Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, October 29, 2013.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, October 29, 2013

11:00 am in 241 Altgeld Hall,Tuesday, October 29, 2013

Cubic theta functions

Daniel Schultz (UIUC Math)

Abstract: Ramanujan's theory of elliptic functions to alternate bases is introduced, and, starting with the theta functions introduced by the Borweins in 1991, the cubic cases of Ramanujan's theory is developed fully.

11:00 am in 345 Altgeld Hall,Tuesday, October 29, 2013

Using `big' spectral data to recover polytopes

Emily Dryden   [email] (Bucknell University)

Abstract: Can one hear the shape of a drum? Popularized by Mark Kac in the 1960s, this question asks whether, given a list of eigenvalues of the Laplace operator associated to a domain in the plane, the domain is uniquely determined. There have been many modifications to Kac's question over the years, and Miguel Abreu brought it into the realm of symplectic geometry around 2000 with his question, "Can one hear the shape of a Delzant polytope?" Delzant polytopes are of interest due to their role in classifying certain symplectic manifolds. We will add some "big" spectral data to the list of eigenvalues and use a new analytic tool to answer a variation of Abreu's question for certain symplectic orbifolds. This is joint work with Victor Guillemin and Rosa Sena-Dias.

11:00 am in Altgeld Hall,Tuesday, October 29, 2013

A variational problem in CR Geometry

John P. D'Angelo (UIUC)

11:00 am in 243 Altgeld Hall,Tuesday, October 29, 2013

Matrix factorizations from the viewpoint of algebraic topology

Anatoly Preygel (Berkeley)

Abstract: The category of matrix factorizations is an invariant in algebraic geometry.  First, we'll give an idiosyncratic definition of matrix factorizations (via E_2 Koszul duality), show how this gives natural candidates for their Hochshild invariants (via the E_2 adjoint action), and remark on pleasant convergence results in characteristic zero. Then, time permitting, we'll discuss an application to constructing "2-periodic" stable categories (over the sphere, not dg-categories).  Using the homotopical definition of matrix factorizations, it is possible to imitate recent work of Dyckerhoff-Kapranov over the sphere spectrum: This allows one to show that the S-construction of a "2-periodic" category carries the structure of cyclic space and to construct the Fukaya category of a Riemann surface as "2-periodic" category.  (These last results come from conversations with J. Lurie.)

1:00 pm in 347 Altgeld Hall,Tuesday, October 29, 2013

Dispersive Estimates for Schrodinger and Wave Equations

Will Green (Rose-Hulman Institute of Technology)

Abstract: In this talk we will investigate recent work on Schrodinger and wave equations. In particular, we will discuss the time decay of the solution operators to these equations. We will survey recent results that show the effect of obstructions at zero energy, eigenfunctions and/or resonances, have on the time decay rates. We will also discuss some applications of these estimates to certain non-linear equations.

2:00 pm in 345 Altgeld Hall,Tuesday, October 29, 2013

Q-polynomial distance-regular graphs and a double affine Hecke algebra of rank one

Jae-Ho Lee (U. Wisconsin, Madison)

Abstract: The concept of Q-polynomial distance-regular graph (DRG) was introduced by Delsarte in 1973. Since then, this concept has been connected to other areas, such as coding theory, Lie theory, and quantum algebras. The double affine Hecke algebra (DAHA) was introduced by Cherednik in 1992. The theory of DAHA has been connected to many other areas of mathematics, such as algebraic geometry, Lie theory, and orthogonal polynomials. I have discovered a relationship between Q-polynomial DRGs and a certain DAHA of rank one. In this talk I will display a certain module for a DAHA of rank one. Using this module and the Terwilliger algebra we show how Q-polynomial DRGs and the DAHA are related.

2:00 pm in Altgeld Hall 347,Tuesday, October 29, 2013

Fluctuations in the Wigner Ensemble

Anna Maltsev (U Bristol Math)

Abstract: I will discuss the fluctuations of the spectral density for the Wigner ensemble on the optimal scale. We study the fluctuations of the Stieltjes transform, and improve the known bounds on the optimal scale. As an application, we derive the semicircle law at the edge of the spectrum. This is joint work with Claudio Cacciapuoti and Benjamin Schlein.

3:00 pm in 243 Altgeld Hall,Tuesday, October 29, 2013

Centers and traces of the categorified affine Hecke algebra (or, some tricks with coherent complexes on the Steinberg variety)

Anatoly Preygel (Berkeley)

Abstract: This is a talk on some tricks and constructions on categories of bounded coherent complexes on nice stacks. The goal of the talk will be to explain how "proper descent with singular-support conditions" gives a framework for getting interesting answers when computing (dg-categorical) invariants of the circle by chopping it into intervals. Our main application will be to the affine Hecke category in geometric representation theory: The Steinberg (derived) stack parametrizes G-local systems on an annulus with B-reductions on the boundary. Its dg-category of bounded coherent complexes is monoidal, and categorifies the affine Hecke algebra in representation theory. We'll see how to identify the trace of this monoidal category with a full subcategory of bounded coherent complexes on Loc_G(torus), cut out by a nilpotent micro support condition. This is joint work with Ben-Zvi and Nadler.

3:00 pm in 241 Altgeld Hall,Tuesday, October 29, 2013

Independent sets in graphs with given minimum degree

Jonathan Cutler   [email] (Montclair State University)

Abstract: There has been quite a bit of recent research involving extremal problems for the enumeration of certain graph substructures. For example, Kahn used an entropy method to show that the number of independent sets in a $d$-regular bipartite graph on $n$ vertices is at most $(2^{d+1}-1)^{n/(2d)}$. Zhao was able to show that this upper bound holds for general regular graphs. Galvin conjectured that if $n\geq 2d$, then the number of independent sets in a graph with $n$ vertices and minimum degree at least $d$ is at most that in $K_{d,n-d}$. Galvin proved that the conjecture is true for a fixed $d$ provided $n$ is large. We were able to prove a strengthened version of Galvin's conjecture, covering the case when $n<2d$ as well. In this talk, we will present the main ideas in the proof of this result. This is joint work with Jamie Radcliffe.

4:00 pm in 243 Altgeld Hall,Tuesday, October 29, 2013

Moduli of Elliptic Curves

Peter Nelson (UIUC Math)

Abstract: I'll talk about various sorts of moduli things of elliptic curves, and how you compute things about them. There might even be some computations!

4:00 pm in 314 Altgeld Hall,Tuesday, October 29, 2013

Trjitzinsky Memorial Lectures

Rodrigo Banuelos (Purdue University)

Abstract: NOTE UNUSUAL TIME AND PLACE: Students should plan to attend the Trjitzinsky Memorial Lecture at 4 p.m. on Tuesday October 29 (given by Rodrigo Banuelos of Purdue University). The lecture series continues on Wednesday and Thursday, in 245 Altgeld Hall. Students in Math 499 are welcome to attend those lectures too, although it is not required.

4:00 pm in 314 Altgeld Hall,Tuesday, October 29, 2013

Lecture 1. The 1966 martingale inequalities, the good- principle, some applications

Rodrigo Banuelos (Purdue University)

Abstract: In 1966, D. L. Burkholder published his seminal paper on the boundedness of martingale transform. While the main results were of considerable importance, the ideas introduced in this paper became the building blocks for many subsequent results in martingales, leading to the celebrated Burkholder-Davis-Gundy inequalities (the bread and butter of modern stochastic analysis) and to numerous applications in analysis, including the solution by Burkholder, Gundy and Silverstein of a longstanding open problem concerning a real variables characterization of the Hardy spaces. In 1984, motivated in part by applications to the geometry of Banach spaces, Burkholder gave a new proof of his 1966 inequality which bypassed L2 theory, introducing a novel and revolutionary new method for proving optimal inequalities in probability and harmonic analysis. This method, commonly referred to now days as the "Burkholder method" (and also as "the Bellman function method"), has had many applications in recent years. The aim of these lectures is to present some of these martingale inequalities and applications, emphasizing ideas rather than technicalities, taking a historical point of view but discussing applications to problems of current interest. 
Lecture 1: The 1966 martingale inequalities, the good-principle, some applications. 
Lecture 2: Sharp inequalities. The "why" and the "how".
Lecture 3: Optimal bounds for singular integrals, Fourier multipliers, connections to rank-one convexity, quasi convexity and weighted norm inequalities.