Monday, March 30, 2015 at 2:30 PM in SAS 4201
Daniel Sage, Louisiana State University
Moy-Prasad filtrations and flat G-bundles on curves
In this talk, I describe a new approach to the study of flat G-bundles
on curves (for complex reductive G) using methods of representation
theory. This approach is based on a geometric version of the
Moy-Prasad theory of minimal K-types (or fundamental strata) for
representations of p-adic groups. In the geometric theory, one
associates a fundamental stratum--data involving an appropriate
filtration on the loop algebra--to a formal flat
G-bundle. Intuitively, this stratum plays the role of the "leading
term" of the flat G-bundle and can be used to define its slope. I
will explain how these ideas can be used to construct well-behaved
moduli spaces of irregular singular flat G-bundles on P^1 and to
realize the isomonodromy equations as an explicit integrable system on
these moduli spaces. Time permitting, I will discuss some
applications (and potential applications) to the geometric Langlands
program, where (unlike the classical situation), fundamental strata
can be associated to objects on both the Galois and automorphic side
of the correspondence. This is joint work with C. Bremer.
Monday, April 6, 2015 at 2:30 PM in SAS 4201
Patrik Noren, NC State
Algebraic graph limits
The theory of graph limits associates random graph models to symmetric measurable functions on the unit square. We investigate what happens when these functions are polynomials. For low degree polynomials the models appearing are familiar and important, for example preferential attachment and Erdos-Renyi. The higher degree polynomials are also useful as any graph limit can be arbitrarily well approximated by a polynomial. We show that this setup could be useful in applications: To determine the parameters of an algebraic graph limit that fits observed data best one can use numerical algebraic geometry efficiently.
Monday, April 13, 2015 at 2:30 PM in SAS 4201
Iana Anguelova, College of Charleston
Monday, April 20, 2015 at 2:30 PM in SAS 4201
Alex Engstrom, Aalto University, Finland
Graph coloring and the total Betti number
The total Betti number of the independence complex of a graph is an intriguing graph invariant. Kalai and Meshulam have raised the question on its relation to cycles and the chromatic number of a graph, and a recent conjecture on that theme was proved by Bonamy, Charbit and Thomasse. We show an upper bound on the total Betti number in terms of the number of vertex disjoint cycles in a graph. The main technique is discrete Morse theory and building poset maps. Ramanujan graphs with arbitrary chromatic number and girth log(n) is a classical construction. We show that any subgraph of them with less than n^0.003 vertices have smaller total Betti number than some planar graph of the same order, although it is part of a graph with high chromatic number.
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