Everytopic Seminar - Spring 2014
Room 226, Goldsmith building
Organizer: Olivier Bernardi
The Everytopic Seminar is a seminar aimed at a broad audience of mathematicians.
The talks are 80 minutes long. The first 40 minutes of the talk should be given in a
colloquium style, and should be accessible to faculty and graduate students
from any field of mathematics. It should allow everyone to grasp the content and
significance of the results being discussed. The last 40 minutes should be in
the style of a research seminar, providing the audience with a deeper understanding
of the results as well as some details of the proofs.
Thursday January 30
Speaker: Kiyoshi Igusa (Brandeis University)
Title: The category of noncrossing partitions.
Abstract: The main purpose of this project to explain my work to people at Brandeis University
by removing the representations of quivers from the definitions.
Many of the theorems that I and co-authors have proven for any Dynkin quiver
(or more generally for finite convex subsets of the preprojective or preinjective partition of
the Auslander-Reiten quiver of any modulated quiver without oriented cycles)
can, in the special case of A_n with straight orientation, be presented purely combinatorially.
I will define the category NP(n) with noncrossing partitions as objects and binary forests (unions of binary trees) as morphisms.
I will use CAT(0) spaces to show that the classifying space of this category is a K(pi,1): It is a finite cubical complex where the link of every
vertex is a flag complex. I will explain using examples what this means.
The fundamental group of BNP(n) is very easy to describe. The explicit construction of its classifying space
allows for the computation of the cohomology of the group. It is free abelian in every degree with rank
given by the ballot numbers.
This is based on 2 joint works. One with Kent Orr, Jerzy Weyman and Gordana Todorov,
another with Gordana Todorov.
Thursday March 6
Speaker: Vadim Gorin (MIT)
Title: 2D stochastic systems and their asymptotics.
The talk is about 2D probabilistic systems which can be analyzed by
algebraic methods. Known examples of such systems include random
stepped surfaces, six-vertex model ("square ice"), spectra of random
matrices, interacting particle systems (e.g.TASEP), and
directed polymers in 2d random media.
Modern results make us believe that all these systems have similar
asymptotic behavior which can be described via new (as compared to
the classical 1d case) limit objects. These objects are the Gaussian
Free Field and Tracy-Widom distributions.
Although we are far from checking the universality of such behavior,
for a certain class of distributions it can be proved. Many rigorous
mathematical results in this direction are based on the relations to
the symmetric functions of representation-theoretic origins and, more
generally, to representations of infinite-dimensional groups.
Thursday March 13
Speaker: Ryan Kinser (Northeastern University)
Title: Finiteness properties in linear algebra and representation theory
Abstract: The first part of the talk will explain how problems linear algebra can often naturally be interpreted in the language of representation theory. The simplest classical example is the correspondence between linear operators and K[x]-modules, where K is a field and x a variable. We will also examine some related geometric constructions such as representation spaces of algebras.
In the second part we will focus on various notions of "finiteness" for classification problems in linear algebra, now using the language of representation theory.
Thursday March 20
Speaker: Han Li (Yale)
Title: Indefinite Integral Quadratic Forms Beyond Classical Reduction Theory
The classical reduction theory of integral quadratic forms was developed by Hermite, Minkowski, Siegel and many others. It is known that a non-degenerate integral quadratic form in n variables is integrally equivalent to a form whose height (the maximum value of the coefficients) is less than its determinant (up to a multiplicative constant), and whose value at (1, 0,...0) is less than the n-th root of its determinant. However, for indefinite forms in at least 3 variables it turns out that neither of the estimates is optimal. In this talk we will discuss some classical results and recent effort in improving these estimates. This is a joint work with Prof. Margulis.
Thursday March 27
Speaker: Rita Jimenez Rolland (Northeastern University)
Title: Stability phenomena in cohomology
Given a sequence of topological spaces Xn we associate a sequence of rational vector spaces Hi(Xn;Q): the ith rational cohomology group of Xn. We will discuss how these spaces change as the parameter n grows.
In the first part of this talk we will consider the configuration space of n ordered and unordered points in the plane to illustrate two possible stability situations. Other examples such as the moduli space of Riemann surfaces and the pure mapping class group will also be discussed. In the second part of the talk, I will introduce the notion of a finitely generated FI-module and show how it explains the behavior of the examples considered before.
Thursday April 3
Speaker: John Baldwin (Boston College)
Title: From knots invariants to bordered Floer homology
Abstract: I'll describe in this talk some motivation for a recent construction of bordered monopole Floer homology (joint work with Jon Bloom). I'll start with a discussion of the classical Jones polynomial of a knot in the 3-sphere. I'll then introduce Khovanov homology, a more sophisticated homological knot invariant which encodes the Jones polynomial but is a stronger invariant in general. In particular, Khovanov homology detects the unknot, whereas the analogous question for the Jones polynomial remains open. This fact was proven using a relationship between Khovanov homology and an even more sophisticated invariant of knots called instanton Floer homology. I'll survey the known relationships between the Khovanov homology of knots and the Floer homology of knots and 3-manifolds. I'll then describe how these sorts of relationships, combined with work of Khovanov on tangle invariants, has motivated a recent construction of bordered monopole Floer homology, which provides invariants of 3-manifolds with parametrized boundary and a pairing formula for computing the monopole Floer homology of a closed 3-manifold from the invariants associated to its pieces.
Thursday April 10
Speaker: John Bergdall (Boston University)
Title: Examples of representations appearing in the completed cohomology of definite unitary group
The local Langlands correspondence for GL(n) gives a direct relationship between certain representations of the group GL_n(K) on p-adic vectors spaces, where K is a non-Archimedean local field of residue characteristic different from p, and n-dimensional p-adic representations of the Galois group Gal(K-bar/K). The proof given by Harris and Taylor illuminates one beautiful aspects of the story: the local correspondence can be realized in the cohomology of certain algebraic varieties.
When K itself is a p-adic field, however, the GL(n)-side of the correspondence is much too coarse to capture some of the intricate behavior on the Galois side. The enriching of the GL(n)-side forms the emerging p-adic Langlands program. The necessary modifications were worked out for n = 2, and K the p-adic number themselves, by Breuil, Colmez, Emerton, Paskunas and many more. The correspondence can be realized inside Emerton's p-adically completed cohomology.
In this talk I will report progress towards conjectures of Christophe Breuil and Florian Herzig (2012) which give a precise prediction on GL(n)-representations appearing inside the cohomology of certain towers of arithmetic manifolds. This represents (very) partial progress towards a formulation of a p-adic local Langlands correspondence beyond the case of GL(2) over the p-adic numbers. Some of our progress was independently by Breuil and Herzig.
This is all joint with Przemyslaw Chojecki.
Map of Brandeis University - the Department of Mathematics (Goldsmith building) is U24 on this map.
For further information please contact:
Olivier Bernardi (bernardi at brandeis dot edu).