# Algebra and Combinatorics 2018 Spring

Spring 2018
Time & Location: All talks are on Friday in Gibson Hall 126 at 3:00 PM unless otherwise noted.
Organizer: Mahir Can

January 26
Algebra and Combinatorics
A closed non-iterative formula for straightening fillings of Young diagram
Reuven HodgesInstitution
Abstract:
Young diagrams are fundamental combinatorial objects in representation theory and algebraic geometry. Many constructions that rely on these objects depend on variations of a straightening process, due to Alfred Young, that expresses a filling of a Young diagram as a sum of semistandard tableaux subject to certain relations. It has been a long-standing open problem to give a non-iterative, closed formula for this straightening process.

In this talk I will give such a formula, as well as a simple combinatorial description of the coefficients that arise. Moreover, an interpretation of these coefficients in terms of paths in a directed graph will be explored. I will end by discussing a surprising application of this formula towards finding multiplicities of irreducible representations in certain plethysms and how this relates to Foulkes' conjecture.

February 9
Topic
SpeakerInstitution
Abstract: TBA

February 16
Algebra and Combinatorics
Counting Borel Orbits in Symmetric Varieties of Types BI and CII
Ozlem UgurluTulane University
Abstract:

February 23
Algebra and Combinatorics
De-noetherizing Cohen-Macaulay rings
Laszlo FuchsTulane university
Abstract:
The Cohen-Macaulay rings play a most important role in commutative algebra and in its applications in algebraic geometry. They are of special kind of noetherian rings, with a rich and fast developing theory. Generalizations to non-noetherian rings are scarce, none is satisfactory, since they preserve only a few selected properties of Cohen-Macaulay rings.  We introduce a new class of commutative non-noetherian rings that is a more natural generalization and enjoys the analogues of almost all of the relevant features of the classical Cohen-Macaulay theory.

The talk is based on three papers (none published as yet), one is a joint work with Luigi Salce, and another with Bruce Olberding.

March 2
Algebra and Combinatorics
Quotients of locally compact abelian groups
Karl HofmannInstitution
Abstract:

March 8
Algebra and Combinatorics
The Separable Quotient Problem for Topological Groups
(joint work with Sidney Morris, and Mikhail Tkachenko)
Arkady LeidermanDepartment of Mathematics, Ben-Gurion University of the Negev, Israel
Abstract: The famous Banach-Mazur problem,  which asks if  every infinite-dimensional Banach space has  an infinite-dimensional separable quotient Banach space, has remained  unsolved for 85 years, though it has been answered in the affirmative for reflexive Banach spaces and even Banach spaces which are duals. The analogous problem for locally convex spaces has been answered in the negative, but has been shown to be true for large classes of locally convex spaces including all non-normable Frechet spaces. We investigated the analogous problem of existing of separable quotients for topological groups.  Indeed there are four natural analogues: Does every non-totally disconnected topological group have a  separable quotient group which is (i) non-trivial; (ii) infinite; (iii) metrizable; (iv) infinite metrizable. All four questions are answered in the negative in our work.  However, positive answers are given for important classes of topological groups including (a) all compact groups; (b) all locally compact abelian groups; (c) all $\sigma$-compact locally compact groups; (d) all abelian pro-Lie groups; (e) all $\sigma$-compact pro-Lie groups; (f) all pseudocompact groups. We observe then that all simple algebraic groups over local fields are separable metrizable groups. Negative answers are proved for abelian precompact groups.
Location: Gibson Hall 308
Time: 1:00 PM

March 16
Algebra and Combinatorics
G2-structures and octonion bundles
Sergey GrigorianSergey Grigorian
Abstract:
The group G2 is the automorphism group of the octonion algebra. Given a G2-structure on a 7-dimensional Riemannian manifold we define an octonion bundle with a fiberwise non-associative product. We then define a metric-compatible octonionic covariant derivative on this bundle that is also compatible with the octonion product. The torsion of the G2-structure is then shown to be an octonionic connection for this covariant derivative with curvature given by the component of the Riemann curvature that lies in the 7-dimensional representation of G2. The choice of a particular G2-structure within the same metric class is then interpreted as a choice of gauge and we show that under a change of this gauge, the torsion transforms as an octonion-valued connection 1-form. We will then discuss further properties of this non-associative structure on 7-manifolds.

March 23
Koszul duality and characters of tilting modules
Pramod AcharLSU
Abstract:
This talk is about the "Hecke category," a monoidal category that appears in various incarnations in geometric representation theory. I will explain some of these incarnations and their roles in solving classical problems, such as the celebrated Kazhdan-Lusztig conjectures on Lie algebra representations. These conjectures (proved in 1981) hinge on the fact that the derived category of constructible sheaves on a flag variety is equipped with an obvious monoidal action of the Hecke category on the right.

It turns out that there is also a second, "hidden" action of the Hecke category on the left. The symmetry between the "hidden" left action and the "obvious" right action leads to the phenomenon known as Koszul duality. In the last part of the talk, I will discuss new results on Koszul duality with coefficients in a field of positive characteristic, with applications to characters of tilting modules for algebraic groups. This is joint work with S. Makisumi, S. Riche, and G. Williamson.