany

Rigidity theorems for A^1-representable theories

We prove that for a large class of A^1-representable theories including

all orientable theories it is possible to construct transfer maps and to

prove rigidity theorems similar to those of Gabber for algebraic

K-theory. This extends rigidity results

of Panin and Yagunov from algebraically closed fields to arbitrary

infinite ones.

Algebra and Number Theory Seminar
Questions or comments?

Posted November 2, 2007

Last modified November 4, 2007

Daniel Sage, Mathematics Department, LSU

Perverse coherent sheaves and special pieces in the unipotent variety

Algebra and Number Theory Seminar
Questions or comments?

Posted September 12, 2007

Last modified November 9, 2007

Zhaohu Nie, Texas A&M Department of Mathematics

Singularities of admissible normal functions

Abstract:

The first proof of the Lefschetz (1,1) theorem was given by

Poincare and Lefschetz using normal functions for a Lefschetz pencil.

The hope to generalize this method to higher codimensional Hodge

conjecture was blocked by the failure of Jacobian inversion. In another

direction, one can hope for an inductive proof of the Hodge conjecture

if for any primitive Hodge class one can find a, necessarily singular,

hypersurface to "capture part of it". Recently Green and Griffiths

introduced the notion of extended normal functions over higher

dimensional bases such that their singular loci corresponds to such

hypersurfaces. In this talk, we will present how to understand

singularities using the viewpoint of admissible normal functions, and

how the Hodge conjecture is then equivalent to the existence of

singularities. This is joint work with P. Brosnan, H. Fang and G.

Pearlstein.

Algebra and Number Theory Seminar
Questions or comments?

Posted November 14, 2007

Last modified November 19, 2007

Daniel Sage, Mathematics Department, LSU

Perverse coherent sheaves and special pieces in the unipotent variety, part 2

Algebra and Number Theory Seminar
Questions or comments?

Posted September 14, 2007

Last modified November 20, 2007

Pramod Achar, Mathematics Department, LSU

Staggered t-structures and equivariant coherent sheaves

Algebra and Number Theory Seminar
Questions or comments?

Posted February 14, 2008

Last modified March 31, 2008

David Treumann, Northwestern University

Staggered t-structures on toric varieties

Achar has introduced a family of t-structures, called staggered t-structures, on the derived category of equivariant coherent sheaves on a G-scheme. These generalize the perverse coherent t-structures of Bezrukavnikov and Deligne, their main point of interest being that they are more often self-dual. We will discuss the example of torus-equivariant sheaves on a toric variety. We will also indicate a similarity between the main new ingredient of Achar’s t-structures – what are called s-structures – and the weight-truncation formalism of Morel.

Algebra and Number Theory Seminar
Questions or comments?

Posted September 15, 2008

Last modified September 19, 2008

Christopher Bremer, Mathematics Department, LSU

Periods of Irregular Singular Connections

Let X be a nonsingular complex projective algebraic curve. Suppose that E is a vector bundle over X with meromorphic connection nabla,
where nabla has poles along a divisor D. If nabla has regular singularities along D, (E, nabla) is uniquely determined by its sheaf of horizontal sections scr(E) on the analytic points of XD. The classification of irregular singular connections requires an additional piece of data: a Stokes filtration on scr(E) defined on sectors around the singular points of nabla.

A theorem of Malgrange (1991) states that there is a quasi-isomorphism between the algebraic de Rham complex associated to (E, nabla), and the `moderate growth' cohomology of scr(E) + Stokes. In this talk, I will describe a method for computing the matrix coefficients, or `periods', of this map. In a later talk, I will discuss the epsilon factorization for the determinant of the period map.

Algebra and Number Theory Seminar
Questions or comments?

Posted September 24, 2008

Last modified October 3, 2008

Christopher Bremer, Mathematics Department, LSU

Periods of Irregular Singular Connections, Part II

Continuation of the September 23 seminar.

Algebra and Number Theory Seminar
Questions or comments?

Posted October 10, 2008

3:40 pm Lockett 235
Charles Neal Delzell, Mathematics Department, LSU

A new, simpler, finitary construction of the real closure of a discrete ordered field

Algebra and Number Theory Seminar
Questions or comments?

Posted October 14, 2008

Last modified October 24, 2008

Jerome W. Hoffman, Mathematics Department, LSU

Motives, algebraic cycles, and Hodge theory

Algebra and Number Theory Seminar
Questions or comments?

Posted October 10, 2008

Last modified November 3, 2008

Alexander Prestel, Universitaet Konstanz

Representing polynomials positive on a semialgebraic set

Algebra and Number Theory Seminar
Questions or comments?

Posted October 14, 2008

Last modified November 10, 2008

Piotr Maciak, Mathematics Department, LSU
Graduate Student

A short journey from Gaussian integers to Drinfeld modules

Algebra and Number Theory Seminar
Questions or comments?

Posted February 27, 2009

3:40 am - 4:30 am Lockett 285
Robert Perlis, Mathematics Department, LSU

The 1-2-3's of Zeta functions of Graphs

Abstract: In 1968, Ihara introduced the zeta function of a finite graph,
with important contributions coming later in papers of Hashimoto, Bass, and Stark and Terras. More recently Mizuno and Sato considered the zeta function of a fully directed graph. (Zeta functions are proliferating like kudzu! Somebody, please make them stop!) In 2003, Sato found a rational expression for the zeta function of a connected, simple, partially directed graph.

This talk will be an elementary introduction to the subject of zeta
functions of graphs (undirected, fully directed, partially directed) and end
with a new theorem giving an Ihara-type formula for the zeta function of any
partially directed graph without Sato's assumptions of connected and simple.

Algebra and Number Theory Seminar
Questions or comments?

Posted March 12, 2009

Last modified March 14, 2009

Robert Peck, Department of Music, Louisiana State University

Applications of Wreath Products to Music Theory

Wreath products are familiar structures in mathematics, but they are relatively new to music theory. This study proposes an investigation into the musical relevance of wreath products, drawing on examples from selected musical literature of the nineteenth and twentieth centuries. We begin by examining a few commonly used groups in music theory, and observe how we may use permutation isomorphism to relate certain orbit restrictions of these groups. Next, we define a direct product of such orbit restrictions. Finally, we allow a permutation of the orbit restrictions themselves, which yields a wreath product. We include examples from Robert Schumann's "Im wunderschoenen Monat Mai," from Dichterliebe, op. 48; Richard Wagner's Siegfried; and Anton Webern's Cantata, op. 29.

Algebra and Number Theory Seminar
Questions or comments?

Posted March 25, 2009

3:40 pm - 4:30 pm Lockett 285
Augusto Nobile, Mathematics Department, LSU

Algorithmic resolution and equiresolution of singularities

We'll review the theory of algorithmic (or constructive) resolution of singularities of algebraic varieties (and some related objects) in characteristic zero and discuss the problem of simultaneous resolution when we have a family, in a way compatible with a chosen resolution algorithm (even in the case when the parameter space is not reduced, e.g. the spectrum of an Artinian ring).

Algebra and Number Theory Seminar
Questions or comments?

Posted April 13, 2009

3:40 pm - 4:30 pm Lockett 285
Augusto Nobile, Mathematics Department, LSU

Algorithmic resolution and equiresolution of singularities II

Algebra and Number Theory Seminar
Questions or comments?

Posted September 2, 2009

3:40 pm Lockett 285
Marco Schlichting, Louisiana State University

Grothendieck-Witt groups and a counterexample to invariance under derived equivalences

Algebra and Number Theory Seminar
Questions or comments?

Posted October 5, 2009

3:40 pm Lockett 285
Helena Verrill, Mathematics Department, LSU

Noncongruence lifts of projective congruence subgroups

Algebra and Number Theory Seminar
Questions or comments?

Posted October 5, 2009

Last modified October 13, 2009

Heather Russell, Mathematics Department, LSU

A combinatorial construction of the Springer representation

Springer varieties are studied because their cohomology carries a natural

action of the symmetric group and their top-dimensional cohomology is

irreducible. In his work on tangle invariants, Khovanov constructed a

family of Springer varieties as subvarieties of a product of spheres. We

show that these varieties can be embedded antipodally in the product of

spheres and that the natural symmetric group action on the product induces

the Springer representation. Our construction admits an elementary (and

geometrically natural) combinatorial description, which we use to prove

that the Springer representation is irreducible in each degree. This work

is joint with Julianna S. Tymoczko at The University of Iowa.

Algebra and Number Theory Seminar
Questions or comments?

Posted October 7, 2009

Last modified October 20, 2009

Christopher Bremer, Mathematics Department, LSU

Moduli for connections of cuspidal type

In my last talk, I described the Riemann-Hilbert correspondence for irregular singular point connections. Although this theory dates back to the work of Malgrange and Sibuya in the 70s, the Riemann-Hilbert map itself was not well understood until recently. In the past decade, Boalch has shown that the Riemann-Hilbert map gives a symplectic isomorphism between a coarse moduli space of connections, and a Poisson Lie group of `Stokes multipliers.' The theory of fundamental strata is a combinatorial tool for describing connections of cuspidal type. Recent work (joint with D S Sage) has shown that the fundamental stratum of a connection can be used to generalize Boalch's work. I will describe our preliminary results, and give some indication of how strata characterize the irregular Riemann-Hilbert map in the cuspidal case.

Algebra and Number Theory Seminar
Questions or comments?

Posted October 5, 2009

Last modified October 26, 2009

Anthony Henderson, School of Mathematics and Statistics, University of Sydney

Enhancing the nilpotent cone

Many features of an algebraic group are controlled by the geometry of its nilpotent cone, which in the case of GL_n(C) is merely the variety N of n x n nilpotent matrices. The study of the orbits of the group in its nilpotent cone leads to combinatorial data relating to the representations of the Weyl group, via the famous Springer correspondence. In the case of GL_n(C), the basic manifestation of this correspondence is the fact that conjugacy classes of nilpotent matrices and irreducible representations of the symmetric group are both parametrized by partitions of n.

Pramod Achar and I have shown that studying the orbits of GL_n(C) in the enhanced

nilpotent cone C^n x N leads to exotic combinatorial data of type B/C (previously studied by Spaltenstein and Shoji). As I will explain, this is closely related to Syu Kato's exotic Springer correspondence for the symplectic group, and also to nilpotent orbits in characteristic 2.

Algebra and Number Theory Seminar
Questions or comments?

Posted October 7, 2009

Last modified November 4, 2009

Jerome W. Hoffman, Mathematics Department, LSU

L-functions and l-adic representations for modular forms

Algebra and Number Theory Seminar
Questions or comments?

Posted November 4, 2009

Last modified November 11, 2009

Moon Duchin, University of Michigan

Limit shapes in groups

Consider larger and larger metric spheres in a group. Under nice circumstances, these converge to a definite "limit shape" as the radius goes to infinity. For instance in finitely generated nilpotent groups one may use a rescaling dilation in the ambient Lie group to shrink down large spheres, and by work of Pansu (extended by Breuillard) this gives a well-defined limit. For a simple example, in the free abelian group Z^2, if we take the standard generating set, the limit shape is a diamond (and the limiting metric, for which this is the unit sphere, is the L^1 metric on the plane). It is natural to ask whether the counting measure on the discrete spheres converges to a measure on the limit shape. I'll discuss our work on this question, and give some ergodic applications and some averaging applications for limit shapes.

Parts of this project are joint work with Samuel Lelièvre, Christopher Mooney, and Ralf Spatzier.

Algebra and Number Theory Seminar
Questions or comments?

Posted October 5, 2009

Last modified November 12, 2009

Jorge Morales, Mathematics Department, LSU

Siegel's mass formula and averages of L-functions over function fields

Algebra and Number Theory Seminar
Questions or comments?

Posted November 25, 2009

Last modified November 30, 2009

Alexander Prestel, Universitaet Konstanz

Axiomatizing the complex unit disc

The Lefshetz-Principle from algebraic geometry says that every algebraic property over the field of complex numbers involving only polynomials, is also tue over any algebraically closed field in characteristic 0. We present a similar transfer principle involving in addition the absolute value of the complex field.

Algebra and Number Theory Seminar
Questions or comments?

Posted February 25, 2010

3:40 pm - 4:30 pm Lockett 285
David Gepner, University of Illinois at Chicago

K-theory and additive functors

Algebra and Number Theory Seminar
Questions or comments?

Posted February 25, 2010

Last modified March 11, 2010

Mark Watkins, University of Sydney

A polynomial version of Hall's conjecture

Hall's conjecture asks for small nonzero values of |x^3-y^2| for integers

x,y. The polynomial analogue is to ask for f(t)^3-g(t)^2 to be of small degree

(compared to that of f,g, which we take to be in \bar Q[t]). The ABC theorem

(of Davenport and Mason) gives an explicit lower bound here. Via the use of

Belyi functions and covers of P^1 (or work of Stothers), we can count the

number of (f,g) that meet this minimal degree, and this turns out to be

related to the Catalan numbers.

This leaves the question of actually exhibiting (f,g) that minimise the

degree. For instance, if there are 14 solutions, we might expect them all

to be Galois conjugate in a number field of degree 14. In joint work with

Noam Elkies, we explicitly construct solutions for many cases, using a

battery of techniques, the most notable of which is multi-dimensional p-adic

Newton iteration to solve polynomial system of equations (or at least find

isolated points on the solution variety). The fields of definition of these

solutions are ramified only at small primes, due to a theorem of Beckmann.

Algebra and Number Theory Seminar
Questions or comments?

Posted February 25, 2010

Last modified March 12, 2010

Robert Fitzgerald, Southern Illinois University

Extending Hurwitz's proof of the four square theorem

The four square theorem says every positive integer can be represented as a sum of four squares. Lagrange (1770) proved this via Euler's four square identity and a descent argument. Hurwitz (1919) gave a proof using a ring of quaternions whose key property is being norm Euclidean. There are six other norm forms that represent all positive integers. I discuss recent work to construct norm Euclidean rings of quaternions for these cases.

Algebra and Number Theory Seminar
Questions or comments?

Posted March 3, 2010

Last modified March 25, 2010

Charles Weibel, Rutgers University

A 1972 Question of Bass and Hochschild homology

Bass asked if R satisfies K_0(R)=K_0(R[x]) then is K_0(R[x,y])

any different? In joint work with Cortinas, Haesemeyer and Walker,

we show that the answer is 'no.'

Algebra and Number Theory Seminar
Questions or comments?

Posted March 22, 2010

Last modified March 29, 2010

Christopher Bremer, Mathematics Department, LSU

A geometric theory of fundamental strata

In this talk, I will describe a theory of fundamental strata for meromorphic connections developed in joint work with D. S. Sage. Fundamental strata were originally used by Bushnell, Kutzko, Howe and Moy to classify cuspidal representations of GL_n over a local field. In the geometric setting, fundamental strata play the role of the `leading term' of a connection. I will introduce the concept of a regular stratum, which generalizes a condition imposed by Boalch (and previously, by Jimbo, Miwa, and Ueno) to study the geometry of the irregular Riemann Hilbert map. Finally, I will describe an application of our theory to a particular case of the wildly-ramified geometric Langlands conjucture.

Algebra and Number Theory Seminar
Questions or comments?

Posted March 19, 2010

Last modified April 12, 2010

Daniel Sage, Mathematics Department, LSU

Moduli Spaces of Irregular Singular Connections

An important problem in the geometric Langlands correspondence is the construction of global meromorphic connections on the projective line with specified local behavior. Boalch has studied the moduli space of such connections in the case where the leading term of the connection is regular semisimple at each singular point. In this talk, I will describe joint work with Bremer in which we show how to construct moduli spaces of connections in much greater generality. I will define a more useful notion of the leading term of a connection in terms of fundamental strata, a concept adapted from the representation theory of p-adic groups. In particular, I will introduce the concept of a regular stratum; a formal connection containing a regular stratum generalizes the naive idea of a connection with regular semisimple leading term. I will then explain how to construct the moduli space of connections on the projective line with specified regular local formal isomorphism classes at a collection of singular points. This moduli space is a symplectic reduction of a direct product of manifolds encoding local data at the singularities. I will also show that this moduli space arises as a symplectic quotient of a smooth manifold by a torus action.

Algebra and Number Theory Seminar
Questions or comments?

Posted March 12, 2010

Last modified April 12, 2010

Jared Culbertson, Mathematics Department, LSU

Perverse Poisson sheaves on the nilpotent cone

Algebra and Number Theory Seminar
Questions or comments?

Posted March 20, 2010

3:40 pm Lockett 285
Jerome W. Hoffman, Mathematics Department, LSU

Infinitesimal structure of Chow groups

Algebra and Number Theory Seminar
Questions or comments?

Posted April 13, 2010

Last modified April 25, 2010

Paulo Lima-Filho, Texas A&M

Integral Deligne cohomology for real varieties and explicit regulators

Abstract: in this talk we introduce a novel version of Deligne cohomology for real

varieties for which bigraded ordinary equivaraint cohomology replaces the role of

singular cohomology in the complex case. We describe explicit "regulator maps" in

the level of complexes from Voevodsky's motivic complexes to Deligne cohomology, and

present several examples.

Algebra and Number Theory Seminar
Questions or comments?

Posted February 25, 2010

Last modified April 25, 2010

Leonardo Mihalcea, Baylor University

Varieties of rational curves in the Grassmannian

Let Ω1 and Ω2 be two Schubert varieties in the Grassmannian, in general position. Given this data, we consider two spaces: the space of rational curves joining Ω1 and Ω2 (a subvariety of the moduli space of stable maps), and the space obtained by taking the union of these curves (a subvariety of the Grassmannian). Both these spaces generalize the much studied Richardson varieties, and play a fundamental role in quantum cohomology. We will study basic properties of these spaces (normality, rationality, singularities), and discuss some applications in quantum K-theory of the Grassmannian and algebraic combinatorics. This is joint work with A. Buch, P.E. Chaput and N. Perrin.

Algebra and Number Theory Seminar
Questions or comments?

Posted April 23, 2010

3:40 pm Lockett 285
Amber Russell, Mathematics Department, LSU

Graham's variety and perverse sheaves on the nilpotent cone

In recent work, Graham has constructed a variety with a map to the nilpotent cone that is similar to the Springer resolution. However, Graham's map differs from the Springer resolution in that it is not in general an isomorphism over the principal orbit, but rather the universal covering map. This map gives rise to a certain semisimple perverse sheaf on the nilpotent cone. In this talk, we will describe the summands of this perverse sheaf via the cohomology of the fibers of Graham's map.

Algebra and Number Theory Seminar
Questions or comments?

Posted September 6, 2010

Last modified September 14, 2010

Sarah Kitchen, University of Utah and Universitaet Freiburg

Harish-Chandra modules and the geometry of partial flag varieties

Cohomological induction gives an algebraic method for constructing representations of a real reductive Lie group G from irreducible representations of parabolic subgroups. Beilinson-Bernstein localization alternatively gives a geometric method for constructing Harish-Chandra modules for G from certain representations of a Cartan subgroup. The duality theorem of Hecht, Milicic, Schmid and Wolf estabilishes a relationship between modules cohomologically induced from Borels and the cohomology of the D-modules on the complex flag variety for G determined by the Beilinson-Bernstein construction. The corresponding geometric constructions on partial flag variety introduce homological complications. In this talk, I will explain the generalization of the duality theorem to partial flag varieties, which fully recovers the composition factors of cohomologically induced modules arising from non-minimal parabolics.

Algebra and Number Theory Seminar
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Posted September 13, 2010

Last modified September 14, 2010

Pramod Achar, Mathematics Department, LSU

Hyperbolic localization and applications

This will be a mostly expository talk about T. Braden's "hyperbolic localization" functor. This is a geometric construction that is defined for varieties equipped with an action of the multiplicative group C^*, and it can be described in an elementary way using the language of ordinary algebraic topology. It turns to have very deep connections with purely algebraic aspects of the representation theory of algebraic groups, and it plays a central role in the proof of the celebrated geometric Satake equivalence.

Algebra and Number Theory Seminar
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Posted September 21, 2010

3:40 pm - 4:30 pm Lockett 240
Greg Muller, Department of Mathematics, LSU

Reflexive and Projective D-modules

I will discuss reflexive and projective D-modules, focusing on the simplest case, the Weyl algebras. They can be reduced to right ideals in D, which can be studied in terms of their images. There is a nice class of ideals on which this image-based technique is effective at producing new results and examples, as well as revealing connections to the `bispectral problem' in differential equations. I will review the general theory, our new results, give some new interesting examples, and discuss the application to the bispectral problem. Joint with Yuri Berest and Oleg Chalykh.

Algebra and Number Theory Seminar
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Posted October 20, 2010

Last modified October 25, 2010

Maria Vega, Mathematics Department, LSU

Twisted Frobenius-Schur Indicators for Hopf Algebras

The classical Frobenius-Schur indicators (FS indicators) for finite groups are virtual characters v_n(V) defined for any representation V and any n>=2. In the familiar case n=2, v_2 partitions the irreducible representations over C into real, complex, and quaternionic representations. In recent years, several generalizations of these invariants have been introduced. Bump and Ginzburg, building on earlier work of Mackey, have defined versions of these indicators which are twisted by an automorphism of the group. In another direction, FS indicators have been constructed for semisimple Hopf algebras; this is due to Linchenko and Montgomery for n=2 and Kashina, Sommerhauser, and Zhu for n>2. We have constructed a twisted version of FS indicators for semisimple Hopf algebras that includes all of the above versions as special cases and have similar properties. For example, the n=2 case leads to a partition of the irreducible representations into three classes. This is joint work with Daniel Sage.

Algebra and Number Theory Seminar
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Posted September 7, 2010

Last modified November 9, 2010

Mahir Can, Tulane University

Unipotent invariant (complete) quadrics

The variety of complete quadrics, which is used by Schubert in his famous computation of the number of space quadrics tangent to 9 quadrics in general position, is a particular compactification of the space of non-singular quadric hypersurfaces in n dimensional complex projective space. In this talk, towards a theory of Springer fibers for complete quadrics, I will describe our recent work on the unipotent invariant complete quadrics. These results involve interesting combinatorics, and in particular, give a

new q-analog of Fibonacci numbers.

This is joint work with Michael Joyce.

Algebra and Number Theory Seminar
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Posted February 17, 2011

Last modified February 24, 2011

Noriko Yui, Queen's University

The modularity of certain K3-fibered Calabi-Yau threefolds over **Q**

We consider certain K3-fibered Calabi-Yau threefolds defined over **Q**. These Calabi-Yau threefolds are constructed using the method of Voisin and Borcea, and are realized as smooth resolutions of quotients of *S* × *E* by some involution. (Here *S* is an algebraic K3 surface and *E* is an elliptic curve.)

First we will discuss the modularity of K3 surfaces *S*. We look into the famous 95 families of K3

surfaces found by Reid and Yonemura. Among them, we will pick K3 surfaces with involution. Our first result is to show that some of these K3 surfaces are of CM type.

Next, we will discuss the modularity of Calabi-Yau threefolds over **Q** obtained from products *S* × *E*. We establish the modularity (automorphicity) of some of these Calabi-Yau threefolds and also thier mirror partners (if exist), in the sense of Arthur and Clozel. Several explicit examples are discussed.

This reports on a joint work in progress with Y. Goto (Hakodate) and R. Livne (Jerusalem).

Algebra and Number Theory Seminar
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Posted February 7, 2011

Last modified March 1, 2011

Skip Garibaldi, Emory University

Matrix groups and diagonalizable matrices

This talk is about groups of matrices over a field, like *GL _{n}* (the group of

*n*-by-

*n*invertible matrices) or a special orthogonal group. Since Elie Cartan's 1894 PhD thesis, at least, the theory of such groups starts with studying a connected abelian subgroup consisting of diagonalizable matrices—we call such a thing a torus. This talk concerns the classical question: If every maximal torus in one group is isomorphic to a maximal torus in another group, are the two groups necessarily isomorphic? This problem is related to questions in differential geometry and classical algebra and there is serious recent progress.

This talk will discuss the recent solution of the problem over number fields (by Gopal Prasad and Andrei Rapinchuk in Pub. Math. IHES, and a small part due to the speaker) and a more elementary version with weaker results over general fields by the speaker and David Saltman.

Algebra and Number Theory Seminar
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Posted March 3, 2011

3:40 pm - 4:30 pm Lockett 277
Sabin Cautis, Columbia University

A categorification of the Heisenberg algebra

Algebra and Number Theory Seminar
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Posted February 17, 2011

Last modified March 11, 2011

Linhong Wang, Southeastern Louisiana University

Noncommutative infinite series rings

Skew power series rings T:=R[[y;tau,delta]], for suitably conditioned right noetherian complete semilocal rings R, automorphisms tau of R, and tau-derivations $delta$ of R, were introduced by Venjakob

in the study of noncommutative Iwasawa theory. In this talk, I will discuss iterated skew power series rings and skew inverse power series rings. With suitable base rings and defining relations, these noncommutative infinite series rings give new examples of local, noetherian, zariskian (in the sense of Li and Van Oystaeyen) domains that are related to quantum algebras. Our study on the q-commutative power series ring k_q[[x]] provides a detailed account of its prime ideal structure. Our results, parallel those found for

quantum affine spaces, include normal separation and finite stratification by commutative noetherian spectra. Combining this normal separation with results of Chan, Wu, Yekutieli, and Zhang, we are able to conclude that k_q[[x]] is catenary. Following the approach of Brown and Goodearl, we also show that links between prime ideals are provided by canonical automorphisms. The new results in this talk are joint work with Edward Letzter.

Algebra and Number Theory Seminar
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Posted February 16, 2011

Last modified March 11, 2011

Matthew Housley, University of Utah

TBA

Algebra and Number Theory Seminar
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Posted March 11, 2011

3:40 pm - 4:30 pm Lockett 277
Aaron Lauda, Columbia University

TBA

Algebra and Number Theory Seminar
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Posted March 9, 2011

3:40 pm - 4:30 pm Lockett 277
David Chapman, Mathematics Department, LSU
Graduate student

TBA

Algebra and Number Theory Seminar
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Posted August 18, 2011

Last modified August 23, 2011

Paul Smith, University of Washington

A 3-Calabi-Yau algebra with G_2 symmetry that is related to the octonions

This talk concerns an associative graded algebra A that is the enveloping algebra of a Lie algebra with exponential growth. The algebra is a coherent, 3-Calabi-Yau, Koszul algebra, and the exceptional group of type G_2 acts as automorphisms of it. The algebra A seems to have first appeared in a physics paper.

A can be defined in many ways. If V is the 7-dimensional irreducible representation of the complex semisimple Lie algebra of type G_2, then A is isomorphic to the tensor algebra T(V) modulo the ideal generated by the submodule of V \otimes V isomorphic to V.

Alternatively, A can be defined as a superpotential algebra derived from a 3-form on R^7 having an open GL(7) orbit and compact isotropy group. A can also be defined in terms of the product on the octonions. A can also be defined in terms of the exterior derivatives of seven 1-forms that appear in E. Cartan's "Five variables" paper.

Classification of the finite-dimensional representations of A is equivalent to classifying square matrices Y with purely imaginary octonion entries such that the imaginary part of Y^2 is zero. There is a derived equivalence relating graded A-modules to representations of a certain quiver (with relations). This equivalence is analogous to Beilinson's equivalence for the derived category of coherent sheaves on P^n.

A can also be defined in terms of the incidence relations for the Fano plane, the projective plane over the field of two elements. These incidence relations give the simplest example of a Steiner triple system. Mariano Suarez-Alvarez has shown that every Steiner triple system gives rise to an algebra analogous to A that is also coherent, 3-Calabi-Yau, and Koszul, though these more general algebras do not seem to have an interesting Lie group acting as automorphisms.

Algebra and Number Theory Seminar
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Posted August 18, 2011

Last modified August 23, 2011

Peter Fiebig, Universität Erlangen-Nürnberg

Moment graphs in topology and representation theory

Moment graphs originated in the work of Goresky, Kottwitz and MacPherson on the equivariant topology of complex varieties with a torus action. In particular, they showed how one can calculate the hypercohomology of a large class of equivariant sheaves using only their restriction to the 1-skeleton (i.e. the moment graph) of the torus action. Building on these ideas, Braden and MacPherson gave an explicit description of the equivariant intersection cohomology of certain complex varieties using sheaves on the moment graph. Now these sheaves also appear in the study of multiplicity questions in representation theory. When combined, one obtains proofs of fundamental conjectures of Lusztig and Kazhdan-Lusztig. In the talk I will explain these ideas in some detail.

Algebra and Number Theory Seminar
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Posted August 29, 2011

3:40 pm - 4:30 pm Lockett 240
Pramod Achar, Mathematics Department, LSU

Geometric Satake, Springer correspondence, and small representations

Let G be a reductive group, and let W be its Weyl group. (For example, take G = GL_n and W = S_n.) In this talk, I will explain how to construct a commutative diagram relating the following four things:

(1) Representations of W

(2) Geometry of the nilpotent cone for G

(3) Representations of G

(4) Geometry of the affine Grassmannian for G

Some parts of the commutative diagram are well-known: (1) and (2) are related by the Springer correspondence; (3) and (4) are related by the geometric Satake isomorphism; and there is a functor from (3) to (1) that can be described as "take the zero weight space." So the main point of the talk will to explain how to related (2) and (4). This is joint work with A. Henderson.

Algebra and Number Theory Seminar
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Posted August 22, 2011

Last modified September 11, 2011

Karl Mahlburg, Department of Mathematics, LSU

Coefficient Asymptotic for Kac-Wakimoto characters

In Kac and Peterson's study of characters for affine Lie algebras, they proved a number of "Denominator identities" that related the weight multiplicities of irreducible submodules to theta functions. They then used modular inversion formulas and Tauberian theorems in order to derive asymptotics for these weight multiplicities; one of the simplest examples of affine Lie algebras leads to Hardy and Ramanujan's famous formula for the asymptotics of p(n), the integer partition function.

In this talk I will present joint work with K. Bringmann on the characters for affine Lie superalgebras that were later introduced by Kac and Wakimoto. In this setting, the characters are products of theta functions and Appell-type sums, which have recently been studied using developments in the theory of mock modular forms and harmonic Maass forms. We find asymptotic series expansions for the coefficients of the characters with polynomial error.

Algebra and Number Theory Seminar
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Posted August 22, 2011

Last modified September 13, 2011

Elizabeth Dan-Cohen, Department of Mathematics, LSU

A Koszul category of representations of finitary Lie algebras

We find an interesting category of representations of the three simple finitary Lie algebras. The modules in question are weight modules for every splitting Cartan subalgebra. We describe the injective modules in this category, and show that the category is antiequivalent to the category of locally unitary finite-dimensional modules over a direct limit of finite-dimensional Koszul algebras. Joint with Ivan Penkov and Vera Serganova.

Algebra and Number Theory Seminar
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Posted August 30, 2011

3:40 pm - 4:30 pm Lockett 240
Greg Muller, Department of Mathematics, LSU

TBA

Algebra and Number Theory Seminar
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Posted October 11, 2011

3:40 pm - 4:30 pm Lockett 240
Jerome W. Hoffman, Mathematics Department, LSU

Galois representations and Humbert surfaces

Algebra and Number Theory Seminar
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Posted September 8, 2011

Last modified October 17, 2011

Amber Russell, Mathematics Department, LSU

Graham's variety and perverse sheaves on the nilpotent cone: Results in type A_n

In recent work, Graham has constructed a variety with a map to the nilpotent cone that is similar to the Springer resolution. However, Graham's map differs from the Springer resolution in that it is not in general an isomorphism over the principal orbit, but rather the universal covering map. This map gives rise to a certain semisimple perverse sheaf on the nilpotent cone. In this talk, we discuss the problem of describing the summands of this perverse sheaf. For type $A$, a key tool is a description of the affine paving of Springer fibers given by Tymozcko that lends itself nicely to understanding the fibers of Graham's map.

Algebra and Number Theory Seminar
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Posted August 30, 2011

3:40 pm - 4:30 pm Lockett 240
Christopher Bremer, Mathematics Department, LSU

TBA

Algebra and Number Theory Seminar
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Posted November 21, 2011

3:40 pm - 4:30 pm Lockett 240
Christopher Bremer, Mathematics Department, LSU

Flat $G$-bundles and regular strata

Let $G$ be a reductive group over a Laurent series (or $p$-adic) field. Broadly speaking, a fundamental stratum is a pair $(K, b)$ consisting of a ``compact'' subgroup $K < G$ and a character $b$ of $K$ that satisfies a non-degeneracy condition. The theory of fundamental strata was originally developed by Bushnell, Howe, Moy, and Prasad to study wildly ramified representations of $p$-adic groups, and this theory plays an important role in the parameterization of admissible GL_n representations. In this talk I will describe recent work with Sage on applications of fundamental strata to the study of flat $G$-bundles. One of our primary innovations is the notion of a ``regular'' stratum, which satisfies a graded version of regular semi-simplicity. I will first discuss results on the Deligne-Simpson problem and isomonodromic deformations of irregular singular flat GL_n bundles, and then indicate how this theory generalizes to the reductive case.

Algebra and Number Theory Seminar
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Posted November 23, 2011

1:40 pm - 3:30 pm Lockett 240
Pramod Achar, Mathematics Department, LSU

Introduction to the Hitchin fibration, Part I

I will repeat a talk given by Olivier Schiffmann at the Université de Caen on November 15, 2011, in preparation for the virtual seminar on November 29. Please contact me for lecture notes.

Algebra and Number Theory Seminar
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Posted November 23, 2011

8:45 am - 10:45 am Lockett 233
Olivier Schiffmann, Université Paris-Sud

Introduction to the Hitchin fibration, Part II

This will be a virtual seminar, joint with the "Groupe de travail en théorie de représentations" at the Université de Caen.

Algebra and Number Theory Seminar
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Posted November 28, 2011

3:40 pm - 4:30 pm Lockett Hall 240
Jorge Morales, Mathematics Department, LSU

Generic polynomials and Frobenius modules

I will begin by giving a brief historical introduction to the classical Noether problem on fields of invariants of finite groups and its relation with the inverse Galois problem. Then I will define the notion of generic polynomial and discuss some modern approaches to their construction, in particular the use of Matzat's "lower bound" theorem on Frobenius modules for the explicit construction of generic polynomials in characteristic p for groups of rational points of algebraic groups. This is (developing) joint work with REU student D. Tseng (MIT); it will be accessible to everyone.

Algebra and Number Theory Seminar
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Posted February 6, 2012

3:40 pm - 4:30 pm Lockett 277
Carl Mautner, Harvard University

Parity sheaves

One approach to Lie theory involves relating categories of representations to sheaves on singular algebraic varieties. This is advantageous in that sheaves can be studied locally. This technique has been quite successful in studying representations over fields of characteristic zero.

The usefulness of this approach often hinges on deep theorems about a class of objects called intersection cohomology sheaves. These theorems generalize classical results on the rational cohomology of smooth projective varieties.

One would like to be able to make use of this technique for representations over fields of positive characteristic. However, in this case, the theorems valid over characteristic zero no longer hold.

We consider a class of objects, parity sheaves, which tend to coincide with intersection cohomology sheaves in characteristic zero and have better behavior over fields of positive characteristic.

(Joint with D. Juteau and G. Williamson.)

Algebra and Number Theory Seminar
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Posted February 8, 2012

Last modified March 2, 2012

Abhinav Kumar, Massachusetts Institute of Technology

Hilbert modular surfaces and K3 surfaces

I will outline an approach to compute equations for Hilbert modular surfaces $Y_{-}(D)$, which are moduli spaces of principally polarized abelian surfaces with real multiplication by the full ring of integers of $Q(sqrt{D})$, based on moduli spaces of elliptic K3 surfaces. Using it we are able to calculate these surfaces for all fundamental discriminants less than 100, and analyze various arithmetic properties, such as rational points and curves which we can use to produce explicit genus 2 curves (or 1-parameter families of these) whose Jacobians have real multiplication. This is joint work with Noam Elkies.

Algebra and Number Theory Seminar
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Posted March 12, 2012

Last modified March 19, 2012

Laura Rider, Department of Mathematics, LSU
Graduate Student

A derived Springer correspondence for mixed perverse sheaves

The simplest case of the Springer correspondence can be understood with linear algebra and knowledge of the representation theory of the symmetric group. We write down the correspondence in this case and then review a geometric method for realizing the relationship. In this setting, the Springer correspondence can be realized as an equivalence between a certain category of perverse sheaves and the category of representations of the Weyl group. We explain how to extend this to a derived equivalence between modules over a graded ring related to W and a certain category of mixed perverse sheaves on the nilpotent cone.

Algebra and Number Theory Seminar
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Posted February 1, 2012

Last modified March 20, 2012

Inka Klostermann, University of North Carolina

Generalization of the Macdonald formula for Hall-Littlewood polynomials

Gaussent and Littelmann developed a formula for Hall-Littlewood polynomials in terms of one-skeleton galleries in the affine building. In type $A_n$, $B_n$ and $C_n$ these galleries can be described by using certain Young tableaux. In this talk I will explain how to translate the Gaussent-Littelmann formula into an easy purely combinatorial formula in terms of Young tableaux. It turns out that the resulting so-called combinatorial Gaussent-Littelmann formula coincides with the well-known Macdonald formula for Hall-Littlewood polynomials in type $A_n$.

Algebra and Number Theory Seminar
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Posted March 15, 2012

Last modified March 28, 2012

Peter Fiebig, Universität Erlangen-Nürnberg

Periodic structures in the affine category O at positive level

The categorical structure of the affine category O at positive level can be described in terms of affine moment graphs. Recently, Martina Lanini exhibited a periodic structure on moment graphs associated to maximally singular affine blocks, which yields a categorification of the stabilization phenomenon of parabolic affine KL-polynomials. I will report on the representation theoretic implications of Lanini's result and, in particular, I will explain how it is connected to a still conjectural structure of the affine category O at the critical level.

Algebra and Number Theory Seminar
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Posted March 13, 2012

Last modified April 13, 2012

Harold Williams, UC Berkeley

Loop Groups and Cluster Integrable Systems

While double Bruhat cells in simple algebraic groups have played a key role in the development of cluster algebras, their counterparts in general Kac-Moody groups have been less studied. In this talk I will explain how the double Bruhat decomposition of an affine Kac-Moody group can be used to construct a new class of completely integrable Hamiltonian systems. As an example we obtain the relativistic periodic Toda system, which we will see leads to a connection with recent work of Goncharov and Kenyon on integrable systems related to dimer models.

Algebra and Number Theory Seminar
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Posted March 15, 2012

3:40 pm - 4:30 pm Lockett 277
James Madden, Mathematics Department, LSU

Sheaves of ratios

Book V of Euclid's Elements is said to be an exposition of Eudoxus theory of ratio. In 1900, Hölder wrote a paper analyzing Book V, and in this connection he proved the fundamental theorem that every archimedean totally-ordered group is isomorphic to a subgroup of the additive reals. A theorem of Yosida (1942) states that every archimedean vector-lattice is a vector lattice of almost-everywhere real functions on a compact space. We can recover Yosida's Theorem by viewing Hölder's Theorem in an appropriate topos. This point of view also leads to improved versions of Yosida's Theorem. The talk illustrates how ideas that are taught in elementary school may, if analyzed with sufficient depth, have a bearing on research questions.

Algebra and Number Theory Seminar
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Posted April 24, 2012

3:40 pm Lockett 277
Zhibin Liang, Capital Normal University, Beijing

The non-commutative Iwasawa theory of modular forms

In this talk, we will discuss some new conjectures on critical values of L-functions twisted by a non-commutative Artin representation. We talk about some explicit computations and how this may contribute to a non-commutative Iwasawa theory.

Algebra and Number Theory Seminar
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Posted September 20, 2012

Last modified September 24, 2012

Greg Muller, Department of Mathematics, LSU

Superunital domains of cluster algebras

A cluster algebra is a type of commutative algebra with a set of distinguished generators, called cluster variables, with desirable combinatorial properties. For a given cluster algebra, we consider its cluster variables as functions on an octant in n-dimensions. The `superunital domain' is the subset of this octant on which each cluster variable is greater than or equal to one. The topology of the boundary encodes many of the desirable properties of the cluster variables. When the cluster algebra is `finite-type', the superunital domain is compact and possesses a natural volume form; I will mention some preliminary results with Joel Geiger and Karl Mahlburg on the integral of this form.

Algebra and Number Theory Seminar
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Posted October 5, 2012

3:30 pm - 4:30 pm Lockett 235
Pramod Achar, Mathematics Department, LSU

Schur-Weyl duality, nilpotent orbits, and tilting modules

This mostly expository talk will be about connections between the following four topics: (1) representations of the symmetric group S_n; (2) representations of the general linear group GL_n; (3) topology of the set of nilpotent matrices; (4) topology of the set of lattices in a vector space over the field of Laurent series. Some of these connections are very old: Issai Schur discovered a link between (1) and (2) more than 100 years ago. The topological aspects have been developed mostly since the mid-1970's, by Springer, Lusztig, Ginzburg, Mirkovic-Vilonen, and others; a nice unifying result has been proved by Carl Mautner. I will try to explain these ideas with concrete examples, and give one application: a new geometric proof, via Fourier transform on the nilpotent variety, of Ringel self-duality for Schur algebras. This last result is joint work of myself and C. Mautner.

Algebra and Number Theory Seminar
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Posted November 3, 2012

Last modified November 6, 2012

Myron Minn-Thu-Aye, Department of Mathematics, LSU
Graduate Student

Multiplicity formulas for perverse coherent sheaves on the nilpotent cone

Bezrukavnikov has shown that the category of perverse coherent sheaves on the nilpotent cone of a complex reductive algebraic group is quasi-hereditary. The Andersen-Jantzen sheaves play an important role, analogous to that of the Verma modules in category O. We describe progress towards computing multiplicities of simple objects in Andersen-Jantzen sheaves. The main tool is an equivalence between perfect complexes on the nilpotent cone and mixed sheaves on the affine Grassmannian.

Algebra and Number Theory Seminar
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Posted October 24, 2012

Last modified November 5, 2012

Joel Geiger, Department of Mathematics, LSU
Graduate Student

Noncommutative prime spectra of quantum Schubert cell algebras

The quantum Schubert cell algebras defined by De Concini, Kac, and Procesi and independently by Lusztig comprise a large and versatile collection of subalgebras of the positive part of the quantized universal enveloping algebras. In this talk I will outline two major approaches to understanding the noncommutative prime spectra of the quantum Schubert cell algebras --- a ring theoretic approach due to Gerard Cauchon and a representation theoretic approach due to Milen Yakimov. We answer two questions of Cauchon and Mérieaux, thereby unifying the two seemingly disparate approaches. Time permitting we will also investigate a result relating this unified approach to the theory of quantum cluster algebras. This work is joint with Milen Yakimov.

Algebra and Number Theory Seminar
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Posted January 21, 2013

Last modified January 28, 2013

Christopher Dodd, University of Toronto

Modules over algebraic quantizations and representation theory

Recently, there has been a great deal of interest in the theory of modules over algebraic quantizations of so-called symplectic resolutions. In this talk I'll discuss some new work—joint, and very much in progress—that open the door to giving a geometric description to certain categories of such modules; generalizing classical theorems of Kashiwara and Bernstein in the case of *D*-modules on an algebraic variety.

Algebra and Number Theory Seminar
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Posted March 5, 2013

3:30 pm - 4:20 pm Lockett 277
Ulrica Wilson, Morehouse College

Noncommutative division rings from Hamiliton to Albert to now

Thanks in large part to Galois, much is known about commutative division rings (aka fields). In this talk we will present some of the history, recent results, and open problems in the study of noncommutative division rings.

Algebra and Number Theory Seminar
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Posted March 6, 2013

3:30 pm - 4:20 pm Lockett 277
Amber Russell, University of Georgia

Cuspidal local systems and a decomposition involving perverse sheaves on the nilpotent cone

In a recent paper, Achar uses hyperbolic localization to give an orthogonal decomposition of the category of constructible sheaves on the nilpotent cone. In particular, he decomposes this category into those arising from the Springer sheaf and those not. In this talk, I will discuss the ongoing project to refine this decomposition using central character properties and Lusztig's cuspidal local systems. This is joint work with Laura Rider.

Algebra and Number Theory Seminar
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Posted March 12, 2013

Last modified April 22, 2013

Peter Samuelson, University of Toronto

Skein modules and the double affine Hecke algebra

The Kauffman bracket skein module is a vector space K_{q}(M) associated to a 3-manifold M and a parameter q ∈ C^{*}. We recall an old theorem which states that the colored Jones polynomials J_{n}(q, K) ∈ C[q,q^{−1}] of a knot K in S^{3} can be computed from K_{q}(S^{3} K). We also describe a theorem of Frohman and Gelca which shows that K_{q}(S^{3}K) is a module over the Z_{2}-invariant subalgebra of the quantum torus A_{q}. This subalgebra is the specialization at t=1 of the double affine Hecke algebra H(q,t), which is a 2-parameter family of algebras. We discuss deformations of K_{q}(S^{3} K) to a 2-parameter family of modules over H(q,t). Conjecturally, these lead to 2-variables polynomials J_{n}(q,t,K) which specialize to the colored Jones polynomials when t=1. (All terms in this abstract will be
defined, and this is work in progress with Yuri Berest.)

Algebra and Number Theory Seminar
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Posted May 17, 2013

Last modified May 24, 2013

Daniel Sternheimer, Rikkyo University & Institut de Mathématiques de Bourgogne

Altneuland in mathematical particle physics: back to the drawing board??

We describe work in progress and outline a "framework for conjectural frameworks" based on Flato's deformation philosophy, on joint works with or by Flato and coworkers (especially Fronsdal) since the 60's, and on discussions with many mathematicians and physicists in the past years. Namely we return to the old problem of connection between external (Poincaré group) and internal (unitary) symmetries of elementary particles but with a (Drinfeld) twist, suggesting that the internal symmetries might emerge from deforming to Anti de Sitter SO(2,3) and quantizing that (possibly in a new generalized manner) at root of unity. That raises challenging problems, both on the mathematical part and for particle physics.

For a longer version of the abstract, click here.

Algebra and Number Theory Seminar
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Posted September 25, 2013

Last modified October 7, 2013

Jacob Matherne, Department of Mathematics, LSU

Computing Upper Cluster Algebras

Cluster algebras are commutative unital domains generated by distinguished elements called cluster variables. These generators are grouped into sets called clusters, and a process called mutation allows movement between the clusters. Many notable varieties (Grassmannians, partial flag varieties, and others) are equipped with cluster structures where certain regular functions play the role of cluster variables.

From a geometric perspective, there is a more natural algebra to consider: the upper cluster algebra. In this talk, we study cluster algebras and upper cluster algebras using algebraic geometry, which leads to an algorithm for producing presentations of upper cluster algebras in terms of generators and relations.

This is joint work with Greg Muller.

Algebra and Number Theory Seminar
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Posted September 25, 2013

Last modified October 14, 2013

Holly Swisher, Oregon State University

Modularity of *k*-rank difference functions

Rank difference functions were used by Atkin and Swinnerton-Dyer to prove the well-loved Ramanujan congruences for the partition function modulo 5 and 7. In 2008, Ahlgren and Treneer recognized rank difference functions for partitions as modular or mock modular objects. Here, we similarly investigate *k*-component multipartitions (also called *k*-colored partitions). Ultimately, we relate restricted *k*-rank generating functions and *k*-rank difference functions to weakly holomorphic modular forms.

Algebra and Number Theory Seminar
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Posted September 25, 2013

Last modified October 16, 2013

Simon Riche, CNRS / Université de Clermont-Ferrand

Perverse sheaves on affine Grassmannians, asymptotic Verma modules, and differential operators on the basic affine space

The Geometric Satake equivalence is an equivalence of categories between certain perverse sheaves on the affine Grassmannian of a reductive algebraic group and representations of the dual reductive group (in the sense of Langlands). The general philosophy underlying this equivalence is that representation theoretic properties of a representation are reflected in topological properties of the corresponding perverse sheaf. In this talk we will explain how one can describe equivariant cohomology of the costalks of these perverse sheaves, together with their natural symmetries, in terms of morphisms between universal Verma modules for the dual Lie algebra, and also in terms of differential operators on the basic affine space of the dual group. This is joint work with Victor Ginzburg.

Algebra and Number Theory Seminar
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Posted September 25, 2013

Last modified October 18, 2013

Matt Papanikolas, Texas A&M University

Special points and L-values in positive characteristic

Going back to Dirichlet and Kummer one knows that special values of Dirichlet L-functions at s=1 can be expressed in terms of logarithms of circular units in cyclotomic fields and Gauss sums, and moreover these identities can be used to show that the group of circular units is of finite index in full group of units. The aim of the present talk is to investigate analogues of these results in positive characteristic for Goss L-functions of Dirichlet type, which take values in function fields of characteristic p. Anderson showed that values of these L-functions at s=1 are found from Carlitz logarithms of special points on the Carlitz module and investigated their properties. We will consider extensions of these results to s > 1, which involve modules of special points on tensor powers of the Carlitz module and log-algebraicity identities.

Algebra and Number Theory Seminar
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Posted October 14, 2013

Last modified November 5, 2013

Ling Long, Mathematics Department, LSU

On p-adic analogues of Ramanujan type formulae for 1 over pi

In this talk, we will give some general backgrounds of hypergeometric series, elliptic curves, and Ramanujan type formulae for 1 over pi. Then we will discuss some p-adic analogues of these formulae which was conjectured by van Hamme for special cases and by Zudilin more generally. This is a joint work with Sarah Chisholm, Alyson Deines, Gabriele Nebe, and Holly Swisher.

Algebra and Number Theory Seminar
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Posted September 30, 2013

Last modified November 5, 2013

Ravi Ramakrishna, Cornell University

Hida Families of modular forms

The basic idea of Hida theory is that certain modular forms live in p-adic analytic families where all the forms are congruent mod p. Even though they have been studied for some 30 years, much remains mysterious Hida theory. This talk will recall relevant aspects of the theory, raise some (hopefully!) interesting and fundamental questions and explain work in progress towards answering some of these.

Algebra and Number Theory Seminar
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Posted January 9, 2014

Last modified January 31, 2014

Tom Lenagan, University of Edinburgh

Algebras with restricted growth

We survey recent and not-so-recent results on growth of algebras, with special emphasis on small values.

Algebra and Number Theory Seminar
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Posted January 31, 2014

Last modified February 12, 2014

Ivan Dimitrov, Queen's University (Canada)

Integrable weight modules of gl(∞)

I will present a theorem classifying the irreducible integrable weight modules with finite dimensional weight spaces over the Lie algebra gl(∞) consisting of finitary infinite matrices. Every such module belongs to one of the following three classes: highest weight modules, infinite symmetric powers of the natural representations, and modules which are not highest weight but whose weights are dominated by a single weight. For the modules in the new third class I will present different realizations and will provide explicit parametrization. I will define all necessary terms and will state the problem and the main result.

Algebra and Number Theory Seminar
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Posted January 31, 2014

Last modified February 21, 2014

Peter Schauenburg, Université de Bourgogne

Module categories of finite Hopf algebroids, and self-duality

The notion of Hopf algebroid generalizes that of a Hopf algebra; the key property the former shares with the latter is that modules over a Hopf algebroid admit a tensor product, much like representations of a group or a Lie algebra. Put in more abstract terms, the module category over a Hopf algebroid is a tensor category. There is already a long list of results going in the other direction: Given a category with an abstract tensor product, the aim is to reconstruct a Hopf algebra (or a more general object such as a quasi-Hopf algebra, or a weak Hopf algebra, or a Hopf algebroid) whose module category is equivalent (or at least closely related) to that category. Many variants exist according to the properties required of the category one starts with, the closeness of the relation obtained between the category and the (co)module category of the reconstructed Hopf-like object, and the properties one can obtain for the latter. I will present a version that gives a completely intrinsic characterization of the module categories of suitably "finite" Hopf algebroids, and which, moreover, admits a rather simple proof. Then I will show that in many situations the Hopf algebroid thus attached to a tensor category is self-dual (after suitably clarifying what self-duality might mean for a Hopf algebroid); this generalizes a result of Pfeiffer on self-duality of certain fusion categories.

Algebra and Number Theory Seminar
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Posted March 25, 2014

3:30 pm - 4:20 pm Lockett 285
Atul Dixit, Tulane University

Some identities of Ramanujan in connection with the circle and divisor problems

On page 336 in his lost notebook, S. Ramanujan proposes an identity that may have been devised to attack a divisor problem. Unfortunately, the identity is vitiated by a divergent series appearing in it. We prove here a corrected version of Ramanujan's identity. While finding a plausible explanation for what may have led Ramanujan to consider a series that appears in this identity, we are led to a connection with a generalization of the famous summation formula of Voronoï. One of the ramifications stemming from this work allows us to obtain a one-variable generalization of two double Bessel series identities of Ramanujan, intimately connected with the circle and divisor problems, and which were proved only recently. This is work in progress and is joint with Bruce C. Berndt, Arindam Roy and Alexandru Zaharescu.

Algebra and Number Theory Seminar
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Posted April 24, 2014

Last modified April 27, 2014

Fang-Ting Tu, National Center for Theoretical Sciences, Taiwan

Automorphic Forms on Shimura Curves of Genus Zero

Our aim is to study the arithmetic properties of automorphic forms on Shimura curves. Recently, Yifan Yang proposed a new method for studying automorphic forms on Shimura curves of genus zero, in which automorphic forms are expressed in terms of solutions of Schwarzian differential equations. We then can use the solutions to study the arithmetic properties of automorphic forms on Shimura curves. In this talk, we will give a quick overview of Yang's results, some applications, and a method to determine Schwarzian differential equations for certain Shimura curves.

Algebra and Number Theory Seminar
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Posted September 17, 2014

3:30 am - 4:30 am 235 Lockett HallEmil Horozov (Sofia University), Calogero-Moser Spaces and Representation Theory

We characterize the phase spaces of both rational and trigonometric Calogero-Moser systems in terms of representations of certain infinite-dimensional Lie algebras. The construction makes use of the theory of bispectral operators. The main result is that the Calogero-Moser spaces (in both cases) coincide with the orbit of the vacuum in this representations of reasonably defined group GL_infinity.

Algebra and Number Theory Seminar
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Posted October 2, 2014

3:30 pm - 4:30 pm 235 Lockett
Susan Montgomery, University of Southern California

On the values of Frobenius-Schur indicators for Hopf algebras

The abstract for this talk is available here.

Algebra and Number Theory Seminar
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Posted October 7, 2014

3:30 pm - 4:30 pm 235 Lockett Hall
Joseph Timmer, Louisiana State University

Bismash Products and Exact Factorizations of S_n

With an exact factorization of a finite group L = FG, one may construct the bismash product Hopf algebra H = kG#kF. If one were to factor the symmetric group Sn = FG, the resulting Hopf algebras have some interesting properties; mostly concerning the indicator values of irreducible modules. In this talk,

present the background of exact factorizations, we present some new results concerning bismashnproducts in general and for those that arise from exact factorizations of Sn.

Algebra and Number Theory Seminar
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Posted October 7, 2014

3:30 pm - 4:30 pm 235 Lockett Hall
Luca Candelori, Louisiana State University

An algebro-geometric theory of vector-valued modular forms, Part 1

In this talk we describe a geometric theory of vector-valued modular forms attached to Weil representations of rank 1 lattices. More specifically, we construct vector bundles over the moduli stack of elliptic curves, whose sections over the complex numbers correspond to vector-valued modular forms attached to rank 1 lattices. The key idea is to construct vector bundles of Schrodinger representations and line bundles of half-forms over appropriate `metaplectic stacks' and then show that their tensor products descend to the moduli stack of elliptic curves. We prove an algebraic version of the Eichler-Zagier Theorem comparing vector-valued modular forms to Jacobi forms. We also give an algebraic notion of q-expansions of vector-valued modular forms, and discuss growth conditions at the cusp at infinity.

Algebra and Number Theory Seminar
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Posted October 7, 2014

3:30 pm - 4:30 pm 235 Lockett Hall
Luca Candelori, Louisiana State University

An algebro-geometric theory of vector-valued modular forms, Part 2

We follow up on our previous talk by describing applications of our geometric theory of vector-valued modular forms. First, we compute algebraic dimension formulas for the spaces of holomorphic vector-valued modular forms over any algebraically closed field (with mild restrictions on the characteristic) by using the Riemann-Roch theorem for Deligne-Mumford stacks. Second, we describe an algebro-geometric theory of modular forms of half-integral weight, as defined in the complex-analytic case by Shimura. Finally, as time allows, we explain how to extend our algebro-geometric theory to vector-valued modular forms attached to Weil representations of positive-definite lattices of higher rank, not just rank 1.

Algebra and Number Theory Seminar
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Posted October 8, 2014

Last modified October 16, 2014

Siu-hung (Richard) Ng, LSU

On Weil Representations of Modular Tensor Categories

Associated to a nondegenerate quadratic function on a finite abelian group is a projective representation of the modular group SL(2,Z) that is known as the Weil representation. This projective representation can be renormalized by the Gauss sum of the quadratic function to an ordinary representation of the metaplectic group Mp(2,Z). In this talk, we will discuss the corresponding analog of the Weil representation of a modular tensor category. This talk is intended to be accessible to graduate students with the knowledge of graduate algebra.

Algebra and Number Theory Seminar
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Posted October 8, 2014

Last modified October 30, 2014

James Zhang, University of Washington

Homological identities concerning Hopf algebra actions on Artin-Schelter regular algebras

The Nakayama automorphism of an Artin-Schelter regular algebra controls the class of Hopf algebras that act on the algebra. This can be interpreted as a homological identity. Several applications of homological identities will be given. The talk is based on recent work of K. Chan, J.-F. Lu, X.-F. Mao, M. Reyes, D. Rogalski and C. Walton.

Algebra and Number Theory Seminar
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Posted October 8, 2014

Last modified November 26, 2014

Mahir Can, Tulane University

Maximal chains of weak order posets of symmetric varieties

The W-set of an element of a weak order poset is useful in the cohomological study of the closures of spherical subgroups in generalized flag varieties. In this talk, after explaining various applications of this notion, we describe in a purely combinatorial manner the W-sets of the weak order posets of three classical symmetric spaces associated to the general linear group. In particular, we give a complete characterization of the maximal chains of an arbitrary lower order ideal in any of these three posets. This is a joint work with Michael Joyce and Ben Wyser.

Algebra and Number Theory Seminar
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Posted December 28, 2014

2:30 pm - 3:20 pm 235 Lockett Hall
Taiki Shibata, University of Tsukusba

Modular representations of Chevalley supergroups

An affine group scheme over a field k is a representable functor from the category of commutative k-algebras to the category of groups. Replacing ``algebras'' with ``superalgebras'' (=Z_2-graded algebras), we obtain the notion of an affine supergroup scheme (or, simply, a supergroup). An important example is a Chevalley supergroup introduced by R. Fioresi and F. Gavarini. I will talk about Hopf algebraic techniques applied to the modular representation theory of Chevalley supergroups. In joint work with A. Masuoka, we showed that, for a Chevalley supergroup G, there is a one-to-one correspondence between the G-modules and the integrable hy(G)-modules. Here, hy(G) is a generalization of the Lie superalgebra of G, called the hyper-superalgebra, due to M. Takeuchi. In my recent work, I obtained a super-analogue of the Steinberg tensor product theorem for Chevalley supergroups, which is a fundamental result in the modular representation theory of algebraic groups. In this talk, I would like to explain these results.

Algebra and Number Theory Seminar
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Posted December 28, 2014

3:30 pm - 4:20 pm 235 Lockett Hall
Kenichi Shimizu, Nagoya University

The distinguished invertible object of a finite tensor category and related topics

Fusion categories are an important class of tensor categories, and their non-semisimple generalizations -- finite tensor categories -- are also an interesting subject. I will talk about some properties of ``the distinguished invertible object'' of a finite tensor category introduced by Etingof, Nikshych and Ostrik. This is a categorical analogue of the modular function of a Hopf algebra (or, going back further, a locally compact group). As the modular function does in the theory of Hopf algebras, the distinguished invertible object plays an important role in the theory of finite tensor categories. In my talk, I will introduce recent my results on the distinguished invertible object, especially its relation with the monoidal center construction and applications to topological invariants.

Algebra and Number Theory Seminar
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Posted January 28, 2015

4:00 pm - 5:00 pm 235 Lockett Hall
Thomas Lam, University of Michigan

Whittaker functions and geometric crystals

I will talk about a formula for Archimedean Whittaker functions as integrals over Berenstein and Kazhdan's geometric crystals. This formula is a geometric analogue of the expression for an irreducible character of a complex semisimple Lie algebra as a sum over Kashiwara's crystals. The formula is closely related to mirror symmetry phenomenona for flag varieties, and to the study of directed polymers in probability.

Algebra and Number Theory Seminar
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Posted February 13, 2015

3:30 pm - 4:20 pm Lockett 235
Henry Tucker, University of Southern California

Frobenius-Schur indicators for near group fusion categories

Fusion categories are C-linear, rigid, semisimple tensor categories. They appear in a diverse range of mathematics, including representation theory of quantum groups, subfactor theory, and conformal field theory. The classical Frobenius-Schur indicator was first defined for representations of a finite group -- the most well-known example of a fusion category. The definition of the indicator has been extended to objects in a general fusion category by work of Ng-Schauenburg. This talk will report on progress toward computation of these indicators for near group fusion categories, which are fusion categories with one non-invertible object.

Algebra and Number Theory Seminar
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Posted February 25, 2015

3:30 pm - 4:30 pm 235 Lockett Hall
Daqing Wan, UC Irvine

Slopes of Modular Forms

The p-adic valuation of the p-th coefficient of a normalized modular eigenform is called the slope of the modular form. Understanding the slope distribution and variation is a major intriguing arithmetic problem in modern number theory and arithmetic geometry. In this talk, I will present a simple introduction to this fascinating subject, ending with our recent joint work with Liang Xiao, Jun Zhang and Ruochuan Liu.

Algebra and Number Theory Seminar
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Posted March 4, 2015

3:30 pm - 4:20 pm Lockett 235
Ben Webster, University of Virginia

Quantizations and duality for symplectic singularities

Since they were introduced about 2 decades ago, symplectic singularities have shown themselves to be a remarkable branch of algebraic geometry. They are much nicer in many ways than arbitrary singularities, but still have a lot of interesting nooks and crannies.

I'll talk about these varieties from a representation theorist's perspective. This might sound like a strange direction, but remember, any interesting symplectic structure is likely to be the classical limit of an equally interesting non-commutative structure, whose representation theory we can study. While this field is still in its infancy, it includes a lot of well-known examples like universal enveloping algebras and Cherednik algebras, and has led a lot of interesting places, including to categorified knot invariants and a conjectured duality between pairs of symplectic singularities. I'll give a taste of these results, in particular on very recent progress in constructing this duality.

Algebra and Number Theory Seminar
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Posted March 23, 2015

3:30 pm - 4:30 pm 235 Lockett Hall
Alexander Garver, University of Minnesota

Combinatorics of Exceptional Sequences in Type A

Exceptional sequences are certain ordered sequences of quiver representations with applications to noncrossing partitions, factorizations of Coxeter elements, cluster algebras, and the representation theory of algebras. We introduce a class of combinatorial objects called strand diagrams that we use to classify exceptional sequences of representations of type A Dynkin quivers. We also use variations of the model to classify c-matrices of type A Dynkin quivers, to interpret exceptional sequences as linear extensions of certain posets, and to give an elementary bijection between exceptional sequences and certain chains in the lattice of noncrossing partitions. This is part of ongoing work with Kiyoshi Igusa, Jacob Matherne, and Jonah Ostroff.

Algebra and Number Theory Seminar
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Posted April 2, 2015

3:30 pm - 4:20 pm Lockett 235
Liang Chang, Texas A&M University

Generalized Frobenius-Schur Indicators and Kuperberg 3-manifold Invariants

Frobenius-Schur indicators were defined originally for finite groups and generalized for Hopf algebras. They are examples of gauge invariants for Hopf algebras, which are useful for the category of representations. Recently, the generalized indicators turned out to coincide with Kuperberg 3-manifold invariants for Lens spaces, which provides topology interpretation for Hopf algebra invariants. In this talk, I will explain these algebraic and topological invariants and recent work on their relation.

Algebra and Number Theory Seminar
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Posted April 15, 2015

3:30 pm - 4:30 pm Lockett 235
Tom Lenagan, University of Edinburgh

The totally nonnegative grassmannian

This will be a survey talk about the cell decomposition of the totally nonnegative grassmannian. Points in the classical kxn grassmannian are k-dimensional subspaces in an n-dimensional vector space. Such points are described by their Plucker coordinates. The totally nonnegative grassmannian consists of those points for which all Plucker coordinates are nonnegative. There is a cell decomposition of the totally nonegative grassmannian given by specifying the vanishing pattern of the Plucker coordinates. In order to describe this cell decomposition, Postnikov introduced several interesting combinatorial devices and we will mention some of these. If time permits, connections between the cell structure of the totally nonnegative grassmannian and the invariant prime spectrum of the quantum grassmannian algebra will be discussed.

Algebra and Number Theory Seminar
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Posted May 17, 2015

3:30 pm - 4:20 pm Lockett 235
Cris Negron, University of Washington

Braided structures and the Gerstenhaber bracket on Hochschild cohomology

Given a finite dimensional Hopf algebra H acting on an algebra A, we can form an intermediate cohomology H˙(H, A) which comes equipped with a natural right H-action, and recovers the Hochschild cohomology of the smash product A#H after taking invariants. In fact, the cohomology H˙(H, A) is a Yetter-Drinfeld module over H and is a braided commutative algebra under the natural braiding induced by the Yetter-Drinfeld structure. This multiplicative structure has proved useful in verifying finite generation of Hopf cohomology, and has been studied extensively by Forest-Greenwood, Shepler, and Witherspoon. Supposing H has finite exponent, I will discuss how one can produce a braided antisymmetric bracket on H˙(H, A) which lifts the Gerstenhaber bracket to this braided setting, in the sense that it recovers the Gerstenhaber bracket after taking invariants.

Algebra and Number Theory Seminar
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Posted October 18, 2015

Last modified October 21, 2015

Ling Long, Mathematics Department, LSU

Hypergeometric functions and their finite field analogues I

Hypergeometric functions are an important class of special functions and they play important roles in many aspects of number theory. In this talk, we will review definitions and basic properties of classical hypergeometric functions and define their analogues over finite fields. This is a joint work with Jenny Fuselier, Ravi Ramakrishna, Holly Swisher and Fang-Ting Tu.

Algebra and Number Theory Seminar
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Posted October 18, 2015

Last modified November 2, 2015

Ling Long, Mathematics Department, LSU

Hypergeometric functions and their finite field analogues II

In this talk, we will discuss the Galois perspective of hypergeometric functions over finite fields. In particular we will associate Galois representations to the classical 2F1 hypergeometric functions with rational parameters via the generalized Legendre curves. Then we will use the Galois perspective to translate several types of classical hypergeometric formulas to the finite field settings. This is a joint work with Jenny Fuselier, Ravi Ramakrishna, Holly Swisher and Fang-Ting Tu.

Algebra and Number Theory Seminar
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Posted October 18, 2015

Last modified October 25, 2015

Cris Negron, Mathematics Department, LSU

A new approach to the Gerstenhaber bracket on Hochschild cohomology and applications

I will discuss a new approach to the Gerstenhaber bracket on Hochschild cohomology, and illustrate this new approach with a particular example related to finite group actions on affine space. The Hochschild cohomology of an algebra, along with the Gerstenhaber bracket, is (the cohomology of) a (dg) Lie algebra controlling the formal deformation theory of that algebra. In the talk I will focus on the aforementioned example in order to explain how our new results relate to, and in this case advance, both classical and current understandings of the Gerstenhaber bracket in geometric contexts. This is joint work with Sarah Witherspoon.

Algebra and Number Theory Seminar
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Posted October 18, 2015

3:30 pm - 4:20 pm Lockett 277
Xingting Wang, Temple University

Quantum p-Groups and Their Classification in Low Dimensions

The classification of p-groups is a fundamental but notoriously difficult problem in group theory. In this talk, the speaker will introduce quantum p-groups as a generalization of p-groups.

Discussion during the talk will be focusing on the recent progress in the complete classification of quantum p-groups in low dimensions. Such classification is part of the classification on finite-dimensional quantum groups in positive characteristic, and also contributes to the understanding of unipotent group schemes in positive characteristic.

Relations between isomorphism classes of quantum p-groups and ordinary p-groups will also be illustrated, which opens a door to using geometric methods in the study of classification of p-groups.

Algebra and Number Theory Seminar
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Posted January 15, 2016

Last modified January 25, 2016

Jesse Levitt, LSU

Classifying connected Hopf algebras of finite GK dimension via finite Drinfeld quantizations

The classification problem for Hopf Algebras of finite GK dimension has attracted a lot of interest in recent years. We will describe a new perspective to it via deformation theory. In 1983 Drinfeld constructed quantizations of all triangular r-matrices. We expand on work of Etingof and Gelaki showing that the ones that are finite define connected Hopf algebras of finite GK dimension. Hopf algebras constructed in this way are isomorphic, as algebras, to universal enveloping algebras. This construction recovers almost all of the known connected Hopf algebras of finite GK dimension, leads to many new examples from the general point of view of quasi-Frobenius Lie algebras, and enables preexisting Lie theoretic classification results to be brought to bear on the question at hand. This is a joint work with Milen Yakimov.

Algebra and Number Theory Seminar
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Posted December 2, 2015

Last modified February 15, 2016

Mehmet Kıral, Texas A&M University

The Voronoi formula and double Dirichlet series

A Voronoi formula is an identity where on one side, there is a weighted sum of Fourier coefficients of an automorphic form twisted by additive characters, and on the other side one has a dual sum where the twist is perhaps by more complicated exponential sums. It is a very versatile tool in analytic studies of L-functions. In joint work with Fan Zhou we come up with a proof of the identity for L-functions of degree N. The proof involves an identity of a double Dirichlet series which in turn yields the desired equality for a single Dirichlet coefficient. The proof is robust and applies to L-functions which are not yet proven to come from automorphic forms, such as Rankin-Selberg L-functions.

Algebra and Number Theory Seminar
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Posted February 2, 2016

Last modified March 14, 2016

Stefan Kolb, Newcastle University

Universal K-matrix for coideal subalgebras

Quantum groups provide a uniform setting for solutions of the quantum Yang-Baxter equation. These solutions are realized via a universal R-matrix which lies at the heart of the origins of quantum groups in the theory of quantum integrable systems. For systems with boundary, additionally, the reflection equation enters the picture. It is expected that solutions of the reflection equation are obtained via a universal K-matrix. A general construction of a universal K-matrix for Hopf algebras was given by Donin, Kulish, and Mudrov. In this talk I will suggest to base the construction of a universal K-matrix on coideal subalgebras of Hopf algebras. I will then discuss examples from the theory of quantum symmetric pairs, based on joint work with Martina Balagovic.

Algebra and Number Theory Seminar
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Posted March 31, 2016

Last modified April 5, 2016

Mark Reeder, Boston College

Adjoint Swan Conductors

Langlands parameters may be regarded as arithmetically enhanced elements in complex Lie groups. The Adjoint Swan Conductor of a parameter is an arithmetic analogue of the dimension of a Springer fiber. The latter satisfy an inequality which becomes an equality for regular elements. I will discuss the analogue of this inequality for Langlands parameters.

Algebra and Number Theory Seminar
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Posted February 22, 2016

Last modified April 11, 2016

Peng Yu, University of Wisconsin-Madison

Special Cycles on Shimura Varieties of Orthogonal Type

I will start with describing the famous Gross-Zagier formula as a relation between Neron-Tate height of Heegner points and central derivative of L-functions. Then I will define special cycles on Shimura variety constructed by Kudla in the setting of orthogonal type. And in the special case of O(1,2), I will show how these cycles be viewed as a generalization of Heegner points and Bruinier, Kulda, Yang and other authors' works on finding Faltings height of these cycles as a generalization of Gross-Zagier formula.

Algebra and Number Theory Seminar
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Posted February 24, 2016

Last modified April 25, 2016

Viswambhara Makam, University of Michigan

Polynomial degree bounds for matrix semi-invariants

Even though the invariant ring for a representation of a reductive group is finitely generated, finding strong bounds for the degree of generators has proved to be extremely difficult. We focus on the left-right action of SL(n) x SL(n) on m-tuples of n-by-n matrices. We show that invariants of degree at most n(n-1) define the null cone, and that consequently invariants of degree at most n^6 generate the invariant ring in characteristic 0. If time permits, we shall discuss the ramifications of our bound to algebraic complexity theory, such as a poly-time algorithm for non-commutative rational identity testing.

Algebra and Number Theory Seminar
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Posted April 4, 2016

Last modified April 18, 2016

Nick Ramsey, DePaul University

p-adic modular forms of half-integral weight and applications to L-values

I'll survey my work on p-adic modular forms of half-integral weight. In particular, I'll explain how to interpolate the Shimura lifting across the eigencurve and give an application to the p-adic interpolation of square roots of special values of L-functions.

Algebra and Number Theory Seminar
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Posted April 15, 2016

Last modified April 25, 2016

Ivan Losev, Northeastern University

Deformations of symplectic singularities and the orbit method

Symplectic singularities were introduced by Beauville in 2000. These are especially nice singular Poisson algebraic varieties that include symplectic quotient singularities and the normalizations of orbit closures in semisimple Lie algebras. Poisson deformations of conical symplectic singularities were studied by Namikawa who proved that they are classified by points of a vector space. Recently I have proved that quantizations of conical symplectic singularities are still classified by the points of the same vector spaces. I will explain these results and then apply them to establish a version of Kirillov's orbit method for semisimple Lie algebras.

Algebra and Number Theory Seminar
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Posted August 11, 2016

Last modified September 6, 2016

Holly Swisher, Oregon State University

Quantum mock modular forms arising from eta-theta functions

In 2013, Lemke Oliver classified all eta-quotients which are theta functions. In this work we construct mock modular forms from the eta-theta functions with even characters, such that the shadows of these mock modular forms are given by the eta-theta functions with odd characters. We further prove that the constructed mock modular forms are quantum modular forms. As corollaries, we establish simple finite hypergeometric expressions which may be used to evaluate Eichler integrals of the odd eta-theta functions, as well as some curious algebraic identities. If time allows, we will address some recent extensions of this work from our recent summer REU project.

This work is joint with: Amanda Folsom, Sharon Garthwaite, Soon-Yi Kang, Stephanie Treneer (AIM SQuaRE project) and Brian Diaz, Erin Ellefsen (OSU REU project).

Algebra and Number Theory Seminar
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Posted September 13, 2016

3:30 pm - 4:20 pm Lockett 284
Kiran Kedlaya, University of Californa, San Diego

Multiplicities of mod 2 Hecke algebras

Abstract:

This is a report on joint work in progress with Anna Medvedovsky (MPI, Bonn). Motivated by computational issues arising in the tabulation of rational newforms (as in Cremona's tables of elliptic curves), we ask about the extent to which the multiplicity of eigenvalues of the Hecke operator T_2 on a space of newforms of odd level is explained by known facts (e.g., Serre's conjecture). In weight 2, we have compiled a massive data set and compared it against known results; this yields partial agreement, but there is still some room for improvement.

Algebra and Number Theory Seminar
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Posted September 13, 2016

Last modified September 28, 2016

Fang-Ting Tu, LSU Mathematics Department

Hypergeometric functions over finite fields and their applications

Abstract: For a hypergeometric algebraic variety, we can express the number of it rational points over finite fields in terms of the so-called hypergeometric functions over finite fields. We have many transformation and evaluation formulas of finite field hypergeometric functions, which are parallel to the results of the classical case. As applications, we can study the arithmetic of hypergeometric varieties using these formulas.

Algebra and Number Theory Seminar
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Posted September 13, 2016

Last modified September 28, 2016

Xingting Wang, Temple University

Quantum groups associated to a pair of preregular forms

Abstract: In this talk, we study universal quantum groups that simultaneously coact on a pair of N-Koszul Artin-Schelter regular algebras. This work leads to a recovery of many well-known examples of quantum groups defined by various authors in the literature. Moreover, we show these quantum groups have surprisingly nice presentations in terms of the twisted superponentials associated to the underlining graded algebras, respectively. In particular, we will discuss the universal quantum group associated to a pair of three-dimensional Sklyanin algebras, whose ring-theoretic and homological behaviors need further investigation. This is a joint work with Alexandru Chirvasitu and Chelsea Walton.

Algebra and Number Theory Seminar
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Posted September 28, 2016

Last modified October 20, 2016

Simon Riche, CNRS / Université de Clermont-Ferrand

Character formulas in the modular representation theory of reductive algebraic groups

Abstract

In this talk I will present a project (including joint works with Pramod

Achar, Shotaro Makisumi, Carl Mautner, and Geordie Williamson) which

aims at providing a character formula for simple representations of

reductive algebraic groups over fields of positive characteristic. This

formula is inspired by Lusztig's conjecture, but different, and is

expected to hold in all characteristics bigger than the Coxeter number.

We expect to prove this formula using a geometric approach involving

coherent sheaves on the Springer resolution and constructible sheaves on

the affine flag variety and the affine Grassmannian of the Langlands

dual group.

Algebra and Number Theory Seminar
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Posted September 16, 2016

Last modified October 20, 2016

Nathan Green, Texas A&M

Special Values of L-functions and the Shtuka Function

Abstract: We study the arithmetic of coordinate rings of elliptic curves in finite characteristic and analyze their connection with Drinfeld modules. Using the functional equation for the shtuka function, we find identities for power sums and twisted power sums over these coordinate rings which allow us to express function field zeta values in terms of the shtuka function and the period of the exponential function. Joint with M. Papanikolas.

Algebra and Number Theory Seminar
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Posted September 13, 2016

Last modified October 20, 2016

Chelsea Walton, Temple University

PBW deformations of braided doubles

Abstract: I'll present new examples of deformations of smash product algebras that arise from Hopf algebra actions on pairs of Koszul module algebras. This construction generalizes several 'double' constructions appearing in the literature, including Weyl algebras and some types of Cherednik algebras, and it complements the braided double construction of Bazlov and Berenstein. There will probably be more questions than answers in this talk. This is joint work with Sarah Witherspoon.

Algebra and Number Theory Seminar
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Posted October 17, 2016

Last modified January 11, 2017

Shotaro Makisumi, Stanford University

A new approach to modular Koszul duality

The Koszul duality of Beilinson-Ginzburg-Soergel is a derived equivalence involving the BGG category O, which plays a central role in the study of highest weight modules of a semisimple complex Lie algebra. Geometrically, this may be viewed as a derived equivalence relating Langlands dual flag varieties. In this talk, I will discuss a new approach to a modular (positive characteristic) analogue of this result proved by Pramod Achar and Simon Riche.

Prerequisites will be kept to a minimum: once I have motivated the result, I will not work with representations or perverse/parity sheaves, instead giving an algebraic/combinatorial model (moment graph sheaves, Soergel bimodules) for these objects, focusing on the case of SL2. This will be enough for illustrating the key ingredient, a new construction of a left monodromy action.

I will also briefly report on joint work in progress with Pramod Achar, Simon Riche, and Geordie Williamson, in which we plan to extend the result to Kac-Moody flag varieties. The latter result would imply the Riche-Williamson conjecture on characters of tilting modules of reductive groups.

Algebra and Number Theory Seminar
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Posted January 15, 2017

Last modified February 5, 2017

William Hardesty, Louisiana State University

Support varieties for algebraic groups and the Humphreys conjecture

Abstract: In this talk I will begin by recalling the notion of a support variety

for a module over a finite group scheme. This will be followed by a

brief overview of classical results and calculations in the case when

the finite group scheme is the first Frobenius kernel of a reductive

algebraic group G. In 1997, J. Humphreys conjectured that the support

varieties of indecomposable tilting modules for G (a very important

class of modules) is controlled by a combinatorial bijection, due to G.

Lusztig, between nilpotent orbits and a certain collection of subsets of

the affine Weyl group called "canonical cells". This later became known

as the "Humphreys conjecture". I will discuss my proof of this

conjecture for when G=GL(n). If time permits, I may also present some of

my recent joint work with P. Achar and S. Riche concerning the Humphreys

conjecture in other types.

Algebra and Number Theory Seminar
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Posted January 15, 2017

Last modified March 6, 2017

Kenny De Commer, Vrije Universiteit Brussel

Heisenberg algebras of quantized enveloping type

Abstract: To any semisimple complex Lie algebra and generic complex number q can be associated two Hopf algebras: the QUEA (quantized universal enveloping algebra) and its dual QFA (quantum function algebra). The first of these can be constructed from a simpler algebra, the QUEA of a Borel subalgebra, by a general process known as the Drinfeld double construction. On the other hand, there exists an intermediate algebra, known as Heisenberg double, linking the Drinfeld double to a tensor product of two Borel QUEA. In this talk, I will explain how the Heisenberg double and the QFA are related, and will explain briefly how this observation can be used to find different spectral realizations of Borel QUEA as (unbounded) operators on a Hilbert space in the case of q real.

Algebra and Number Theory Seminar
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Posted January 11, 2017

Last modified March 9, 2017

Nicholas Cooney, Univ. Clermont-Ferrand

Quantizations of Multiplicative Quiver Varieties at Roots of Unity

Abstract: To a quiver Q with dimension vector d, one can associate an algebra Dq(Q), which is a flat q- deformation of the algebra of differential operators on the space of d-dimensional representations of the quiver, D(Matd(Q)). There is also a quantum moment map q compatible with various degenerations of the source and target to their classical analogues. These algebras and the map q were first constructed by David Jordan, who then studied them in the case where q 2 C is not a root of unity. I will discuss the case where q is a root of unity. Here, the algebra Dq(Q) attains a large centre. For dimension d equal to 1 at each vertex of Q, Dq(Q) is locally a matrix algebra generically on Spec(Z). One can associate quiver varieties to Q with this dimension vector that are multiplicative versions of hypertoric varieties. In this case, quantum Hamiltonian reductions of Dq(Q) along q are quantizations of these multiplicative hypertoric varieties which are again locally matrix algebras. The category of coherent sheaves of modules for these algebras is derived equivalent to that of modules over the global sections algebra - an instance of derived Beilinson- Bernstein localisation. In the first part of the talk, I will give the necessary background and context, explaining how this work can be framed as an instance of a paradigm which is prevalent in geometric representation theory. The second part will consist of a more detailed treatment of the root of unity case, including a discussion of possible extensions of some of these results to higher dimension vectors. This is joint work with Iordan Ganev and David Jordan.

Algebra and Number Theory Seminar
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Posted January 15, 2017

Last modified March 13, 2017

Stefan Kolb, Newcastle University

The center of quantum symmetric pair coideal subalgebras -- revisited

Abstract: Drinfeld-Jimbo quantised enveloping algebras (QUE) have a younger sibling, the theory of quantum symmetric pairs, which is as rich in structure as QUE themselves. In finite type, the center of QUE can be described in terms of their universal R-matrix. In this talk I will explain how the recently constructed universal K-matrix for quantum symmetric pairs can be employed in a similar fashion to describe a basis of the center of quantum symmetric pair coideal subalgebras. This simplifies joint work with G. Letzter from 2006.

Algebra and Number Theory Seminar
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Posted January 15, 2017

Last modified March 22, 2017

Peter Jorgensen, Newcastle University

Thick subcategories of d-abelian categories

Let d be a positive integer. The notion of d-abelian categories was introduced by Jasso. Such a category does not have kernels and cokernels, but rather d-kernels and d-cokernels which are longer complexes with weaker universal properties. Canonical examples of d-abelian categories are d-cluster tilting subcategories of abelian categories. We introduce the notion of thick subcategories of d-abelian categories. We show that functorially finite thick subcategories of d-cluster tilting subcategories are in bijection to so-called d-rigid epimorphisms. This generalises a classic result by Geigle and Lenzing. We apply this to show a classification of the thick subcategories of a family of d-abelian categories associated to quivers of type A_n. This is a report on joint work with Martin Herschend and Laertis Vaso.

Algebra and Number Theory Seminar
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Posted January 15, 2017

Last modified March 29, 2017

Dima Arinkin, University of Wisconsin

Geometry of linear ODEs

Abstract: There is a classical correspondence between systems of n linear ordinary differential equations (ODEs) of order one and linear ODEs of order n. (The correspondence may be viewed as a kind of `canonical normal form' for systems of ODEs.) The correspondence can be restated geometrically: given a Riemann surface C, a vector bundle E on C, and a connection nabla on E, it is possible to find a rational basis of E such that nabla is in the canonical normal form. All of the above objects have a version for arbitrary semisimple Lie group G (with the case of systems of ODEs corresponding to G=GL(n)): we can consider differential operators whose `matrices' are in the Lie algebra of G, and then try to `change the basis' so that the `matrix' is in the `canonical normal form.' However, the statement turns out to be significantly harder. In my talk, I will show how the geometric approach can be used to prove the claim for any G.

Algebra and Number Theory Seminar
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Posted January 23, 2017

Last modified March 17, 2017

Jie Zhou, Perimeter institute

Periods and Gromov-Witten invariants

Abstract: The mirror symmetry conjectures asserts that the generating series of Gromov-Witten invariants (curve counting) of a Calabi-Yau variety are identical to some "universal" differential polynomials of period integrals of its mirror Calabi-Yau variety. I will explain in detail how these "universal" polynomials can be read off from the Picard-Fuchs system of the mirror Calabi-Yau variety, for the genus zero and one cases which are so far the only cases proved rigorously in mathematics. I will also discuss some nice ingredients (e.g., generating series of point counting, polylogarithms, Feynman diagrams and manipulation on Picard-Fuchs equations) which seem to have a motivic nature. A particularly interesting example of Calabi-Yau 3-fold will be emphasized, in which modular forms arise naturally.

Algebra and Number Theory Seminar
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Posted April 11, 2017

3:00 pm - 3:50 pm Lockett 277
Ha Tran, University of Calgary

On reduced ideals of a number field

Let F be a number field. The reduced ideals of F can be used for computing its class group and regulator. In this talk, we will introduce reduced ideals first for quadratic fields then for an arbitrary number field. Next, we will discuss a generalization of reduced ideals using the LLL-algorithm. Finally, some open problems relating to this topic will be presented.

Algebra and Number Theory Seminar
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Posted April 22, 2017

Last modified May 18, 2017

David Lax, Virginia Tech

Order Filter Model for Minuscule Plucker Relations

Abstract: The Plucker relations which define the Grassmann manifolds as projective varieties interact nicely with a natural order on the projective coordinates; the resulting homogeneous coordinate ring is an algebra with straightening law. This is a property shared by all minuscule flag manifolds. The order structures on their projective coordinates share common properties and are called minuscule lattices. We study their generalized Plucker relations independent of Lie type through the minuscule lattices. To do this we combinatorially model the Plucker coordinates based on Wildberger's construction of minuscule Lie algebra representations; it uses the colored partially ordered sets known as minuscule posets. We obtain, uniformly across Lie type, descriptions of the Plucker relations of ``extreme weight''. We show that these are supported by ``double-tailed diamond'' sublattices of minuscule lattices. From this, we obtain a complete set of Plucker relations for the exceptional minuscule flag manifolds. These Plucker relations are straightening laws for their respective coordinate rings.

Algebra and Number Theory Seminar
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Posted August 2, 2017

Last modified September 12, 2017

Matthew Lee, University of California, Riverside

Global Weyl modules for non-standard maximal parabolics of twisted affine Lie algebras

Abstract: In this talk we will discuss the structure of non standard maximal parabolics of twisted affine Lie algebras, global Weyl modules and the associated commutative associative algebra, $\mathbf{A}_\lambda$. Since the global Weyl modules associated with the standard maximal parabolics have found many applications the hope is that these non-standard maximal parabolics will lead to different, but equally interesting applications.

Algebra and Number Theory Seminar
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Posted October 21, 2017

3:10 pm - 4:00 pm 276 Lockett
Bach Nguyen, Louisiana State University

Noncommutative discriminants via Poisson geometry and representation theory

The notion of discriminant is an important tool in number theory, algebraic geometry and noncommutative algebra. However, in concrete situations, it is difficult to compute and this has been done for few noncommutative algebras by direct methods. In this talk, we will describe a general method for computing noncommutative discriminants which relates them to representation theory and Poisson geometry. As an application we will provide explicit formulas for the discriminants of the quantum Schubert cell algebras at roots of unity. If time permits, we will also discuss this for the case of quantized coordinate rings of simple algebraic groups and quantized universal enveloping algebras of simple Lie algebras. This is joint work with Kurt Trampel and Milen Yakimov.

Algebra and Number Theory Seminar
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Posted September 9, 2017

Last modified October 20, 2017

Anna Romanov, University of Utah

A Kazhdan-Lusztig algorithm for Whittaker modules

The category of Whittaker modules for a complex semisimple Lie algebra generalizes the category of highest weight modules and displays similar structural properties. In particular, Whittaker modules have finite length composition series and all irreducible Whittaker modules appear as quotients of certain standard Whittaker modules which are generalizations of Verma modules. Using the localization theory of Beilinson-Bernstein, one obtains a beautiful geometric description of Whittaker modules as twisted sheaves of D-modules on the associated flag variety. I use this geometric setting to develop an analogue of the Kazhdan-Lusztig algorithm for computing the multiplicities of irreducible Whittaker modules in the composition series of standard Whittaker modules.

Algebra and Number Theory Seminar
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Posted August 31, 2017

Last modified November 9, 2017

Yifan Yang, National Taiwan University

Rational torsion points on the generalised Jacobian of a modular curve with cuspidal modulus

In this talk we consider the generalised Jacobian of the modular curve X_0(N) with respect to the reduced divisor given by the sum of cusps. When N is a prime power >3, we show that the group of rational torsion points on the generalised Jacobian tends to be much smaller than the classical Jacobian. This is a joint work with Takao Yamazaki.

Algebra and Number Theory Seminar
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Posted December 11, 2017

Last modified January 14, 2018

Yinhuo Zhang, Universiteit Hasselt

Finite-dimensional quasi-Hopf algebras of Cartan type

In this talk, we present a general method for constructing finite-dimensional quasi-Hopf algebras from finite abelian groups and braided vector spaces of Cartan type. The study of such quasi-Hopf algebras leads to the classification of finite-dimensional radically graded basic quasi-Hopf algebras over abelian groups with dimensions not divisible by 2,3,5,7 and associators given by abelian 3-cocycles. As special cases , the small quasi-quantum groups are introduced and studied. Many new explicit examples of finite-dimensional genuine quasi-Hopf algebras are obtained.

Algebra and Number Theory Seminar
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Posted November 30, 2017

Last modified January 23, 2018

William Casper, Louisiana State University

Algebras of Differential Operators and Algebraic Geometry with Applications

The algebro-geometric structure of the centralizer of a differential operator has a strong influence over the value of the operator itself. This principle serves as the basis of the theory of soliton solutions of the Korteweg-de Vries equation. Furthermore, these ideas have been shown to have purely algebraic applications in the context of the Schottky's problem of characterizing Jacobian varieties. In this talk, we relate some of the historical highlights in the study of centralizers of differential operators. Following this, we describe some recent applications in the classification of bispectral differential operators. (The latter is based on joint work with Milen Yakimov and results from the author's Ph.D. thesis)

Algebra and Number Theory Seminar
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Posted November 30, 2017

Last modified January 26, 2018

William Casper, Louisiana State University

The Prolate Spheroidal Phenomenon, Bispectrality, and Growth of Algebras

The prolate spheroidal phenomenon is the property that certain integral operators possess commuting differential operators. It has been long conjectured that integral operators possessing the prolate spheroidal property are closely related to bispectral functions. In this talk we demonstrate a general connection between the two topics by establishing a natural bi-filtration on the algebra of bispectral operators and measuring the growth rate. By obtaining an estimate for the growth rate, we are able to show that the bispectral operator algebra contains a differential operator commuting with an integral operator.

Algebra and Number Theory Seminar
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Posted November 30, 2017

Last modified February 19, 2018

Peng-Jie Wong, PIMS-University of Lethbridge

Holomorphy of L-functions and distribution of primes

The analytic properties of L-functions have been one of the central topics in number theory as they have many arithmetic applications. For example, the distribution of prime numbers has a deep connection with the properties of the Riemann zeta function. In general, for any number field, there are primes and L-functions of similar nature. In this talk, we shall discuss the holomorphy of such L-functions and its applications to the distributions of the associated primes.

Algebra and Number Theory Seminar
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Posted January 15, 2018

Last modified February 22, 2018

Wen-Ching Winnie Li, Pennsylvania State University

colloquium this week

Algebra and Number Theory Seminar
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Posted February 27, 2018

Last modified March 6, 2018

Laura Rider, University of Georgia

An Iwahori-Whittaker model for the Satake category

The geometric Satake equivalence gives a topological incarnation of the representation theory of a connected, reductive algebraic group over any field. This description uses so-called "spherical" perverse sheaves on the affine Grassmannian. In my talk, I'll discuss an Iwahori-Whittaker model for this category. This model takes advantage of a cellular stratification of the affine Grassmanian, and as a result, allows for some nice applications of the equivalence. This work is joint with Roman Bezrukavnikov, Dennis Gaitsgory, Ivan Mirkovic, and Simon Riche.

Algebra and Number Theory Seminar
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Posted March 1, 2018

Last modified March 6, 2018

Daniel Sternheimer, Rikkyo University & Institut de Mathématiques de Bourgogne

The reasonable effectiveness of mathematical deformation theory in physics

New fundamental physical theories can, so far a posteriori, be seen as emerging from existing ones via some kind of deformation. That is the basis for Flato's "deformation philosophy", of which the main paradigms are the physics revolutions from the beginning of the twentieth century, quantum mechanics (via deformation quantization) and special relativity.

On the basis of these facts we explain how symmetries of hadrons (strongly interacting elementary particles) could "emerge" by deforming in some sense (including quantization) the Anti de Sitter symmetry (AdS), itself a deformation of the Poincare group of special relativity.

The ultimate goal is to base on fundamental principles the dynamics of strong interactions, which originated over half a century ago from empirically guessed "internal" symmetries.

We start with a rapid presentation of the physical (hadrons) and mathematical (deformation theory) contexts, including a possible explanation of photons as composites of AdS singletons and of leptons as similar composites. Then we present a "model generating" framework in which AdS would be deformed and quantized (possibly at root of unity and/or in manner not yet mathematically developed with noncommutative "parameters").

That would give (using deformations) a space-time origin to the "internal" symmetries of elementary particles, on which their dynamics were based, and either question, or give a conceptually solid base to, the Standard Model, in line with Einstein's quotation: "The important thing is not to stop questioning. Curiosity has its own reason for existing."

Algebra and Number Theory Seminar
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Posted January 16, 2018

Last modified February 25, 2018

Rina Anno, Kansas State University

Non-split P^n-twists

P^n-objects were introduced in 2005 by Huybrechts and Thomas as objects E in D^b(Coh X) for a smooth projective X satisfying certain conditions, one of which is Ext^*(E,E) being isomorphic as a graded ring to H^*(P^n,C). These objects induce autoequivalences of D^b(Coh X) called P^n-twists. In 2011, Addington proposed a definition for P^n-functors that also define autoequivalences of the target category. One of the requirements in his definition is that if F is a P^n-functor and R is its right adjoint, RFsimeq oplus H^i, where H is an autoequivalence of the source category of F. We are going to introduce the definition of a P^n-functor where RF is isomorphic to a repeated extension of id by H^i (a convolution of a complex of H^i's in some DG enhancement, which generalizes the direct sum), and provide a class of examples. This is joint work with Timothy Logvinenko.

Algebra and Number Theory Seminar
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Posted March 13, 2018

Last modified March 19, 2018

Neal Livesay, University of California, Riverside

Moduli spaces of irregular singular connections

A classical problem in mathematics is that of classifying singular differential operators. An algebro-geometric variant of this problem involves the construction of moduli spaces of connections on vector bundles over P^1 with singularities x_1,...,x_k. Locally (i.e., around a singularity x_i), a selection of a basis for the vector bundle induces a matrix form for the connection. The study of matrices associated to connections is analogous to the study of matrices associated to linear maps. In this talk, I will discuss a construction of moduli spaces of connections on P^1 which are locally diagonalizable, along with recent generalizations made by C. Bremer, D. Sage, and N. Livesay.

Algebra and Number Theory Seminar
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Posted March 1, 2018

LSU Spring Break

Algebra and Number Theory Seminar
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Posted January 15, 2018

Last modified April 8, 2018

Li Guo, Rutgers University at Newark

A Locality Principle of Renormalization via Algebraic Birkhoff Factorization

An interpretation of the locality principle in renormalization is that a locality property is preserved in the process of renormalization. We establish such a principle in the framework of the algebraic approach of Connes and Kreimer to quantum field renormalization, by working with their algebraic Birkhoff factorization. More precisely we show that if a regularization map is a locality map, then so is the corresponding renormalization map from the algebraic Birkhoff factorization. For this purpose, we introduce locality for various algebraic structures including those of a Hopf algebra, a Rota-Baxter algebra and a regularization map between the two algebras. For applications, we consider the exponential generating function of lattice points in a convex cone, giving rise to a meromorphic function with linear poles.

Algebra and Number Theory Seminar
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Posted February 27, 2018

Last modified April 18, 2018

Soumya (Shom) Banerjee, Tulane University

Revisiting the Variety of Complete Quadrics

The variety of complete quadrics is a family of smooth projective variety that has a long and interesting history which rivals Grassmanninan variety and Flag varieties. However, surprisingly little is known about its geometric structure. In this talk, I will explain our attempts to understand the geometry of this variety in an explicit way. This is a joint work with Mahir Can and Mike Joyce.

Algebra and Number Theory Seminar
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Posted April 30, 2018

Last modified May 6, 2018

Xingting Wang, Temple University

Representations of 4-dimensional Sklyanin algebras through Poisson geometry.

In 1982, Sklyanin constructed a certain noncommutative graded algebra A(E,tau) depending on an elliptic curve E embedded in P^3 and a point tau in E related to the Yang-Baxter equation in "quantum inverse scattering method". It was shown by Smith and Stafford that these so-called 4-dimensional Sklyanin algebras have the same Hilbert series as the polynomial algebra on four variables and possess excellent homological property. When tau is torsion-free, Smith and Staniszkis proved that there are exactly 4-parametric families of non-trivial irreducible representations at each dimension of k >= 1. In this talk, we give all irreducible representations of A(E, tau) when tau is of finite order n>4 with the help of Poisson geometry and deformation quantization. This is a joint work of Chelsea Walton and Milen Yakimov.

Algebra and Number Theory Seminar
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Posted September 7, 2018

Last modified September 30, 2018

Armin Straub, University of South Alabama

The congruences of Fermat, Euler, Gauss and stronger versions thereof

The Gauss congruences are a natural generalization of the more familiar Fermat and Euler congruences. Interesting families of combinatorial and number theoretic sequences are known to satisfy these congruences. Though a general classification remains wide open, Minton characterized constant recursive sequences satisfying Gauss congruences. We consider the natural extension of this question to Laurent coefficients of multivariate rational functions. One of the motivations for studying Gauss congruences lies in the fact that a certain interesting class of sequences, related to Ap'ery-like constructions of linear forms in zeta values, conjecturally satisfies stronger versions of these congruences. We outline this story and indicate recent developments. The first part of this talk is based on joint work with Frits Beukers and Marc Houben, while the second part includes joint work with Dermot McCarthy and Robert Osburn.

Algebra and Number Theory Seminar
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Posted November 10, 2018

Last modified November 15, 2018

Yilong Wang, Louisiana State University

Higher Gauss sums of modular categories

In this talk, we will introduce the notion of a modular category with an emphasis on the Galois group action such a category. Then we will discuss a family of categorical invariants of a modular category called the higher Gauss sums as generalizations of the classical quadratic Gauss sums.

Algebra and Number Theory Seminar
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Posted November 10, 2018

Last modified November 25, 2018

Scott Baldridge, Louisiana State University

From Spatial Trivalent Graphs with Rigid Perfect Matchings to Categories of Cobordisms and Frobenius Algebras

In the first part of this talk, we introduce the notion of an "interlocking crossbar web", which generalizes knots and links to spatial trivalent graphs with rigid perfect matchings. We then define a new cohomology theory that is invariant of these webs and show how to compute it using simple examples. When the web is a knot (i.e., no crossbars), this cohomology theory reduces to the usual Khovanov Homology of the knot. When the web is planar, this cohomology is a recently-discovered invariant of the planar trivalent graph with its perfect matching. In the second part of the talk, we attempt to interpret this cohomology in terms of TQFTs: What is the category of cobordisms (2Cob) for this theory? In particular, what are examples of generators and relations in the category? How do these generators and relations relate to Frobenius Algebras? The second part of the talk is hoped to be more of a fruitful discussion between participants than a lecture.

Algebra and Number Theory Seminar
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Posted November 3, 2018

Last modified January 28, 2019

Ignacio Nahuel Zurrian, Universidad Nacional de Cordoba (National University of Cordoba)

Completeness of the Bethe Ansatz for an open q-boson system with integrable boundary interaction

We employ a discrete integral-reflection representation of the

double affine Hecke algebra of type $C^V$$C$ at the critical level

$q=1$, to endow the open finite q-boson system with integrable

boundary interactions at the lattice ends. It is shown that the Bethe

Ansatz entails a complete basis of eigenfunctions for the commuting

quantum integrals in terms of Macdonald's three-parameter hyproctahedral

Hall-Littlewood polynomials. This is a joint work with J.F. van Diejen and

E. Emsiz.

Algebra and Number Theory Seminar
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Posted January 28, 2019

3:10 pm - 4:00 pm 232 Lockett
Wenbin Guo, University of Science and Technology of China

Recent some progress on finite groups

In this talk, we will discuss some recent developments of the theory of finite groups, which include F-hypercenter and its generalizations, the theory of quasi-F-groups, the generalization of Schur-Zassenhaus theorem, Hall theorem and Chunihin theorem and answers to two Wielandt''s open problems.

Algebra and Number Theory Seminar
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Posted November 3, 2018

Last modified February 3, 2019

Iván Angiono, Universidad Nacional de Cordoba (National University of Cordoba)

On Nichols algebras

Nichols algebras appeared naturally when several authors, led by Andruskiewitsch and Schneider, looked for the classification of (a family of) non semisimple finite dimensional Hopf algebras. They are a universal quotient of the tensor algebra of a braided vector space. The aim of this talk is to introduce Nichols algebras, present several examples, and finally give some properties when the braided vector space is of diagonal type.

Algebra and Number Theory Seminar
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Posted November 3, 2018

Last modified February 9, 2019

Ignacio Nahuel Zurrian, Universidad Nacional de Cordoba (National University of Cordoba)

Time-Band-Limiting for Matrix-valued functions

The subject of time-band-limiting, originating in signal processing, is dominated by the "miracle" that a naturally appearing integral operator admits a commuting differential one allowing for a numerically efficient way to compute its eigenfunctions. Bispectrality is an effort to dig into the reasons behind this miracle. This search has revealed unexpected connections with several parts of mathematics. In this talk consider a matrix valued version of bispectrality and give a general condition under which we can display a constructive and simple way to obtain the commuting differential operator. Furthermore, we will build an operator that commutes with both the time-limiting operator and the band-limiting operators.

Algebra and Number Theory Seminar
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Posted November 3, 2018

Last modified February 22, 2019

Nicolas Andruskiewitsch, Universidad Nacional de Cordoba (National University of Cordoba)

The classification of Hopf algebras with finite Gelfand-Kirillov dimension

The classification of Hopf algebras with finite Gelfand-Kirillov dimension has received attention recently. Nichols algebras play an important role in this question that will be explained in the talk together with an overview of examples and partial results.

Algebra and Number Theory Seminar
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Posted December 27, 2018

Last modified February 22, 2019

Eric Rowell, Texas A&M

Representations of Mapping Class Groups and Motion Groups

(2+1)TQFTs and their algebraic counterparts (modular categories) provide finite dimensional representations

of mapping class groups, such as the braid group and SL(2,Z). Analogously, one expects to (3+1)TQFTs to

give us representations of motion groups, such as the loop braid group--the motions of the n-component unlink.

I will describe a few questions related to these representations, some of which are motivated by topological quantum

computation, and what is currently known about their answers.

Algebra and Number Theory Seminar
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Posted December 11, 2018

Mardi Gras Holiday

Algebra and Number Theory Seminar
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Posted October 1, 2018

Last modified October 26, 2018

Luca Candelori, Wayne State University

Transcendence of Periods and Endomorphism Algebras of Jacobian Varieties

In this talk I will describe a new method to bound the number of linear relations with algebraic coefficients between the periods of an algebraic curve. As shown by Shiga and Wolfart, these bounds provide information regarding the dimension of the endomorphism algebra of the corresponding Jacobian variety. I will explain how to employ these new bounds to explore two of the many open questions about endomorphism algebras of Jacobians: which Jacobians have complex multiplication, and which Jacobians are totally decomposable.

Algebra and Number Theory Seminar
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Posted December 17, 2018

Last modified March 17, 2019

Timo Richarz, Technische Universitat Darmstadt

Smoothness of Schubert varieties in affine Grassmannians

The geometry in the reduction of Shimura varieties, respectively moduli spaces of Drinfeld shtuka plays a central role in the Langlands program, and it is desirable to single out cases of smooth reduction. This question reduces to the corresponding Schubert variety which is defined in terms of linear algebra, and thus easier to handle.

We consider Schubert varieties which are associated with a reductive group G over a Laurent series local field, and a special vertex in the Bruhat-Tits building. If G splits, a strikingly simple classification is given by a Theorem of Evans-Mirkovic and Malkin-Ostrik-Vybornov. If G does not split, the analogue of their theorem fails: there is a single surprising additional case of "exotic smoothness". In my talk, I explain how to obtain a complete list of the smooth and rationally smooth Schubert varieties. This is joint work with Thomas J. Haines from Maryland.

Algebra and Number Theory Seminar
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Posted February 12, 2019

Last modified April 8, 2019

Peter Jorgensen, Newcastle University

Model categories of quiver representations

This is a report on joint work with Henrik Holm.

Let R be a k-algebra. Given a cotorsion pair (A,B) in Mod(R), Gillespie's Theorem shows how to construct a model category structure on C(Mod R), the category of chain complexes over Mod(R). There is an associated homotopy category H.

If (A,B) is the trivial cotorsion pair (projective modules, everything), then H is the derived category D(Mod R). Several other important triangulated categories can also be obtained from the construction.

Chain complexes over R are the Mod(R)-valued representations of a certain quiver with relations: Linearly oriented A double infinity modulo the composition of any two consecutive arrows. We show that Gillespie's Theorem generalises to arbitrary self-injective quivers with relations, providing us with many new model category structures.

Algebra and Number Theory Seminar
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Posted December 11, 2018

Spring Break

Algebra and Number Theory Seminar
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Posted May 7, 2019

3:10 am - 4:00 pm 232 Lockett
Liang Chang, Nankai University

On 3-manifold invariants from Hopf algebras

There are two main approaches for defining quantum invariants of closed 3-manifolds from Hopf algebras. The first one is to use the Heegaard diagram presentation of 3-manifolds. The other one uses the presentation of 3-manifolds by surgery along links. In this talk, we will review these two types of quantum invariants constructed by Kuperberg and Hennings, and report the recent work on their relationship.

Algebra and Number Theory Seminar
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Posted August 27, 2019

Last modified September 5, 2019

Moises Herradon Cueto, Louisiana State University

The local type of difference equations

D-modules allow us to study differential equations through the lens of algebraic geometry. They are widely studied and have been shown to be full of structure. In contrast, the case of difference equations is lacking some of the most basic constructions. We focus on the following question: D-modules have a clear notion of what it means to restrict to a (formal) neighborhood of a point, namely extension of scalars to a power series ring. However, what does it mean to restrict a difference equation to a neighborhood of a point? I will give an answer which encompasses the intuitive notions of a "zero" and a "pole" of a difference equation, but further it is consistent in two more ways. First of all, we can show that restricting a difference equation to a point and to its complement is enough to recover the difference equation. Secondly, there exists a local Mellin transform analogous to the local Fourier transform. The local Fourier transform describes singularities of a D-module on the affine line in terms of the singularities of its Fourier transform. Similarly, the Mellin transform is an equivalence between D-modules on the punctured affine line and difference modules on the line, and we can relate singularities on both sides via this local Mellin transform. I will also talk about how to apply the same ideas to other kinds of difference equations, such as elliptic equations, which generalize difference and differential equations at once.

Algebra and Number Theory Seminar
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Posted September 7, 2019

Last modified September 27, 2019

Solly Parenti, University of Wisconsin, Madison

Unitary CM Fields and the Colmez Conjecture

In 1993, Pierre Colmez conjectured a formula for the Faltings height of a CM abelian variety in terms of logarithmic derivatives of certain L-functions. I will discuss how we can extend the known cases of the conjecture to a class of unitary CM fields using the recently proven average version of the conjecture.

Algebra and Number Theory Seminar
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Posted September 14, 2019

Last modified October 5, 2019

Chenliang Huang, Indiana University-Purdue University Indianapolis (IUPUI)

The solutions of gl(m|n) Gaudin Bethe ansatz equation, rational pseudodifferential operators, and the gl(m|n) spaces

We consider the gl(m|n) Gaudin Bethe ansatz equation associated to a tensor product of polynomial modules. Given a solution we describe a reproduction procedure which produces a family of new solutions which we call a population of solutions. We also write a rational pseudodifferential operator invariant under the reproduction procedure. We expect that the coefficients of the expansion of this operator are eigenvalues of the higher Gaudin Hamiltonians acting on the corresponding Bethe vector. The kernels of the numerator and denominator of the rational differential operator consist of rational functions and form a super space. Then we show that the population is canonically identified with the set of complete factorizations of the rational pseudodifferential operator, and with the variety of full super flags in the super space of rational functions. We conjecture that the eigenvectors of the Gaudin Hamiltonians are in a bijection with super spaces of rational functions with the prescribed properties which we call the gl(m|n) spaces.

Algebra and Number Theory Seminar
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Posted August 19, 2019

Last modified October 5, 2019

Changningphaabi Namoijam, Texas A&M

Transcendence of Hyperderivatives of Logarithms and Quasi-logarithms of Drinfeld Modules

In 2012, Chang and Papanikolas proved the transcendence of certain logarithms and quasi-logarithms of Drinfeld Modules. We extend this result to transcendence of hyperderivatives of these logarithms and quasi-logarithms. To do this, we construct a suitable t-motive and then use Papanikolas' results on transcendence degree of the period matrix of a t-motive and dimension of its Galois group.

Algebra and Number Theory Seminar
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Posted October 11, 2019

Last modified November 15, 2019

Ignacio Nahuel Zurrian, Universidad Nacional de Cordoba (National University of Cordoba)

Some applications of discrete harmonic analysis

We will discuss some concrete applications of discrete integrable systems through certain representations of Double Affine Hecke Algebras.

Algebra and Number Theory Seminar
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Posted October 11, 2019

Last modified November 27, 2019

Kent Vashaw, Louisiana State University

Noncommutative tensor triangular geometry

We describe a general theory of the prime spectrum of non-braided monoidal triangulated categories. These notions are a noncommutative analogue to Paul Balmer's prime spectra of symmetric tensor-triangulated categories. Noncommutative monoidal triangulated categories appear naturally as stable module categories for non-quasitriangular Hopf algebras and as derived categories of bimodules of noncommutative algebras. In stable module categories of Hopf algebras, the support theory of the category, as described by Benson-Iyengar-Krause, is linked to the Balmer spectrum, which is shown to be the final support datum. We will describe how this connection can be used to compute Balmer spectra in general, and we will compute the Balmer spectra for stable module categories of the small quantum group of a Borel subalgebra at a root of unity, and the stable module categories for smash coproduct Hopf algebras of group algebras with coordinate rings of groups. This is joint work with Daniel Nakano and Milen Yakimov.