Algebra Seminar 2019

Current contacts: Vasily Dolgushev, Ed Letzter or Martin Lorenz.

The Seminar usually takes place on Mondays at 1:30 PM in Room 617 on the sixth floor of Wachman Hall. Click on title for abstract.

• Monday January 28, 2019 at 13:30, Wachman 617
Paschke Categories, K-homology and the Riemann-Roch Transformation

Khashayar Sartipi, University of Illinois at Chicago

For a separable C^*-algebra A, we introduce an exact C^*-category called the Paschke Category of A, which is completely functorial in A, and show that its K-theory groups are isomorphic to the topological K-homology groups of the C^*-algebra A. Then we use the Dolbeault complex and ideas from the classical methods in Kasparov K-theory to construct an acyclic chain complex in this category, which in turn, induces a Riemann-Roch transformation in the homotopy category of spectra, from the algebraic K-theory spectrum of a complex manifold X, to its topological K-homology spectrum. This talk is based on the preprint https://arxiv.org/abs/1810.11951

• Monday February 4, 2019 at 13:30, Wachman 617
Detecting free objects in associative algebras: A survey

Edward Letzter, Temple University

In the 1970s, Lichtman asked whether or not the multiplicative group of units of a noncommutative division algebra contains a free subgroup and Makar-Limanov asked whether or not a finitely generated infinite dimensional noncommutative division algebra must contain a free subalgebra. These questions are still open in general, even if many important special cases have been resolved, and have recently received renewed attention. (These questions can be considered in analogy to the Tits Alternative for linear groups as well as Gromov's Theorem on groups with polynomial growth.) My talks will survey both older and newer results.

• Monday February 11, 2019 at 13:30, Wachman 617
Detecting free objects in associative algebras, II

Edward Letzter, Temple University

A discussion of more recent results, and still-open questions, on free subalgebras and free multiplicative subsemigroups of associative algebras.

• Monday February 18, 2019 at 13:30, Wachman 617
On the Replacement Property for $PSL(2,p)$

Aidan Lorenz, Temple University

The replacement property (or Steinitz Exchange Lemma) for vector spaces has a natural analog for finite groups and their generating sets. For the special case of the groups $PSL(2,p)$, where $p$ is a prime larger than 5, first partial results concerning the replacement property were published by Benjamin Nachman. The main goal of this talk is to outline the methods involved in providing a complete answer for $PSL(2,p)$ (which was accomplished during the Summer of 2018). This talk is based on a paper in preparation joint with Baran Zadeoglu and David Cueto Noval.

• Monday February 25, 2019 at 13:30, Wachman 617
Prime Torsion of the Brauer Group of an Elliptic Curve

Charlotte Ure, Michigan State University

The Brauer group of an elliptic curve $E$ is an important invariant with intimate connections to cohomology and rational points. Elements of this group can be described as Morita equivalence classes of central simple algebras over the function field. The Merkurjev-Suslin theorem implies that these classes can be written as tensor product of symbol (or cyclic) algebras. In this talk, I will describe an algorithm to calculate generators and relations of the $q$-torsion ($q$ a prime) of the Brauer group of $E$ in terms of these tensor products over any field of characteristic different from $2$,$3$, and $q$, containing a primitive $q$-th root of unity. This is work in progress.

• Monday March 11, 2019 at 13:30, Wachman 617
Catalan Functions and k-Schur functions

Anna Pun, Drexel University

Li-Chung Chen and Mark Haiman studied a family of symmetric functions called Catalan (symmetric) functions which are indexed by pairs consisting of a partition contained in the staircase (n-1, ..., 1,0) (of which there are Catalan many) and a composition weight of length n. They include the Schur functions, the Hall-Littlewood polynomials and their parabolic generalizations. They can be defined by a Demazure-operator formula, and are equal to GL-equivariant Euler characteristics of vector bundles on the flag variety by the Borel-Weil-Bott theorem. We have discovered various properties of Catalan functions, providing a new insight on the existing theorems and conjectures inspired by Macdonald positivity conjecture.

A key discovery in our work is an elegant set of ideals of roots that the associated Catalan functions are k-Schur functions and proved that graded k-Schur functions are G-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We exposed a new shift invariance property of the graded k-Schur functions and resolved the Schur positivity and k-branching conjectures by providing direct combinatorial formulas using strong marked tableaux. We conjectured that Catalan functions with a partition weight are k-Schur positive which strengthens the Schur positivity of Catalan function conjecture by Chen-Haiman and resolved the conjecture with positive combinatorial formulas in cases which capture and refine a variety of problems. This is joint work with Jonah Blasiak, Jennifer Morse and Daniel Summers. Here are the links to the papers on ArXiv: https://arxiv.org/abs/1804.03701, https://arxiv.org/abs/1811.02490

• Monday March 18, 2019 at 13:30, Wachman 617
Classifying Actions of $T_n \otimes T_n$ on Path Algebras of Quivers

Delaney Aydel, Temple University

Let $T_n$ denote the $n$th Taft algebra. We fully classify inner-faithful actions of $T_n \otimes T_n$ on four-vertex Schurian quivers as extensions of the actions of $\mathbb{Z}_n \times \mathbb{Z}_n$. One example will be presented in full, with the remaining results briefly given.

• Monday March 25, 2019 at 13:30, Wachman 617
Topics in Galois Theory, I

Martin Lorenz, Temple University

This is the first lecture in a minicourse (probably three lectures) surveying some topics in Galois Theory that are not typically covered in the graduate algebra course (Math 8011/12): inverse Galois theory, Noether's rationality problem, the Chebotarev density theorem,... The Galois Theory portion of Math 8011/12 will be sufficient background for the material presented in this minicourse; so it will be accessible to all students in my current Math 8012 class. No proofs will be given; the goal is to describe some research directions that are of current interest.

• Monday April 1, 2019 at 13:30, Wachman 617
Topics in Galois Theory, II

Martin Lorenz, Temple University The second talk in this series will be devoted to the behavior of Galois groups under reduction mod primes. More specifically, given an polynomial $f \in \mathbf{Z}[x]$, I will discuss the question what the (cyclic!) Galois groups of the reductions of $f$ mod various primes tell us about the Galois group of $f$.

• Monday April 8, 2019 at 13:30, Wachman 617
Topics in Galois Theory, III

Martin Lorenz, Temple University

First, I will finish (after some reminders) the proof of the reduction-mod-primes recipe for Galois groups from last time. Then I will address the following deficiency of the reduction method: while the full symmetric group is easily detected in this way, small Galois groups require further tools. I will explain a probabilistic method that is based on the Tchebotarov Density Theorem.

• Monday April 15, 2019 at 13:30, Wachman 617
What is so algebraic about the algebraic fundamental group?

Vasily Dolgushev, Temple University

A careful definition of the fundamental group in the realm of algebraic geometry requires a lot of effort. In Chapter 4 of his book "Galois groups and fundamental groups", Tamas Szamuely gives a gentle introduction to this topic for algebraic curves. In my two lectures, I will follow Tamas's presentation from this Chapter. Most of proofs will be omitted but I will try give examples. My lectures are partially inspired by Martin Lorenz's recent mini-course.

• Monday April 22, 2019 at 13:30, Wachman 617
Kazhdan-Lusztig polynomials of matroids

Jacob Matherne, IAS, Princeton

Kazhdan-Lusztig (KL) polynomials for Coxeter groups were introduced in the 1970s, providing deep relationships among representation theory, geometry, and combinatorics. In 2016, Elias, Proudfoot, and Wakefield defined analogous polynomials in the setting of matroids. In this talk, I will compare and contrast KL theory for Coxeter groups with KL theory for matroids.

I will also associate to any matroid a certain ring whose Hodge theory can conjecturally be used to establish the positivity of the KL polynomials of matroids as well as the "top-heavy conjecture" of Dowling and Wilson from 1974 (a statement on the shape of the poset which plays an analogous role to the Bruhat poset). Examples involving the geometry of hyperplane arrangements will be given throughout. This is joint work with Tom Braden, June Huh, Nick Proudfoot, and Botong Wang.

• Monday April 29, 2019 at 13:30, Wachman 617
What is so algebraic about the algebraic fundamental group? Part 2

Vasily Dolgushev, Temple University

This is the second lecture devoted to Chapter 4 of Tamas Szamuely's book "Galois Groups and Fundamental Group". I will define the algebraic fundamental group of a curve and talk about the outer Galois action on the algebraic fundamental group. Examples will be given.

• Monday September 9, 2019 at 13:30, Wachman 617
A brief introduction to operads I

Vasily Dolgushev, Temple University

The operad PaB is closely related to the Grothendieck-Teichmueller group GT introduced by Vladimir Drinfeld in 1990. This is the first talk in the mini-course devoted to the operad PaB, GT-shadows and their action on child's drawings. In this talk, I will introduce operads and give various examples. This mini-course should be accessible to first year graduate students.

• Monday September 16, 2019 at 13:30, Wachman 617
A brief introduction to operads II

Vasily Dolgushev, Temple University

In the second talk of this series, I will give more examples of operads. I will also talk about one of the central objects of this series, the operad of parenthesized braids PaB. This is an operad in the category of groupoid and it is "assembled from" Artin's braid groups. This operad was introduced by Dmitry Tamarkin in 1998 and a very similar object was introduced by Dror Bar-Natan in 1996.

• Monday October 7, 2019 at 13:30, Wachman 617
The operad PaB of parenthesized braids I

Vasily Dolgushev, Temple University

After a brief review of the operad PaB, I will talk about compatible equivalence relations on the truncation of PaB. A large supply of such equivalence relations come from finite index normal subgroups of $B_4$ which are contained in $PB_4$.

• Monday October 14, 2019 at 13:30, Wachman 617
Highest-weight representations and global Weyl modules: from classical Lie algebras to Yangians

Prasad Senesi, The Catholic University of America

Highest-weight representations play a prominent role in the representation theory of Lie algebras and quantum groups. Particular examples of highest-weight representations of certain infinite-dimensional Lie algebras called the Weyl modules (for loop and quantum algebras) were introduced by Chari and Pressley in 2000. In this introductory talk, we proceed by example from the classical structure and representation theory of the special linear algebra in dimensions 2 and 3, to that of the corresponding loop algebras and quantum groups. Along the way, the utility of highest-weight representations, and of the (local and global) Weyl Modules, in all of these settings will be described. We will conclude with a discussion of the Yangian, its relation to the quantum loop algebra, and some recent work concerning its global Weyl modules. This is joint work with Bach Nguyen (Temple University) and Matt Lee (University of Illinois at Chicago).

• Monday October 21, 2019 at 13:30, Wachman 617
An approach toward supersymmetric cluster algebras

Ashish K. Srivastava, Saint Louis University

In this talk we will propose the notion of cluster superalgebra which is a supersymmetric version of the classical cluster algebra introduced by Fomin and Zelevinsky. We show that the symplectic-orthogonal supergroup $SpO(2|1)$ admits a cluster superalgebra structure and as a consequence of this, we deduce that the supercommutative superalgebra generated by all the entries of a superfrieze is a cluster superalgebra. We also show that the coordinate superalgebra of the super Grassmannian $G(2|0; 4|1)$ of chiral conformal superspace (that is, $(2|0)$ planes inside the superspace $\mathbb C^{4|1}$) is a quotient of a cluster superalgebra.

• Monday October 28, 2019 at 13:30, Wachman 617
The operad PaB of parenthesized braids II

Vasily Dolgushev, Temple University

After a brief reminder of the operad PaB, I will talk about the compatible equivalence relations coming from finite index normal subgroups N in $B_4$ which are contained in the pure braid group $PB_4$ on 4 strands. If time will permit, I will introduce GT-shadows.

• Monday November 4, 2019 at 13:30, Wachman 617
$k$-Schur and Catalan functions

Jennifer Morse, University of Virginia

We will discuss the inception, subsequent developments, and resolution of a symmetric function conjecture from the 1990's. The $k$-Schur functions arose via computer experimentation with symmetric functions called Macdonald polynomials; they are symmetric functions with coefficients involving a single $t$-parameter. Conjectures that they satisfy many strong and beautiful positivity properties compelled further study. In the special case when $t=1$, it was unexpectedly discovered that $k$-Schur functions are geometrically significant in an area called affine Schubert calculus and for computing Gromov-Witten invariants. However, the intricate combinatorics behind $k$-Schur functions involving the Bruhat order on the affine symmetric group made progress with generic $t$ extremely hard to come by.

We recently discovered a new approach to the study of $k$-Schur functions; they are a subclass of Catalan functions, $G$-equivariant Euler characteristics of vector bundles on the flag variety defined by raising operators and indexed by Dyck paths. This perspective led us to settle decades old conjectures, providing tableaux enumeration formulas to do so. Joint work with Blasiak, Pun, and Summers.

• Monday November 11, 2019 at 13:30, Wachman 617
GT-shadows and their action on child's drawings I

Vasily Dolgushev, Temple University

Last time, I introduced GT-shadows and showed that GT-shadows are morphisms in a groupoid whose objects are compatible equivalence relations on PaB. This time, I will talk about child's drawings subordinate to a given compatible equivalence relation on PaB. I will explain in what sense GT-shadows act on child's drawings.

• Monday December 2, 2019 at 13:30, Wachman 617
GT-shadows and their action on child's drawings II

Vasily Dolgushev, Temple University

I will introduce child's drawings which are subordinate to a given compatible equivalence relation on PaB. I will introduce the action of GT-shadows on child's drawings as a (co)functor from the groupoid of GT-shadows to the category of finite sets. If time will permit, I will talk about the inverse subordination problem for child's drawings and infinite chains in the poset $NFI_{PB_4}(B_4)$.

• Monday December 9, 2019 at 13:30, Wachman 617
Brill-Noether theory and degeneracy loci

Linda Chen, Swarthmore College

Degeneracy loci of morphisms between vector bundles have been used in a wide range of situations, including classical approaches to the Brill-Noether theory of special divisors on curves. I will give an introduction to these connections and describe recent developments, including new K-theoretic formulas for degeneracy loci and their applications to Brill-Noether loci. These recover the formulas of Eisenbud-Harris, Pirola, and Chan-Lopez-Pflueger-Teixidor for Brill-Noether curves. This is joint work with Dave Anderson and Nicola Tarasca.