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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.
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.
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.
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.
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.
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 $f$.
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.
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.
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
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.
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.
Edward Letzter, Temple University
A discussion of more recent results, and still-open questions, on free subalgebras and free multiplicative subsemigroups of associative algebras.
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.
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
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