Title: Schubert Calculus and Cohomology of Grassmannians

Date: 03/12/2018

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

In this first lecture, we shall begin by constructing a CW-structure for the complex Grassmannian. We will then begin our exploitation of this CW-structure with the goal of answering questions in enumerative geometry. These questions include “How many lines lie on the intersection of two quadric hypersurfaces in P^4?” and “Given four smooth curves in P^3, how many lines will intersect all four curves?”

Title: Real applications of non-real numbers: Ramanujan graphs (First Phillips Lecture)

Date: 03/12/2018

Time: 5:30 PM - 6:30 PM

Place:

The real numbers form a completion of the field of rational numbers. We will describe the fields of p-adic numbers which are different completions of the rationals. Once they are defined, one can study analysis and geometry over them. While being very abstract, the main motivation for studying them came from number theory. Developments in the last 2-3 decades shows various applications to the real world: communication networks, etc. This is done via expander graphs and Ramanujna grpahs which are "Riemann surfaces over these p-adic fields". All notions will be explained.

Title: High dimensional expanders: From Ramanujan graphs to Ramanujan complexes (Second Phillips Lecture)

Date: 03/13/2018

Time: 4:00 PM - 5:00 PM

Place: 115 International Center

Expander graphs in general, and Ramanujan graphs, in particular, have played a major role in combinatorics and computer science in the last 4 decades and more recently also in pure math. Approximately 10 years ago, a theory of Ramanujan complexes was developed by Li, Lubotzky-Samuels-Vishne and others. In recent years a high dimensional theory of expanders is emerging. The notions of geometric and topological expanders were defined by Gromov in 2010 who proved that the complete d dimensional simplicial complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d greater than 1. Ramanujan complexes were shown to be geometric expanders by Fox-Gromov-Lafforgue-Naor-Pach in 2013, but it was left open if they are also topological expanders. By developing new isoperimetric methods for “locally minimal small” F_2-co-chains, it was shown recently by Kaufman- Kazdhan- Lubotzky for small dimensions and Evra-Kaufman for all dimensions that the d-skeletons of (d+1)-dimensional Ramanujan complexes provide bounded degree topological expanders. This answers Gromov’s original problem, but still leaves open whether the Ramanujan complexes themselves are topological expanders. We will describe these developments and the general area of high dimensional expanders and some of its open problems.

Title: Groups' approximation, stability and high dimensional expanders (Third Phillips Lecture)

Date: 03/14/2018

Time: 10:00 AM - 11:00 AM

Place: C304 Wells Hall

Several well-known open questions (such as: are all groups sofic or hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the unitary groups U(n) (in the hyperlinear case)?
In the case of U(n), the question can be asked with respect to different metrics and norms. We answer, for the first time, one of these versions, showing that there exist fintely presented groups which are not approximated by U(n) with respect to the Frobenius (=L_2)norm.
The strategy is via the notion of “stability”: some higher dimensional cohomology vanishing phenomena is proven to imply stability and using higher dimensional expanders, it is shown that some non-residually finite groups (central extensions of some lattices in p-adic Lie groups) are Frobenious stable and hence cannot be Frobenius approximated.
All notions will be explained. Joint work with M. De Chiffre, L. Glebsky and A. Thom.

Title: Hiring Cycle and Expectations for Faculty at Liberal Arts Schools

Date: 03/14/2018

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

This seminar is aimed at postdocs and graduate students who will be on the job market soon, but all are welcome.
The seminar will start with a short (20 minute) presentation on Chris Marx about the hiring cycle and expectations for faculty at liberal arts schools. Afterwards, there will be a discussion/question and answer period.

Title: Correction term, diagonalization theorem and the sliceness of 2-bridge knots

Date: 03/14/2018

Time: 4:10 PM - 5:00 PM

Place: C204A Wells Hall

About a decade ago, Lisca classified which 2-bridge knots are smoothly slice using an obstruction derived from Donaldson's diagonaliztion theorem. It is known that the diagonalization theorem can be proved using the Heegaard Floer correction term. Moreover, this correction term can also be used to construct a slicing obstruction for knots. In this expository talk, I will explain Josh Greene's proof that these two slicing obstructions actually coincide for 2-bridge knots.

Title: Dependence of the density of states on the probability distribution for discrete random Schrödinger operators

Date: 03/15/2018

Time: 11:00 AM - 12:00 PM

Place: C304 Wells Hall

We prove the Hölder-continuity of the density of states measure (DOSm) and the integrated density of states (IDS) with respect to the probability distribution for discrete random Schrödinger operators with a finite-range potential. In particular, our result implies that the DOSm and the IDS for smooth approximations of the Bernoulli distribution converge to the corresponding quantities for the Bernoulli-Anderson model. Other applications of the techniques are given to the dependency of the DOSm and IDS on the disorder, and the continuity of the Lyapunov exponent in the weak-disorder regime. The talk is based on recent joint work with Peter Hislop (Univ. of Kentucky).

Title: Connected Heegaard Floer homology and homology cobordism

Date: 03/15/2018

Time: 2:00 PM - 3:00 PM

Place: C304 Wells Hall

We study applications of Heegaard Floer homology to homology cobordism. In particular, to a homology sphere Y, we define a module HF_conn(Y), called the connected Heegaard Floer homology of Y, and show that this module is invariant under homology cobordism and isomorphic to a summand of HF_red(Y). The definition of this invariant relies on involutive Heegaard Floer homology. We use this to define a new filtration on the homology cobordism group, and to give a reproof of Furuta's theorem. This is joint work with Jen Hom and Tye Lidman.

Title: Unstable entropy and pressure for partially hyperbolic systems

Date: 03/15/2018

Time: 3:00 PM - 4:00 PM

Place: C517 Wells Hall

We study ergodic properties caused by the unstable part of partially hyperbolic systems. We define unstable metric entropy, topological entropy and pressures, and prove the corresponding variational principles. For unstable metric entropy we obtain affineness, upper semi-continuity and a version of Shannon-McMillan-Breiman theorem. We also obtain existence of Gibbs u-states, differentiability properties of unstable pressure, such as tangent functionals, Gateaux differentiability and Frechet differentiability.