Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, October 21, 2014.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, October 21, 2014

11:00 am in 241 Altgeld Hall,Tuesday, October 21, 2014

#### Zagier polynomials and modified Nörlund polynomials

###### Atul Dixit   [email] (Tulane University)

Abstract: In 1998, Don Zagier studied the numbers $B_{n}^{*}$ which he called 'modified Bernoulli numbers'. They satisfy amusing variants of the properties of the ordinary Bernoulli numbers. Recently, Victor H. Moll, Christophe Vignat and I studied an obvious generalization of the modified Bernoulli numbers, which we call 'Zagier polynomials'. These polynomials are also rich in structure, and a theory parallel to that of ordinary Bernoulli polynomials exists. One thing that was missing was a generalization of Zagier's beautiful exact formula for $B_{2n}^{*}$ for the Zagier polynomials. In an ongoing joint work with M. L. Glasser and K. Mahlburg, we have been able to obtain this generalization which involves Chebyshev polynomials and infinite series of Bessel function $Y_{n}$. I will mainly focus on these results. In the second part of my talk, I will discuss another generalization of the modified Bernoulli numbers that we studied along with A. Kabza, namely 'modified Nörlund polynomials' $B_{n}^{(\alpha)*}$ , $\alpha\in\mathbb{N}$, and obtain their generating function along with applications.

11:00 am in 243 Altgeld Hall,Tuesday, October 21, 2014

#### QX as a Hopf algebra

###### Charles Rezk (UIUC)

Abstract: The suspension spectrum of the infinite loop space QX admits a product and a coproduct. I'll discuss a theorem of Kuhn, which describes this structure with respect to the Snaith splitting. Then I'll talk about how things simplify after a suitable localization (e.g., with respect to Morava K-theory.)

1:00 pm in 243 Altgeld Hall,Tuesday, October 21, 2014

#### Recurrent Weil-Petersson geodesics with non-uniquely ergodic ending laminations.

###### Babak Modami (Illinois)

Abstract: A well-known result of H. Masur guarantees that any Teichmuller geodesic which is recurrent to a compact part of the moduli space of Riemann surfaces has a uniquely ergodic vertical foliation. In contrast, we construct recurrent Weil-Petersson geodesics with non-uniquely ergodic ending lamination. This is joint work with Jeffrey Brock. View talk at http://youtu.be/GHczwQwJ3Tg

1:00 pm in 347 Altgeld Hall,Tuesday, October 21, 2014

#### Unit Distance Problems

###### Richard Oberlin (Florida State U. Math)

Abstract: We consider a continuous version of the Erdos unit distance problem (joint w/ D. Oberlin).

2:00 pm in Altgeld Hall 347,Tuesday, October 21, 2014

Abstract: This talk will detail the stability of Markov chains. One measure of stability of a time-homogeneous Markov chain is a mixing time. I will define similar measures for special types of time-inhomogeneous Markov chains called the adiabatic and stable adiabatic times. I will discuss the use of these Markov chains and I will discuss how the adiabatic and stable adiabatic times relate to mixing times. This talk is an exploration of linear algebra, analysis and probability.

3:00 pm in 241 Altgeld Hall,Tuesday, October 21, 2014

#### Grid Ramsey problem and related questions

###### Choongbum Lee   [email] (MIT)

Abstract: The Hales--Jewett theorem is one of the pillars of Ramsey theory, from which many other results follow. A celebrated theorem of Shelah says that Hales--Jewett numbers are primitive recursive. A key tool used in his proof, now known as the cube lemma, has become famous in its own right. In its simplest form, this lemma says that if we color the edges of the Cartesian product $K_n \times K_n$ in $r$ colors then, for $n$ sufficiently large, there is a rectangle with both pairs of opposite edges receiving the same color. Shelah's proof shows that $n = r^{\binom{r+1}{2}} + 1$ suffices, and more than twenty years ago, Graham, Rothschild and Spencer asked whether this bound can be improved to a polynomial in $r$. We show that this is not possible by providing a superpolynomial lower bound in $r$. We will also discuss a deep connection between this problem and generalized Ramsey numbers, and present a solution to a problem of Erdős and Gyárfás on the transition of asymptotics of generalized Ramsey numbers. Joint work with David Conlon (Oxford), Jacob Fox (MIT), and Benny Sudakov (ETH Zurich)

3:00 pm in 243 Altgeld Hall,Tuesday, October 21, 2014

#### Interpolation problems and the birational geometry of moduli spaces of sheaves

###### Jack Huizenga (UIC Math)

Abstract: Questions like the Nagata conjecture seek to determine when certain zero-dimensional schemes impose independent conditions on sections of a line bundle on a surface. Understanding analogous questions for vector bundles instead amounts to studying the birational geometry of moduli spaces of sheaves on a surface. We explain how to use higher-rank interpolation problems to compute the cone of effective divisors on any moduli space of sheaves on the plane. This is joint work with Izzet Coskun and Matthew Woolf.

4:00 pm in 2 Illini,Tuesday, October 21, 2014

#### Comonotonic Approximations of Risk Measures for Variable Annuity Guaranteed Benefits

###### Xiaochen Jing (UIUC Math)

Abstract: The computations of various risk metrics are essential to the quantitative risk management of variable annuity guaranteed benefits. The current market practice of Monte Carlo simulation often requires intensive computations, which can be very costly to implement. In an attempt to find low-cost solutions, we explore the techniques of comonotonic bounds to produce approximation of risk measures. This is joint work with Dr. Runhuan Feng.

4:00 pm in 245 Altgeld Hall,Tuesday, October 21, 2014

#### Mathematical Analysis of Neuronal Network Dynamics in the Brain

###### David Cai (Courant Institute and Shanghai Jiao Tong University)

Abstract: From the perspective of nonlinear dynamical systems, nonequilibrium statistical physics, causal inference, and scientific modeling, we will describe some recent developments of mathematical methods used in analysis of the dynamics of neuronal networks arising from the brain. We will outline some theoretical difficulties in characterization of neuronal network dynamics for the understanding of information processing in the brain.