LSU College of Science
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Calendar


Time interval:   Events:

Tuesday, September 16, 2003

Algebra and Number Theory Seminar  Questions or comments?

Posted September 10, 2003
Last modified January 27, 2004

3:40 pm - 4:30 pm Lockett 237

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

Tuesday, September 30, 2003

Algebra and Number Theory Seminar  Questions or comments?

Posted September 8, 2003
Last modified September 17, 2003

3:40 pm - 4:30 pm Lockett 282

Helena Verrill, Mathematics Department, LSU
Examples of rigid Calabi-Yau 3-folds

Tuesday, October 14, 2003

Algebra and Number Theory Seminar  Questions or comments?

Posted September 11, 2003
Last modified September 17, 2003

3:40 pm - 4:30 pm Lockett 282

Paul van Wamelen, Mathematics Department, LSU
Analytic Jacobians in Magma

Tuesday, October 21, 2003

Algebra and Number Theory Seminar  Questions or comments?

Posted October 15, 2003

3:40 pm - 4:30 pm Room 239

Paul van Wamelen, Mathematics Department, LSU
Analytic Jacobians in Magma II

Tuesday, October 28, 2003

Algebra and Number Theory Seminar  Questions or comments?

Posted September 12, 2003
Last modified September 17, 2003

3:40 pm - 4:30 pm Lockett 282

Charles Neal Delzell, Mathematics Department, LSU
A generalization of Polya's theorem to signomials with rational exponents

Tuesday, November 11, 2003

Algebra and Number Theory Seminar  Questions or comments?

Posted November 3, 2003

3:40 pm - 4:30 pm Lockett 282

Eric Baxter, University of New Orleans
Prime time

Monday, April 19, 2004

Algebra and Number Theory Seminar  Questions or comments?

Posted March 23, 2004
Last modified April 16, 2004

2:40 pm - 3:30 pm Lockett 235 Originally scheduled for 3:40 pm

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

Friday, April 30, 2004

Algebra and Number Theory Seminar  Questions or comments?

Posted April 26, 2004
Last modified April 27, 2004

3:40 pm - 4:30 pm Lockett 235

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

Tuesday, September 14, 2004

Algebra and Number Theory Seminar  Questions or comments?

Posted September 9, 2004
Last modified September 13, 2004

3:10 pm - 4:00 pm Lockett 282

Helena Verrill, Mathematics Department, LSU
Finding the Picard Fuchs differential equations of certain families of Calabi-Yau varieties

Tuesday, September 21, 2004

Algebra and Number Theory Seminar  Questions or comments?

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

Tuesday, September 28, 2004

Algebra and Number Theory Seminar  Questions or comments?

Posted September 9, 2004
Last modified September 13, 2004

3:10 pm - 4:00 pm Lockett 282

Marco Schlichting, Louisiana State University
Higher algebraic K-theory of forms and Karoubi's fundamental theorem

Tuesday, October 12, 2004

Algebra and Number Theory Seminar  Questions or comments?

Posted September 9, 2004
Last modified September 30, 2004

3:10 pm - 4:00 pm Tuesday, October 5, 2004 Lockett 282

Ambar Sengupta, Mathematics Department, LSU
Calculus Reform, or How (super)Algebra simplifies Calculus (on manifolds)

Wednesday, October 20, 2004

Algebra and Number Theory Seminar  Questions or comments?

Posted September 17, 2004
Last modified October 19, 2004

3:40 pm - 4:30 pm Lockett 282

Pramod Achar, Mathematics Department, LSU
Hecke algebras and complex reflection groups

Tuesday, November 2, 2004

Algebra and Number Theory Seminar  Questions or comments?

Posted September 10, 2004
Last modified October 22, 2004

3:10 pm - 4:00 pm Tuesday, October 26, 2004 Lockett 282 Originally scheduled for 3:40 pmMonday, October 4, 2004

Robert Perlis, Mathematics Department, LSU
Disconnected thoughts on Klein's four group

Sunday, January 9, 2005

Algebra and Number Theory Seminar  Questions or comments?

Posted December 9, 2005

Locket 285

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

Tuesday, March 8, 2005

Algebra and Number Theory Seminar  Questions or comments?

Posted March 4, 2005

2:30 pm - 3:30 pm Lockett 282

Augusto Nobile, Mathematics Department, LSU
Algorithmic equiresolution

Tuesday, March 15, 2005

Algebra and Number Theory Seminar  Questions or comments?

Posted March 9, 2005
Last modified March 11, 2005

3:10 pm - 4:00 pm Lockett 282

Jean Bureau, Louisiana State University
The Four Conjecture

Tuesday, March 29, 2005

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

Tuesday, April 5, 2005

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.

Tuesday, April 12, 2005

Algebra and Number Theory Seminar  Questions or comments?

Posted April 4, 2005
Last modified April 7, 2005

2:40 pm - 3:30 pm Lockett 282

Debra Czarneski, LSU
Zeta Functions of Finite Graphs

Abstract

Tuesday, April 19, 2005

Algebra and Number Theory Seminar  Questions or comments?

Posted April 11, 2005
Last modified April 14, 2005

3:10 pm - 4:00 pm Lockett 282

James Madden, Mathematics Department, LSU
Ways of ordering real algebras

Abstract

Tuesday, April 26, 2005

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

Abstract

Monday, September 12, 2005

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

Monday, September 19, 2005

Algebra and Number Theory Seminar  Questions or comments?

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

Monday, October 3, 2005

Algebra and Number Theory Seminar  Questions or comments?

Posted September 21, 2005
Last modified September 28, 2005

3:35 pm - 4:30 pm Monday, September 26, 2005 Locket 285

Jerome W. Hoffman, Mathematics Department, LSU
Koszul Duality III

continuation of previous algebra seminar

Monday, October 10, 2005

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

Monday, October 17, 2005

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

Monday, October 24, 2005

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

Monday, November 7, 2005

Algebra and Number Theory Seminar  Questions or comments?

Posted October 15, 2005
Last modified November 2, 2005

3:40 pm Locket 285

Jorge Morales, Mathematics Department, LSU
Quaternion orders, ternary quadratic forms and hyperelliptic curves

Monday, November 14, 2005

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

Monday, November 21, 2005

Algebra and Number Theory Seminar  Questions or comments?

Posted November 9, 2005

3:40 pm Locket 285

Pramod Achar, Mathematics Department, LSU
How I learned to stop worrying and love stacks

Monday, November 28, 2005

Algebra and Number Theory Seminar  Questions or comments?

Posted November 4, 2005
Last modified November 9, 2005

3:30 pm Locket 285 Originally scheduled for 3:30 pm, Monday, November 14, 2005

Planning meeting to decide graduate courses in algebra for next year

Monday, December 5, 2005

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.

Tuesday, January 10, 2006

Algebra and Number Theory Seminar  Questions or comments?

Posted December 9, 2005
Last modified December 18, 2005

2:30 pm

Marie-José Bertin, Université Pierre et Marie Curie, Paris
Lehmer's problem and Mahler measure

Monday, January 30, 2006

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

Monday, February 6, 2006

Algebra and Number Theory Seminar  Questions or comments?

Posted February 1, 2006

2:30 pm Locket 276

James Madden, Mathematics Department, LSU
Orderings of commutative rings with nilpotents

Monday, March 6, 2006

Algebra and Number Theory Seminar  Questions or comments?

Posted January 31, 2006
Last modified February 14, 2006

2:30 am Locket 276

Edith Adan-Bante, University of Southern Mississippi Gulf Coast
On Conjugacy Classes and Finite Groups

abstract

Monday, March 13, 2006

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

Monday, March 27, 2006

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

Monday, April 3, 2006

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)

Monday, April 17, 2006

Algebra and Number Theory Seminar  Questions or comments?

Posted April 5, 2006
Last modified April 6, 2006

3:45 pm Locket 276

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.

Monday, April 24, 2006

Algebra and Number Theory Seminar  Questions or comments?

Posted April 5, 2006
Last modified April 6, 2006

3:45 pm Locket 276

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.

Monday, May 1, 2006

Algebra and Number Theory Seminar  Questions or comments?

Posted March 13, 2006
Last modified April 5, 2006

2:30 pm Locket 276

Edith Adan-Bante, University of Southern Mississippi Gulf Coast
On Characters and Finite Groups

abstract

Tuesday, September 12, 2006

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

Tuesday, September 19, 2006

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

Tuesday, February 27, 2007

Algebra and Number Theory Seminar  Questions or comments?

Posted February 6, 2007
Last modified February 7, 2007

3:40 pm - 4:30 pm Lockett 243

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.

Tuesday, May 1, 2007

Algebra and Number Theory Seminar  Questions or comments?

Posted April 24, 2007
Last modified April 27, 2007

3:40 pm Lockett 136

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.

Tuesday, November 6, 2007

Algebra and Number Theory Seminar  Questions or comments?

Posted November 2, 2007
Last modified November 4, 2007

3:40 pm - 4:30 pm Lockett 277

Daniel Sage, Mathematics Department, LSU
Perverse coherent sheaves and special pieces in the unipotent variety

Tuesday, November 13, 2007

Algebra and Number Theory Seminar  Questions or comments?

Posted September 12, 2007
Last modified November 9, 2007

3:40 pm - 4:30 pm Lockett 240

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.

Monday, November 19, 2007

Algebra and Number Theory Seminar  Questions or comments?

Posted November 14, 2007
Last modified November 19, 2007

11:40 am - 12:30 pm Lockett 241

Daniel Sage, Mathematics Department, LSU
Perverse coherent sheaves and special pieces in the unipotent variety, part 2

Tuesday, November 27, 2007

Algebra and Number Theory Seminar  Questions or comments?

Posted September 14, 2007
Last modified November 20, 2007

3:40 pm - 4:30 pm Lockett 239

Pramod Achar, Mathematics Department, LSU
Staggered t-structures and equivariant coherent sheaves

Tuesday, April 1, 2008

Algebra and Number Theory Seminar  Questions or comments?

Posted February 14, 2008
Last modified March 31, 2008

1:40 pm - 2:30 pm Lockett 111

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.

Tuesday, September 23, 2008

Algebra and Number Theory Seminar  Questions or comments?

Posted September 15, 2008
Last modified September 19, 2008

3:40 pm Lockett 235

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.

Tuesday, October 7, 2008

Algebra and Number Theory Seminar  Questions or comments?

Posted September 24, 2008
Last modified October 3, 2008

2:40 pm Lockett 235 Originally scheduled for 3:30 pmMonday, September 29, 2008

Christopher Bremer, Mathematics Department, LSU
Periods of Irregular Singular Connections, Part II

Continuation of the September 23 seminar.

Tuesday, October 14, 2008

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

Tuesday, October 28, 2008

Algebra and Number Theory Seminar  Questions or comments?

Posted October 14, 2008
Last modified October 24, 2008

3:40 pm Lockett 235

Jerome W. Hoffman, Mathematics Department, LSU
Motives, algebraic cycles, and Hodge theory

Tuesday, November 4, 2008

Algebra and Number Theory Seminar  Questions or comments?

Posted October 10, 2008
Last modified November 3, 2008

3:40 pm Lockett 276

Alexander Prestel, Universitaet Konstanz
Representing polynomials positive on a semialgebraic set

Tuesday, November 11, 2008

Algebra and Number Theory Seminar  Questions or comments?

Posted October 14, 2008
Last modified November 10, 2008

3:40 pm Lockett 235

Piotr Maciak, Mathematics Department, LSU Graduate Student
A short journey from Gaussian integers to Drinfeld modules

Tuesday, March 3, 2009

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.

Tuesday, March 24, 2009

Algebra and Number Theory Seminar  Questions or comments?

Posted March 12, 2009
Last modified March 14, 2009

3:40 pm - 4:30 pm Lockett 235

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.

Tuesday, March 31, 2009

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).

Tuesday, April 14, 2009

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

Tuesday, September 8, 2009

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

Tuesday, October 6, 2009

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

Tuesday, October 13, 2009

Algebra and Number Theory Seminar  Questions or comments?

Posted October 5, 2009
Last modified October 13, 2009

3:40 pm Lockett 285

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.

Tuesday, October 20, 2009

Algebra and Number Theory Seminar  Questions or comments?

Posted October 7, 2009
Last modified October 20, 2009

3:40 pm Lockett 285

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.

Tuesday, October 27, 2009

Algebra and Number Theory Seminar  Questions or comments?

Posted October 5, 2009
Last modified October 26, 2009

3:40 pm Lockett 285

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.

Tuesday, November 10, 2009

Algebra and Number Theory Seminar  Questions or comments?

Posted October 7, 2009
Last modified November 4, 2009

3:40 pm Lockett 285

Jerome W. Hoffman, Mathematics Department, LSU
L-functions and l-adic representations for modular forms

Friday, November 13, 2009

Algebra and Number Theory Seminar  Questions or comments?

Posted November 4, 2009
Last modified November 11, 2009

3:40 pm Lockett 285

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.

Tuesday, November 17, 2009

Algebra and Number Theory Seminar  Questions or comments?

Posted October 5, 2009
Last modified November 12, 2009

3:40 pm Lockett 285

Jorge Morales, Mathematics Department, LSU
Siegel's mass formula and averages of L-functions over function fields

Tuesday, December 1, 2009

Algebra and Number Theory Seminar  Questions or comments?

Posted November 25, 2009
Last modified November 30, 2009

4:00 pm Lockett 285

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.

Monday, March 8, 2010

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

Friday, March 12, 2010

Algebra and Number Theory Seminar  Questions or comments?

Posted February 25, 2010
Last modified March 11, 2010

3:40 pm Lockett 285

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.

Tuesday, March 23, 2010

Algebra and Number Theory Seminar  Questions or comments?

Posted February 25, 2010
Last modified March 12, 2010

3:40 pm Lockett 285

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.

Friday, March 26, 2010

Algebra and Number Theory Seminar  Questions or comments?

Posted March 3, 2010
Last modified March 25, 2010

3:40 pm - 4:30 pm Lockett 237

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.'

Tuesday, March 30, 2010

Algebra and Number Theory Seminar  Questions or comments?

Posted March 22, 2010
Last modified March 29, 2010

3:40 pm Lockett 285

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.

Tuesday, April 13, 2010

Algebra and Number Theory Seminar  Questions or comments?

Posted March 19, 2010
Last modified April 12, 2010

3:40 pm Lockett 285

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.

Monday, April 19, 2010

Algebra and Number Theory Seminar  Questions or comments?

Posted March 12, 2010
Last modified April 12, 2010

3:40 pm Lockett 285

Jared Culbertson, Mathematics Department, LSU
Perverse Poisson sheaves on the nilpotent cone

Tuesday, April 20, 2010

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

Tuesday, April 27, 2010

Algebra and Number Theory Seminar  Questions or comments?

Posted April 13, 2010
Last modified April 25, 2010

3:40 pm Lockett 285

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.

Friday, April 30, 2010

Algebra and Number Theory Seminar  Questions or comments?

Posted February 25, 2010
Last modified April 25, 2010

3:40 pm Lockett 285 Originally scheduled for 3:30 pmFriday, March 12, 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.

Thursday, May 6, 2010

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.

Tuesday, September 14, 2010

Algebra and Number Theory Seminar  Questions or comments?

Posted September 6, 2010
Last modified September 14, 2010

3:40 pm - 4:30 pm Lockett 240

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.

Tuesday, September 21, 2010

Algebra and Number Theory Seminar  Questions or comments?

Posted September 13, 2010
Last modified September 14, 2010

3:40 pm - 4:30 pm Lockett 240

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.

Tuesday, September 28, 2010

Algebra and Number Theory Seminar  Questions or comments?

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.

Thursday, November 4, 2010

Algebra and Number Theory Seminar  Questions or comments?

Posted October 20, 2010
Last modified October 25, 2010

3:40 pm - 4:30 pm Lockett 240 Originally scheduled for 3:40 pm, Tuesday, October 26, 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.

Tuesday, November 16, 2010

Algebra and Number Theory Seminar  Questions or comments?

Posted September 7, 2010
Last modified November 9, 2010

3:40 pm - 4:30 pm Lockett 240

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.

Friday, February 25, 2011

Algebra and Number Theory Seminar  Questions or comments?

Posted February 17, 2011
Last modified February 24, 2011

3:40 pm - 4:30 pm Lockett 277

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).

Wednesday, March 2, 2011

Algebra and Number Theory Seminar  Questions or comments?

Posted February 7, 2011
Last modified March 1, 2011

3:40 pm - 4:30 pm Lockett 277

Skip Garibaldi, Emory University
Matrix groups and diagonalizable matrices

This talk is about groups of matrices over a field, like GLn (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.

Wednesday, March 9, 2011

Algebra and Number Theory Seminar  Questions or comments?

Posted March 3, 2011

3:40 pm - 4:30 pm Lockett 277

Sabin Cautis, Columbia University
A categorification of the Heisenberg algebra

Tuesday, March 15, 2011

Algebra and Number Theory Seminar  Questions or comments?

Posted February 17, 2011
Last modified March 11, 2011

2:40 pm - 3:30 pm Lockett 284

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.

Tuesday, March 22, 2011

Algebra and Number Theory Seminar  Questions or comments?

Posted February 16, 2011
Last modified March 11, 2011

2:40 pm - 3:30 pm Lockett 284

Matthew Housley, University of Utah
TBA

Tuesday, April 12, 2011

Algebra and Number Theory Seminar  Questions or comments?

Posted March 11, 2011

3:40 pm - 4:30 pm Lockett 277

Aaron Lauda, Columbia University
TBA

Tuesday, April 26, 2011

Algebra and Number Theory Seminar  Questions or comments?

Posted March 9, 2011

3:40 pm - 4:30 pm Lockett 277

David Chapman, Mathematics Department, LSU Graduate student
TBA

Tuesday, August 23, 2011

Algebra and Number Theory Seminar  Questions or comments?

Posted August 18, 2011
Last modified August 23, 2011

3:40 pm - 4:30 pm Lockett 240

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.

Tuesday, August 30, 2011

Algebra and Number Theory Seminar  Questions or comments?

Posted August 18, 2011
Last modified August 23, 2011

3:40 pm - 4:30 pm Lockett 240

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.

Tuesday, September 6, 2011

Algebra and Number Theory Seminar  Questions or comments?

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.

Tuesday, September 13, 2011

Algebra and Number Theory Seminar  Questions or comments?

Posted August 22, 2011
Last modified September 11, 2011

3:40 pm - 4:30 pm Lockett 240

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.

Tuesday, September 20, 2011

Algebra and Number Theory Seminar  Questions or comments?

Posted August 22, 2011
Last modified September 13, 2011

3:40 pm - 4:30 pm Tuesday, September 6, 2011 Lockett 240

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.

Tuesday, September 27, 2011

Algebra and Number Theory Seminar  Questions or comments?

Posted August 30, 2011

3:40 pm - 4:30 pm Lockett 240

Greg Muller, Department of Mathematics, LSU
TBA

Tuesday, October 11, 2011

Algebra and Number Theory Seminar  Questions or comments?

Posted October 11, 2011

3:40 pm - 4:30 pm Lockett 240

Jerome W. Hoffman, Mathematics Department, LSU
Galois representations and Humbert surfaces

Tuesday, October 25, 2011

Algebra and Number Theory Seminar  Questions or comments?

Posted September 8, 2011
Last modified October 17, 2011

3:40 pm - 4:30 pm Lockett 240

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.

Tuesday, November 15, 2011

Algebra and Number Theory Seminar  Questions or comments?

Posted August 30, 2011

3:40 pm - 4:30 pm Lockett 240

Christopher Bremer, Mathematics Department, LSU
TBA

Tuesday, November 22, 2011

Algebra and Number Theory Seminar  Questions or comments?

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.

Monday, November 28, 2011

Algebra and Number Theory Seminar  Questions or comments?

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.

Tuesday, November 29, 2011

Algebra and Number Theory Seminar  Questions or comments?

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  Questions or comments?

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.

Tuesday, February 14, 2012

Algebra and Number Theory Seminar  Questions or comments?

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.)

Friday, March 16, 2012

Algebra and Number Theory Seminar  Questions or comments?

Posted February 8, 2012
Last modified March 2, 2012

3:40 pm - 4:30 pm Lockett 277

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.

Tuesday, March 20, 2012

Algebra and Number Theory Seminar  Questions or comments?

Posted March 12, 2012
Last modified March 19, 2012

3:40 pm - 4:30 pm Lockett 277

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.

Tuesday, March 27, 2012

Algebra and Number Theory Seminar  Questions or comments?

Posted February 1, 2012
Last modified March 20, 2012

3:40 pm - 4:30 pm Lockett 277

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 $A_n$.

Tuesday, April 3, 2012

Algebra and Number Theory Seminar  Questions or comments?

Posted March 15, 2012
Last modified March 28, 2012

3:40 pm - 4:30 pm Lockett 277

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.

Tuesday, April 17, 2012

Algebra and Number Theory Seminar  Questions or comments?

Posted March 13, 2012
Last modified April 13, 2012

3:40 pm - 4:30 pm

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.

Tuesday, April 24, 2012

Algebra and Number Theory Seminar  Questions or comments?

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.

Wednesday, May 2, 2012

Algebra and Number Theory Seminar  Questions or comments?

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.

Tuesday, September 25, 2012

Algebra and Number Theory Seminar  Questions or comments?

Posted September 20, 2012
Last modified September 24, 2012

3:30 pm - 4:30 pm Lockett 235

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.

Tuesday, October 9, 2012

Algebra and Number Theory Seminar  Questions or comments?

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.

Tuesday, November 6, 2012

Algebra and Number Theory Seminar  Questions or comments?

Posted November 3, 2012
Last modified November 6, 2012

3:30 pm - 4:20 pm Lockett 235

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.

Tuesday, November 13, 2012

Algebra and Number Theory Seminar  Questions or comments?

Posted October 24, 2012
Last modified November 5, 2012

3:30 pm - 4:30 pm Lockett 235

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.

Tuesday, January 29, 2013

Algebra and Number Theory Seminar  Questions or comments?

Posted January 21, 2013
Last modified January 28, 2013

3:30 pm - 4:30 pm Lockett 277

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.

Tuesday, March 12, 2013

Algebra and Number Theory Seminar  Questions or comments?

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.

Thursday, March 14, 2013

Algebra and Number Theory Seminar  Questions or comments?

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.

Tuesday, April 23, 2013

Algebra and Number Theory Seminar  Questions or comments?

Posted March 12, 2013
Last modified April 22, 2013

3:30 pm - 4:20 pm Lockett 277

Peter Samuelson, University of Toronto
Skein modules and the double affine Hecke algebra

The Kauffman bracket skein module is a vector space Kq(M) associated to a 3-manifold M and a parameter q ∈ C*. We recall an old theorem which states that the colored Jones polynomials Jn(q, K) ∈ C[q,q−1] of a knot K in S3 can be computed from Kq(S3 K). We also describe a theorem of Frohman and Gelca which shows that Kq(S3K) is a module over the Z2-invariant subalgebra of the quantum torus Aq. 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 Kq(S3 K) to a 2-parameter family of modules over H(q,t). Conjecturally, these lead to 2-variables polynomials Jn(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.)

Monday, June 3, 2013

Algebra and Number Theory Seminar  Questions or comments?

Posted May 17, 2013
Last modified May 24, 2013

2:00 pm - 3:00 pm Lockett 277

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.

Tuesday, October 8, 2013

Algebra and Number Theory Seminar  Questions or comments?

Posted September 25, 2013
Last modified October 7, 2013

3:30 pm - 4:20 pm Lockett 243

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.

Tuesday, October 15, 2013

Algebra and Number Theory Seminar  Questions or comments?

Posted September 25, 2013
Last modified October 14, 2013

3:30 pm - 4:20 pm Lockett 243

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.

Tuesday, October 22, 2013

Algebra and Number Theory Seminar  Questions or comments?

Posted September 25, 2013
Last modified October 16, 2013

3:30 pm - 4:20 pm Lockett 243

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.

Tuesday, October 29, 2013

Algebra and Number Theory Seminar  Questions or comments?

Posted September 25, 2013
Last modified October 18, 2013

3:30 pm - 4:20 pm Lockett 243

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.

Tuesday, November 12, 2013

Algebra and Number Theory Seminar  Questions or comments?

Posted October 14, 2013
Last modified November 5, 2013

3:30 pm - 4:20 pm Lockett 243

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.

Tuesday, November 19, 2013

Algebra and Number Theory Seminar  Questions or comments?

Posted September 30, 2013
Last modified November 5, 2013

3:30 pm - 4:20 pm Lockett 243

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.

Friday, January 31, 2014

Algebra and Number Theory Seminar  Questions or comments?

Posted January 9, 2014
Last modified January 31, 2014

3:30 pm - 4:20 pm Lockett 285 Originally scheduled for 3:30 pm, Tuesday, January 28, 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.

Tuesday, February 18, 2014

Algebra and Number Theory Seminar  Questions or comments?

Posted January 31, 2014
Last modified February 12, 2014

3:30 pm - 4:20 pm Lockett 285

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.

Tuesday, February 25, 2014

Algebra and Number Theory Seminar  Questions or comments?

Posted January 31, 2014
Last modified February 21, 2014

3:30 pm - 4:20 pm Lockett 285

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.

Tuesday, April 8, 2014

Algebra and Number Theory Seminar  Questions or comments?

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.

Tuesday, April 29, 2014

Algebra and Number Theory Seminar  Questions or comments?

Posted April 24, 2014
Last modified April 27, 2014

3:30 pm - 4:20 pm Lockett 285

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.

Tuesday, September 23, 2014

Algebra and Number Theory Seminar  Questions or comments?

Posted September 17, 2014

3:30 am - 4:30 am 235 Lockett Hall

Emil 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.

Wednesday, October 8, 2014

Algebra and Number Theory Seminar  Questions or comments?

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.

Tuesday, October 14, 2014

Algebra and Number Theory Seminar  Questions or comments?

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.

Tuesday, October 21, 2014

Algebra and Number Theory Seminar  Questions or comments?

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.

Tuesday, October 28, 2014

Algebra and Number Theory Seminar  Questions or comments?

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.

Tuesday, November 11, 2014

Algebra and Number Theory Seminar  Questions or comments?

Posted October 8, 2014
Last modified October 16, 2014

3:30 pm - 4:30 pm 235 Lockett Hall

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.

Tuesday, November 18, 2014

Algebra and Number Theory Seminar  Questions or comments?

Posted October 8, 2014
Last modified October 30, 2014

3:30 pm - 4:30 pm 235 Lockett Hall

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.

Tuesday, December 2, 2014

Algebra and Number Theory Seminar  Questions or comments?

Posted October 8, 2014
Last modified November 26, 2014

3:30 pm - 4:30 pm 235 Lockett Hall

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.

Thursday, January 8, 2015

Algebra and Number Theory Seminar  Questions or comments?

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  Questions or comments?

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.

Tuesday, February 3, 2015

Algebra and Number Theory Seminar  Questions or comments?

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.

Thursday, February 19, 2015

Algebra and Number Theory Seminar  Questions or comments?

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.

Thursday, February 26, 2015

Algebra and Number Theory Seminar  Questions or comments?

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.

Tuesday, March 10, 2015

Algebra and Number Theory Seminar  Questions or comments?

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.

Tuesday, March 24, 2015

Algebra and Number Theory Seminar  Questions or comments?

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.

Tuesday, April 14, 2015

Algebra and Number Theory Seminar  Questions or comments?

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.

Tuesday, April 21, 2015

Algebra and Number Theory Seminar  Questions or comments?

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.

Tuesday, May 26, 2015

Algebra and Number Theory Seminar  Questions or comments?

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.

Tuesday, October 27, 2015

Algebra and Number Theory Seminar  Questions or comments?

Posted October 18, 2015
Last modified October 21, 2015

3:30 pm - 4:20 pm Lockett 277

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.

Tuesday, November 3, 2015

Algebra and Number Theory Seminar  Questions or comments?

Posted October 18, 2015
Last modified November 2, 2015

1:30 pm - 2:20 pm Lockett 285

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.

Tuesday, November 10, 2015

Algebra and Number Theory Seminar  Questions or comments?

Posted October 18, 2015
Last modified October 25, 2015

3:30 pm - 4:20 pm Lockett 277

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.

Tuesday, November 24, 2015

Algebra and Number Theory Seminar  Questions or comments?

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.

Tuesday, January 26, 2016

Algebra and Number Theory Seminar  Questions or comments?

Posted January 15, 2016
Last modified January 25, 2016

3:30 pm - 4:20 pm Lockett 277

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.

Tuesday, February 16, 2016

Algebra and Number Theory Seminar  Questions or comments?

Posted December 2, 2015
Last modified February 15, 2016

3:30 pm - 4:20 pm Lockett 277

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.

Tuesday, March 15, 2016

Algebra and Number Theory Seminar  Questions or comments?

Posted February 2, 2016
Last modified March 14, 2016

3:30 pm - 4:20 pm Lockett 277

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.

Thursday, April 7, 2016

Algebra and Number Theory Seminar  Questions or comments?

Posted March 31, 2016
Last modified April 5, 2016

3:30 pm - 4:20 pm Lockett 277

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.

Tuesday, April 12, 2016

Algebra and Number Theory Seminar  Questions or comments?

Posted February 22, 2016
Last modified April 11, 2016

3:30 pm - 4:20 pm Lockett 277

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.

Tuesday, April 26, 2016

Algebra and Number Theory Seminar  Questions or comments?

Posted February 24, 2016
Last modified April 25, 2016

2:30 pm - 3:20 pm Lockett 276

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  Questions or comments?

Posted April 4, 2016
Last modified April 18, 2016

3:30 pm - 4:20 pm Lockett 277

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.

Thursday, April 28, 2016

Algebra and Number Theory Seminar  Questions or comments?

Posted April 15, 2016
Last modified April 25, 2016

2:30 pm - 3:20 pm Lockett 277

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.

Tuesday, September 13, 2016

Algebra and Number Theory Seminar  Questions or comments?

Posted August 11, 2016
Last modified September 6, 2016

3:30 pm - 4:20 pm Lockett 284

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).

Tuesday, September 20, 2016

Algebra and Number Theory Seminar  Questions or comments?

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.

Tuesday, October 4, 2016

Algebra and Number Theory Seminar  Questions or comments?

Posted September 13, 2016
Last modified September 28, 2016

3:30 pm - 4:20 pm Lockett 284

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.

Tuesday, October 18, 2016

Algebra and Number Theory Seminar  Questions or comments?

Posted September 13, 2016
Last modified September 28, 2016

3:30 pm - 4:20 pm Lockett 284

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.

Tuesday, October 25, 2016

Algebra and Number Theory Seminar  Questions or comments?

Posted September 28, 2016
Last modified October 20, 2016

3:30 pm - 4:20 pm Lockett 284

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.

Tuesday, November 1, 2016

Algebra and Number Theory Seminar  Questions or comments?

Posted September 16, 2016
Last modified October 20, 2016

3:30 pm - 4:20 pm Lockett 284

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.

Friday, November 4, 2016

Algebra and Number Theory Seminar  Questions or comments?

Posted September 13, 2016
Last modified October 20, 2016

3:30 pm - 4:20 pm Lockett 277

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.

Tuesday, November 15, 2016

Algebra and Number Theory Seminar  Questions or comments?

Posted October 17, 2016
Last modified January 11, 2017

3:30 pm - 4:20 pm

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.

Tuesday, February 7, 2017

Algebra and Number Theory Seminar  Questions or comments?

Posted January 15, 2017
Last modified February 5, 2017

3:30 pm - 4:20 pm Lockett 277

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.

Tuesday, March 7, 2017

Algebra and Number Theory Seminar  Questions or comments?

Posted January 15, 2017
Last modified March 6, 2017

3:30 pm - 4:20 pm Lockett 277

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.

Tuesday, March 14, 2017

Algebra and Number Theory Seminar  Questions or comments?

Posted January 11, 2017
Last modified March 9, 2017

3:00 pm - 3:50 pm Lockett 277

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.

Tuesday, March 21, 2017

Algebra and Number Theory Seminar  Questions or comments?

Posted January 15, 2017
Last modified March 13, 2017

3:00 pm - 3:50 pm Lockett 277

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.

Tuesday, March 28, 2017

Algebra and Number Theory Seminar  Questions or comments?

Posted January 15, 2017
Last modified March 22, 2017

3:00 pm - 3:50 pm Lockett 277

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.

Tuesday, April 4, 2017

Algebra and Number Theory Seminar  Questions or comments?

Posted January 15, 2017
Last modified March 29, 2017

3:00 pm - 3:50 pm Lockett 277

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.

Tuesday, April 25, 2017

Algebra and Number Theory Seminar  Questions or comments?

Posted January 23, 2017
Last modified March 17, 2017

3:00 pm - 3:50 pm Lockett 277

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.

Tuesday, May 9, 2017

Algebra and Number Theory Seminar  Questions or comments?

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 fi rst for quadratic fi elds then for an arbitrary number fi eld. Next, we will discuss a generalization of reduced ideals using the LLL-algorithm. Finally, some open problems relating to this topic will be presented.

Thursday, May 18, 2017

Algebra and Number Theory Seminar  Questions or comments?

Posted April 22, 2017
Last modified May 18, 2017

3:00 pm - 3:50 pm Lockett 9

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.

Tuesday, September 12, 2017

Algebra and Number Theory Seminar  Questions or comments?

Posted August 2, 2017
Last modified September 12, 2017

3:10 pm - 4:00 pm 276 Lockett

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.



Tuesday, October 31, 2017

Algebra and Number Theory Seminar  Questions or comments?

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.

Tuesday, November 7, 2017

Algebra and Number Theory Seminar  Questions or comments?

Posted September 9, 2017
Last modified October 20, 2017

3:10 pm - 4:00 pm 276 Lockett

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.

Tuesday, November 14, 2017

Algebra and Number Theory Seminar  Questions or comments?

Posted August 31, 2017
Last modified November 9, 2017

3:10 pm - 4:00 pm 276 Lockett

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.

Tuesday, November 28, 2017

Algebra and Number Theory Seminar  Questions or comments?

Posted August 23, 2017
Last modified October 20, 2017

3:00 pm - 3:50 pm
Cancelled

Li Guo, Rutgers University at Newark
TBD


Algebra and Number Theory Seminar  Questions or comments?

Posted November 12, 2017

3:10 pm - 4:00 pm

William Casper, Louisiana State University
TBD