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for events the day of Tuesday, October 10, 2017.

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Tuesday, October 10, 2017

12:00 pm in 243 Altgeld Hall,Tuesday, October 10, 2017

Dynamics, geometry, and the moduli space of Riemann surfaces

Alex Wright (Stanford)

Abstract: The moduli space of Riemann surfaces of fixed genus is one of the hubs of modern mathematics and physics. We will tell the story of how simple sounding problems about polygons, some of which arose as toy models in physics, became intertwined with problems about the geometry of moduli space, and how the study of these problems in Teichmuller dynamics lead to connections with homogeneous spaces, algebraic geometry, dynamics, and other areas. The talk will mention joint works with Alex Eskin, Simion Filip, Curtis McMullen, Maryam Mirzakhani, and Ronen Mukamel.

1:00 pm in 347 Altgeld Hall,Tuesday, October 10, 2017

Recent progress on the pointwsie convergence problem of Schrodinger equations

Xiaochun Li (Illinois Math)

Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We prove that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method.

2:00 pm in 347 Altgeld Hall,Tuesday, October 10, 2017

Limit theorems with weights and applications

Yanghui Liu (Purdue University)

Abstract: The term “limit theorem” is associated with a multitude of statements having to do with the convergence of probability distributions of sums of increasing number of random variables. Given that a limit theorem result holds, “weighted limit theorem” considers the asymptotic behavior of the corresponding weighted random sums. The weighted limit theorem problem has drawn a lot of attention in recent articles due to its key role in topics such as parameter estimations, Ito’s formula in law, time-discrete numerical schemes, and normal approximations, and various “unexpected” weighted limit theorems have been discovered since then. The purpose of this talk is to introduce a general framework and a transferring principle for this problem, and to provide improvement of the existing results in a few aspects.

3:00 pm in 243 Altgeld Hall,Tuesday, October 10, 2017

Koszul duality and characters of tilting modules

Pramod Achar (Louisiana State University)

Abstract: This talk is about the "Hecke category," a monoidal category that appears in various incarnations in geometric representation theory. I will explain some of these incarnations and their roles in solving classical problems, such as the celebrated Kazhdan-Lusztig conjectures on Lie algebra representations. These conjectures (proved in 1981) hinge on the fact that the derived category of constructible sheaves on a flag variety is equipped with an obvious monoidal action of the Hecke category on the right. It turns out that there is also a second, "hidden" action of the Hecke category on the left. The symmetry between the "hidden" left action and the "obvious" right action leads to the phenomenon known as Koszul duality. In the last part of the talk, I will discuss new results on Koszul duality with coefficients in a field of positive characteristic, with applications to characters of tilting modules for algebraic groups. This is joint work with S. Makisumi, S. Riche, and G. Williamson.

3:00 pm in 241 Altgeld Hall,Tuesday, October 10, 2017

Packing chromatic number of cubic graphs

Xujun Liu (Illinois Math)

Abstract: A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\dots, V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $\chi_p(G)$, of a graph $G$ is the minimum $k$ such that $G$ has a packing $k$-coloring. Sloper showed that there are $4$-regular graphs with arbitrarily large packing chromatic number. The question whether the packing chromatic number of subcubic graphs is bounded appears in several papers. We answer this question in the negative. Moreover, we show that for every fixed $k$ and $g\geq 2k+2$, almost every $n$-vertex cubic graph of girth at least $g$ has the packing chromatic number greater than $k$. Joint work with József Balogh and Alexandr Kostochka.

4:00 pm in 314 Altgeld Hall,Tuesday, October 10, 2017

Lecture I. Set theory and trigonometric series

Alexander Kechris (Caltech)

Abstract: The Trjitzinsky Memorial Lectures will be held October 10-12, 2017. A reception will follow this first lecture from 5-6 pm in 239 Altgeld Hall. Alexander Kechris will present: "A descriptive set theoretic approach to problems in harmonic analysis, ergodic theory and combinatorics." Descriptive set theory is the study of definable sets and functions in Polish (complete, separable metric) spaces, like, e.g., the Euclidean spaces. It has been a central area of research in set theory for over 100 years. Over the past three decades, there has been extensive work on the interactions and applications of descriptive set theory to other areas of mathematics, including analysis, dynamical systems, and combinatorics. My goal in these lectures is to give a taste of this area of research, including an extensive historical background. These lectures require minimal background and should be understood by anyone familiar with the basics of measure theory and functional analysis. Also the three lectures are essentially independent of each other.