Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted August 26, 2014

3:00 pm - 4:00 pm Lockett 232
Jacob Matherne, Department of Mathematics, LSU

Representation Theory of the Symmetric Group

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted September 8, 2014

3:00 pm - 4:00 pm Lockett 113
Richard Frnka, Department of Mathematics, LSU
Graduate Student

Farey Sequences, Ford Circles, and Their Application in Rademacher's Theorem for the Partition Function

Abstract:The Farey Sequence of order n on an interval is the complete ordered sequence of reduced fractions whose denominator does not exceed n in the interval. These fractions can be used to generate Ford Circles, which have some very nice properties including a relation to modular forms. For two consecutive fractions in the sequence of order n (called Farey neighbors), the Ford Circles generated by them are tangent at only one point. By taking the arc on a circle between the two tangent points from both of its Farey neighbors for every fraction in the sequence, we can form a periodic, continous path. Rademacher used this path to integrate the generating function for partitions to come up with an exact formula for the partition number, which had only been approximated before. This talk does not require any background, and will be accessible to any graduate/undergraduate students with a basic knowledge of Euclidean geometry.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted September 13, 2014

3:30 pm - 4:30 pm Lockett 113
Lucius Schoenbaum, LSU

Tropical Geometry I

Tropical geometry is a relatively new subject in mathematics which draws connections between algebraic geometry and discrete mathematics and applies them, for example, to enumerative geometry and areas of theoretical physics. In this talk I will introduce the subject and present the graph-theoretic proof of the Riemann-Roch theorem for tropical curves due to Gathmann (2006), based on work of Baker and Norine. This talk should be accessible to any graduate student, or motivated undergraduate familiar with basic abstract algebra and discrete math.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted September 19, 2014

Last modified May 28, 2015

Bach Nguyen, LSU

Derived Categories

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted October 9, 2014

3:30 pm - 4:30 pm Lockett 113
Lucius Schoenbaum, LSU

Tropical Geometry II

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted October 18, 2014

Last modified October 27, 2014

Sean Taylor, LSU

Introduction to Theory of Sheaf, Part I

Sheaves are an important tool in modern mathematics that were introduced by Jean Leray during the 1940's. They have since become an integral part of algebraic geometry due to the work of many, not the least of which are Jean-Pierre Serre and Alexander Grothendieck. However, they are fruitful not only to algebraic geometry, but also to such areas as algebraic topology and representation theory. In this talk I will present the basic definition of sheaves, the category of sheaves on a topological space, and the functors that can be associated with these categories. I will also present a theorem relating certain types of sheaves on a topological space and the fundamental group of that space.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted October 30, 2014

3:30 pm - 4:30 pm Lockett 113
Sean Taylor, LSU

Introduction to Theory of Sheaf, Part II

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted November 9, 2014

3:30 pm - 4:30 pm Lockett 113
Dun Liang, LSU

Hodge Theory

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted November 19, 2014

3:30 pm - 4:30 pm Lockett 113
Dun Liang, LSU

Curves, Jacobian, and Their Moduli

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted February 6, 2015

Last modified February 18, 2015

Eric Bucher, LSU

Maximal Green Sequences

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted March 5, 2015

3:40 pm - 4:30 pm Lockett 233
Joseph Timmer, Louisiana State University

An Introduction to Hopf Algebras

Hopf algebras have grown in usefulness since their introduction and seem to be a pervasive element in many areas of mathematics. They appear in Topology, Representations of Groups, Lie Theory, Category Theory and even Applied Mathematics. In this talk, we introduce the definitions, structures and "well known" theory. We will focus on examples and ideas of the field. The talk will be accessible to first year graduate students. The only assumed knowledge will be very basic ring theory and some linear algebra.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted March 15, 2015

3:30 pm - 4:30 pm Lockett 233
Ian Runnels, LSU

Higher Polytopal Relations for Wigner's 6j-symbol

In the quantum theory of angular momentum, Eugene Wigner introduced a gadget called the 6j-symbol to calculate recoupling coefficients for interactions between several particles. Mathematically, these gadgets found uses in the representation theory of what Wigner called "Simply Reducible Groups", classical groups with the property that each irreducible representation is self-dual and multiplicity free; one such example is the Lie group SU(2,C). Over the next few decades, both physicists and mathematicians worked out many symmetries and relations for 6j-symbols, the most famous of which is called the Elliot-Biedenharn identity (secretly this is just the pentagon identity for tensor categories). In this talk, I will develop the definition of the 6j-symbol through the representation theory of SU(2) and introduce some new identities stemming from the combinatorics of these gadgets.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted August 25, 2015

3:15 pm - 4:15 pm Lockett 233
Jacob Matherne, Department of Mathematics, LSU

The Hilbert scheme of points in the plane

Abstract: We will begin with a few brief notions in algebraic geometry needed to introduce the Hilbert scheme of points in the plane. The Hilbert scheme is a compactification of a configuration space of n distinct particles moving around on a plane. Our main tool for studying it will be the combinatorics of Young diagrams. Time permitting, we may discuss torus actions on the Hilbert scheme and the computation of its cohomology. The talk should be accessible to first-year graduate students (even if the words seem scary at first), so come by!

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted August 27, 2015

3:15 pm - 4:15 pm Lockett 233
Kyle Istvan, LSU

Quantum Invariant Theory

This informal discussion will focus on motivating the use of quantum groups to creating topological invariants, following the perspective of Manin. We will begin with a brief discussion of SL(2,C), its action on C^2, and why this particular group is of interest in geometry, topology, (and vaguely, number theory.) We will then define a new "geometric" object, the quantum complex plane C_q^{2}, and proceed to derive the necessary deformation of SL(2,C) in order to have a useful action on the quantum plane. If time permits, we will see the Kauffman relations (from the study of links and 3-manifolds) appear very naturally in this setting as the quantum analogue of the Cayley-Hamilton Identity, and hopefully motivate the further study of deformations of classical groups. This talk is based on a series of lectures given by Roland van der Veen at Gazi University in August 2015.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted September 21, 2015

3:15 pm - 4:15 pm Lockett 233
Neal Livesay, LSU

An excursion into the mathematics of Jacques Tits

Abstract: The theory of buildings provides a beautiful combinatorial and geometric viewpoint to the theory of algebraic groups. It can be shown that every group G with an algebraic condition (having a BN-pair) corresponds to a simplicial complex (a building) endowed with a compatible action by G. We will discover this correspondence by working out the details for G=GL_2(k) and GL_3(k).

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted October 12, 2015

3:15 pm - 4:15 pm Lockett 233
Bach Nguyen, LSU

A New Look into the Center of the Quantized Enveloping Algebra of a Complex Semi-simple Lie Algebra

Abstract: In the paper \textit{``Local Finiteness of the Adjoint Action for Quantized Enveloping Algebras''} by Anthony Joseph and Gail Letzter, they show that the center of $U_{q}(\mathfrak{g})$ ($\mathfrak{g}$ is the Kac-Moody algbra) is isomorphic to the $W$-invariants in the ring $k(q)[T^0]$, where $W$, the Weyl group, acting by traslation, and $T^{0}=T_{<}^{-1}T_{<}$, where $T_<=-R^+$, and $R^+$ is the intersection of four times the dominant weight with the extended root lattice. Recently, in \textit{``Generalized Joseph's Decompositions,''} Arkady Berenstein and Jacob Greenstein give a new construction for the basis of $\mathcal{Z}(U_q)$ which allows us to identify the center with the ring of symmetric functions. In this talk, we'll be discussing the construction that lead to this new basis of $\mathcal{Z}(U_q)$.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted October 17, 2015

Last modified October 21, 2015

Richard Frnka, Department of Mathematics, LSU
Graduate Student

Partitions, Unimodal Sequences, and some Congruence Relations

Abstract: The partition number p(n) is an important mathematical idea that has applications in Combinatorics, Number Theory, and even Representation Theory. In this talk, we will discuss the generating function for p(n), some important relations linking infinite products to theta functions, and will consider different restrictions on the parts that make up the partitions. The only requirement for this talk will be a basic knowledge of geometric series. The plan is to make this talk very accessible and basic, so we will be able to go deeper into the asymptotics of the partition function and unimodal sequences in a later talk.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted November 9, 2015

3:15 pm - 4:15 pm Lockett 233
Cris Negron, Mathematics Department, LSU

Cohomology for the Young and Restless

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted November 30, 2015

3:15 pm - 4:15 pm Lockett 233
Jesse Levitt, LSU

Nearly Commutative Rings and Algebras: Properties preserved by controlled non-commutativity

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted February 17, 2016

4:30 pm - 5:30 pm Lockett 233
Bach Nguyen, LSU

Poisson Structures arising from Noncommutative Algebras I

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted February 23, 2016

3:40 pm - 4:40 pm Lockett 233
Bach Nguyen, LSU

Poisson Structures arising from Noncommutative Algebras II

We will continue to discuss examples of Poisson Structures coming from noncommutative algebras such as Lie algebra, algebra of polynomial differential, and algebra of quantum matrices.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted March 7, 2016

3:40 pm - 4:10 pm Lockett 233
Neal Livesay, LSU

Regular filtrations on the symplectic loop algebra

Filtrations on the symplectic loop algebra can give explicit normal forms for formal flat $Sp$-bundles with ``toral'' singularities. Conjecturally, these filtrations will be useful for constructing well-behaved moduli spaces, generalizing the work of Bremer and Sage on flat $GL$-bundles. I will discuss what is meant by a ``toral'' singularity and illustrate the theory for some small rank examples.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted March 7, 2016

4:10 pm - 4:40 pm Lockett 233
Lucius Schoenbaum, LSU

Introduction to Topos Theory

What does left exactness of a left R-module over a commutative ring have to do with the existence of atoms in a Boolean algebra? We will learn about this and more in a leisurely, very short introduction to topos theory.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted April 13, 2016

3:40 pm - 4:40 pm Lockett 233
Cris Negron, Mathematics Department, LSU

A Suprised Talk

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted April 20, 2016

3:40 pm - 4:40 pm Lockett 233
Maitreyee Kulkarni, LSU

Introduction to Nakajima's quiver varieties

Kashiwara's crystals give a combinatorial description of characters of irreducible finite dimensional U_q(g)-modules. The situation gets much more complicated in the case of the quantum loop algebras U_q(Lg), as the characters are now polynomials in terms of q. Nakajima has given a geometric description of the irreducible q-characters of U_q(Lg) using his graded quiver varieties. In this talk, we will define these quiver varieties, compute some examples and see how they are related to the q-characters.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted April 27, 2016

1:30 pm - 2:20 pm Lockett 233
Sean Taylor, LSU

An Introduction to Algebraic Stacks

Algebraic stacks constitute an often over-looked area of modern algebraic geometry, because of the seeming abstractness of the subject. In this talk, I will attempt to give a (very) brief and hopefully gentle introduction to stacks and what makes a stack "algebraic." This will hopefully be approachable even for those who do not know what a scheme is, since the audience will be encouraged to think about the geometric objects in their favorite categories (manifolds, complex analytic spaces, etc.). In short, these are powerful geometric (!) objects that are becoming more and more fundamental to algebraic geometry and representation theory among other subjects.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted May 2, 2016

1:30 pm - 2:30 pm Lockett 233
Sean Taylor, LSU

Introduction to Algebraic Stacks Part 2

After introducing what a Grothendieck topology was last week, we will be able to proceed to actually define stacks - which are higher analogues of sheaves - and then algebraic stacks. If time provides, we will look at common examples of these beautiful and powerful geometric objects.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted July 11, 2016

2:00 pm - 3:00 pm Lockett 233
Cris Negron, Mathematics Department, LSU

Gauge invariants from the antipode for a Chevalley Hopf algebras

The antipode of a given Hopf algebra is an easily overlooked, yet mysteriously informative, part of the Hopf algebra structure. (For the uninitiated, the antipode of a Hopf algebra is an algebra anti-automorphism which acts like the inversion operator of a group.) For example, a result of of Larson and Radford states that a finite dimensional Hopf algebra in characteristic 0 is a semisimple ring if and only if the square of the antipode is the identity. For a finite dimensional non-semisimple Hopf algebra we only know that the order of the antipode is some positive even integer. This number can be seen as a measure of non-semisimplicity. In this talk I will discuss gauge invariance of the order of the antipode for a certain class of finite dimensional Hopf algebras, and some other related invariants. Rather, I will discuss how the order of the antipode for a given Hopf algebra can, in some cases, be extricated from its associated tensor category of representations.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted August 23, 2016

3:35 pm - 4:35 pm Lockett 233
Emily Cowie, LSU

Introduction to Lie algebras and their representations

This talk is intended to give an introduction to the basics of Lie algebra representation theory. This talk will provide the necessary background and definitions before describing the combinatorics of the most fundamental Lie algebra, sl(2, C).

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted September 27, 2016

3:30 pm - 4:30 pm Lockett 233
Bach Nguyen, LSU

Orders: "You are out of order!"

Abstract: We will define orders and discuss examples and some of their important properties.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted October 6, 2016

3:30 pm - 4:30 pm Lockett 233
Lucius Schoenbaum, LSU

Cartesian Closed Categories and Lambda Calculus

During the 1960's and 1970's, connections between logic and category theory were discovered through the work of Grothendieck, Lawvere, Lambek, Benabou, and others. In the 1980's, these developments made an impact on areas of computer science, such as functional programming and the design of many functional programming languages. In this talk, I will focus on cartesian closed categories and the (simply-typed) lambda calculus, which are related via the Curry-Howard-Lambek correspondence (I will explain what this is). Prerequisites: No category theory other than a basic idea of what categories and functors are.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted November 9, 2016

3:30 pm - 4:30 pm Lockett 233
Sean Taylor, LSU

Algebraic Geometry over Fields of Characteristic p

Over fields of characteristic p there exist new phenomena that arise. One important example of this is the action of the Galois group, and in particular the Frobenius element, on the geometric points of a scheme and on the etale cohomology groups. In this talk we will discuss some of these structures and more.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted November 16, 2016

4:30 pm - 5:30 pm Lockett 233
Shotaro Makisumi, Stanford University

Introduction to Soergel Bimodules

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted January 11, 2017

3:00 pm - 4:00 pm Lockett 233
Jacob Matherne, University of Massachusetts at Amherst

Combinatorial Fourier transform for quiver representation varieties in type A

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted January 17, 2017

10:00 am - 11:00 am Lockett 233
Sean Taylor, LSU

Perverse Sheaves on Toric Varieties

Toric varieties (or "torus embeddings" as they were originally known) are defined as algebraic varieties that have an algebraic torus as an open dense subset such that the natural action of the torus on itself extends smoothly to the whole variety. They have slowly become of more and more interest to both algebraic geometers and combinatorists. The reason for this is that a large class of toric varieties can be functorially associated to several combinatorial objects, the most well known of which are called combinatorial fans. They are also of interest to algebraic geometers for their own sake - much in the way that algebraic curves or surfaces are - and because, though they obey a whole host of interesting and powerful geometric, topological, and combinatorial properties, they paradoxically turn out to be a fantastic place to test new theorems in algebraic geometry. In this talk, we will discuss recent research associated with the speaker's thesis with Pramod Achar. The ultimate goal is to finish creating a mixed category of perverse sheaves on toric varieties in the sense of Beilinson, Ginzburg, and Soergel. Along the way, however, it has become necessary to make an extended study of perverse sheaves on toric varieties and the special properties that they possess. We will begin with some basic definitions and arrive at some fascinating decompositions of Ext groups on toric varieties.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted January 17, 2017

3:00 pm - 4:00 pm Lockett 233
Sean Taylor, LSU

Representations of SO($\infty$)

One of the primary goals of representation theory and harmonic analysis is to decompose "natural" representations into irreducible pieces. In the 20th century, the regular representation of semisimple Lie groups on symmetric spaces stood out as both an example of extraordinary success of this task and as a model of beauty. In the study of symmetric spaces and their connections to representation theory, we find an interaction between algebra, geometry, and analysis. Since the introduction of Kac-Moody Lie algebras and Kac-Moody Lie groups, infinite-dimensional Lie theory has been an important area of exploration for representation theorists. It is a vista that is still very open, however, and it turns out to be important to consider even "simple" examples of infinite-dimensional Lie groups. In this talk, we will explore recent research of the speaker along with Matthew Dawson, Stephan Merignon, and Gestur Olafsson on a "replacement" for the regular representation of SO($\infty$) and a construction of new representations on these spaces.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted February 28, 2017

3:00 pm - 4:00 pm Lockett 233
Bach Nguyen, LSU

Quantum Groups: Definitions, Motivations/ History and Applications

The term quantum groups is often used in a loosed way to describe objects in mathematics which may or may not related to each other. To clear up some confusion, we will discuss the definitions of quantum groups in various settings such as: universal enveloping algebra of semisimple lie algebras/ Kac-Moody algebras, coordinate ring of simple algebraic groups, C^*- algebras, and infinite dimension quantum groups. If time permits, the history/ motivation and applications of these objects will also be mentioned.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted March 14, 2017

4:30 pm - 5:30 pm Lockett 233
Trey Trampel, LSU

Computing Noncommutative Discriminants via Poisson Primes

We will present a general method for computing discriminants of noncommutative algebras obtained from specialization at roots of unity. This method builds a connection with Poisson geometry and will express the discriminants as products of Poisson primes. The method will be used to compute the discriminants of specializations at roots of unity of algebras of quantum square matrices. We will also evaluate the more general case of specialization of any quantum Schubert cell algebra.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted August 31, 2017

3:30 pm - 4:30 pm Lockett 233
Andrew Alaniz, LSU

Hecke Operators and Ramanujan's tau function

Abstract: The purpose of this talk is, primarily, to show how applying algebraic methods to situations involving analytic objects can often simplify the situation. Often, fourier coefficients encode interesting arithmetic or combinatorial information about a particular generating series considered as a holomorphic function, this is a common theme in number theory. Hecke's revolutionary insight was understanding the universial meaning of constructions like these. We will consider a certain class of operators, the so-called Hecke operators, and Ramanujan's tau function via the theory of modular forms.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted September 6, 2017

3:15 pm - 4:15 pm Lockett 233
William Hardesty, Louisiana State University

Baby Verma modules for p-restricted Lie algebras

Abstract: Will give a brief introduction to the representation of theory of (semi-simple) restricted Lie algebras with an emphasis on an important class of representations called "Baby Verma modules". Various results concerning the structure of these modules will be presented and, if time permits, we will briefly discuss some relevant ongoing joint work between the speaker and V. Nandakumar.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted September 6, 2017

3:30 pm - 4:30 pm Lockett 233
Matt Lee, UC Riverside

Demazure Flags and q-Hypergeometric Series

Abstract: Since the current algebra of sl_2 is not semisimple, we need to understand more than just the irreducible representations. We will look at the filtration for a family of modules parametrized by a partition. Attempts to generalize this filtration to other modules leads to an interesting connection to q-Hypergeometric series and related concepts. This talk is accessible to anyone with basic knowledge of sl_2.

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted November 19, 2017

3:30 pm - 4:20 pm Lockett 233
Sean Taylor, LSU

Mixed Categories of Sheaves on Toric Varities

Student Algebra Seminar
Graduate Student Algebra and Number Theory Seminar

Posted December 7, 2017

3:30 pm
Bach Nguyen, Louisiana State University

Hecke algebra and its interesting applications