Mathematics

Stable subgroups and Morse subgroups of mapping class groups

Heejoung Kim (Illinois Math)

Abstract: The notion of a ``quasiconvex'' subgroup plays of a word-hyperbolic group $G$ plays an important role in the theory of hyperbolic groups. This notion has several equivalent characterizations in that context, in terms of being ``undistorted", in terms of the action on the boundary, in terms of being ``rational" with respect to automatic structures on $G$, in terms of the contracting properties of the projection maps, etc. For an arbitrary finitely generated group $G$, there are two recent generalizations of the notion of a quasiconvex subgroup: a ``stable'' subgroup and a ``Morse'' subgroup. In this talk, we will discuss these two notions and their different properties. We prove that the properties of being Morse and being stable coincide for a subgroup of infinite index in the mapping class group of an oriented, connected, finite type surface with negative Euler characteristic.

Some aspects of simple proofs informed by history, ancient and modern

Victor Pambuccian (Arizona State University Math)

Abstract: We'll look at (i) the existence of a direct proof (even within intuitionistic logic) of the Steiner-Lehmus theorem, stating that a triangle with two congruent angle bisectors must be isosceles, (ii) the most primitive arithmetic capable of proving the irrationality of $\sqrt{2}$, going back to the early Pythagoreans, and why it cannot prove the irrationality of $\sqrt{17}$, the case Theodorus of Cyrene apparently stumbled upon, and (iii) the simplest proof for the fact that 30 is the greatest number all of whose totitives are prime.

Local density estimate for a hypoelliptic SDE

Cheng Ouyang (UIC Math)

Abstract: In a series of three papers in the 80’s, Kusuoka and Stroock developed a probabilistic program in order to obtain sharp bounds for the density function of a hypoelliptic SDE driven by a Brownian motion. We aim to investigate how their method can be used to study rough SDEs driven by fractional Brownian motions. In this talk, I will outline Kusuoka and Stroock’s approach and point out where the difficulties are in our current setting. The talk is based on an ongoing project with Xi Geng and Samy Tindel.

(normally ordered) Tensor product of Tate objects and decomposition of higher adeles

Aron Heleodoro (Northwestern University)

Abstract: In this talk I will introduce the different tensor products that exist on Tate objects over vector spaces (or more generally coherent sheaves on a given scheme). As an application, I will explain how these can be used to describe higher adeles on an n-dimensional smooth scheme. Both Tate objects and higher adeles would be introduced in the talk. (This is based on joint work with Braunling, Groechenig and Wolfson.)

Some important combinatorial sequences

Zoltan Furedi (Renyi Institute of Mathematics, Budapest, Hungary and UIUC)

Abstract: The sequence $a(1), a(2), a(3), \dots$ of reals is called subadditive if $a(n+m) \leq a(n)+ a(m)$ for all integers $n,m$. Fekete's lemma states that the sequence $\{a(n)/n\}$ has a limit (it could be negative infinity). Let $f(n)$ be a non-negative, non-decreasing sequence. De Bruijn and Erdos (1952) called the sequence $(a(n))$ nearly $f$-subadditive if $a(n+m) \leq a(n)+ a(m) + f(n+m)$ holds for all $n\leq m \leq 2n$. They showed that if the error term $f$ is small, $\sum_{ n=1}^{\infty} f(n)/n^2$ is finite, then the limit $a(n)/n$ still exists. Their results is listed in the Bollobas-Riordan book (2006) as one of the most useful tools in Percolation Theory. Among other things we show that the de Bruijn-Erdos condition for the error term in their improvement of Fekete's Lemma is not only sufficient but also necessary in the following strong sense. If $\sum_{ n=1}^{\infty} f(n)/n^2 =\infty$, then there exists an nearly $f$-subadditive sequence $(b(n))$, such that the sequence of slopes $(b(n)/n)$ takes every rational number. This is a joint work with I. Ruzsa.

Introduction to GIT, II

Itziar Ochoa de Alaiza Gracia

Abstract: The aim of this talk is to give the motivation for the GIT quotient. We will do so by introducing different notions of quotients, illustrated by examples. Finally we will define the Affine and projective GIT quotients.