Mathematics

A tame Cantor set

Philipp Hieronymi (UIUC)

Abstract: Let $\overline{\mathbf{R}}:=(\mathbf{R},<,+,\cdot)$ denote the real ordered field. Our focus here is on expansions of $\overline{\mathbf{R}}$ by Cantor sets. For our purposes, a Cantor set is a non-empty, compact subset of $\mathbf{R}$ that has neither interior nor isolated points. We consider the following question due to Friedman, Kurdyka, Miller and Speissegger: is there a Cantor set $K$ and $N\in \mathbf{N}$ such that every set definable from $(\overline{\mathbf{R}},K)$ is $\boldsymbol{\Sigma}_N^1$? I will answer this question positively. In addition to using techniques from model theory, o-minimality and descriptive set theory and previous work of Friedman et al., the work presented in this talk depends crucially on well known results about the monadic second order theory of one successor due to Buechi, Landweber and McNaughton.

Stochastic flows for Levy processes with Holder drifts

Renming Song   [email] (UIUC Math)

Abstract: In this talk I will present some new results on the following SDE in $R^d$: $$ dX_t=b(t, X_t)dt+dZ_t, \quad X_0=x, $$ where $Z$ is a Levy process. We show that for a large class of Levy processes $Z$ and Holder continuous drift $b$, the SDE above has a unique strong solution for every starting point $x\in R^d$. Moreover, these strong solutions form a $C^1$-stochastic flow. In particular, we show that, when $Z$ is a symmetric $\alpha$-stable process with $\alpha\in (0, 1]$ and $b$ is $\beta$-Holder continuous with $\beta \in (1-\alpha/2, 1)$, the SDE above has a unique strong solution.

The Drinfeld-Lafforgue-Vinberg degeneration of the stack of G-bundles

Simon Schieder (Harvard)

Abstract: We study the singularities of the Drinfeld-Lafforgue-Vinberg compactification of the moduli stack of G-bundles on a smooth projective curve for a reductive group G. The definition of this compactification is due to Drinfeld and relies on the Vinberg semigroup of G. We will mostly focus on the case $G=SL_2$; in this case the compactification can alternatively be viewed as a canonical one-parameter degeneration of the moduli stack of $SL_2$-bundles. We then study the singularities of this one-parameter degeneration via the associated nearby cycles construction. Time permitting, we might sketch a generalization to the case of an arbitrary reductive group G and the relation to Langlands duality.

On degree sequences forcing the square of a Hamilton cycle

Andrew Treglown   [email] (School of Mathematics, University of Birmingham)

Abstract: Many famous results in extremal graph theory give minimum degree conditions that force some substructure. For example, Dirac's classical theorem characterises the minimum degree that ensures a Hamilton cycle in a graph. However, sometimes it is possible to obtain stronger results via degree sequence conditions. For example, Pósa gave a significant strengthening of Dirac's theorem: if $d_1 \le d_2 \le \dots \le d_n$ is the degree sequence of $G$ and $d_i \geq i+1$ for all $i < n/2$, then $G$ contains a Hamilton cycle. A famous conjecture of Pósa gives a minimum degree condition that ensures a graph contains the square of a Hamilton cycle. This was proved for large graphs by Komlós, Sárközy and Szemerédi. In this talk we consider a degree sequence analogue of this theorem. This is joint work with Katherine Staden.

The Arithmetic of Ergodic Systems

Diaaeldin Taha (UIUC Math)

Abstract: In this talk we discuss the ergodic theoretical equivalent of the integer multiplication operator and the property of one number being a factor of another. A natural question then arises: What is the "correct" ergodic theoretical equivalent for the co-prime property? H. Furstenberg answered that question in 1967 by introducing the concept of disjointness in his seminal paper "Disjointness in Ergodic Theory, Minimal Sets, and a Problem in Diophantine Approximation". We will present the foundation of the theory of joining of ergodic systems, and discuss how disjointness is used as a classification tool. The main idea is to take a collection of ergodic systems and forcing them to live in the same space and see how they interact. This is closely related to the probability theory technique of coupling. Familiarity with basic measure theory and the definition of a probability space will be assumed. Everything else will be defined along the way.

Department Awards Banquet