# Seminars & Colloquia

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This series of lecture will introduce the study of groups acting on the circle and the line, the moduli spaces of such actions, and the role of these spaces in questions of geometric topology, dynamics, and foliation theory. I will focus on new rigidity results in low regularity, surveying important techniques, geometrically motivated examples, and open problems.

We propose a novel class of Hawkes-based model that assesses two types of systemic risk in high-frequency price processes: the endogenous systemic risk within a single process and the interactive systemic risk between a couple of processes. We examine the existence of systemic risk at a microscopic level via an empirical analysis of the futures markets of the West Texas Intermediate (WTI) crude oil and gasoline and perform a comparative analysis with the conditional value-at-risk as a benchmark measure of the proposed model. Throughout the analysis, we uncover remarkable empirical findings in terms of the high-frequency structure of the two markets: for the past decade, the level of endogenous systemic risk in the WTI market was significantly higher than that in the gasoline market. Moreover, the level at which the gasoline price affects the WTI price was constantly higher than in the opposite case. Although the two prices interact with each other at the transaction-unit level, the degree of relative influences on the two markets, that is, from the WTI to the gasoline and vice versa, was very asymmetric, but that difference has reduced gradually over time.

This series of lecture will introduce the study of groups acting on the circle and the line, the moduli spaces of such actions, and the role of these spaces in questions of geometric topology, dynamics, and foliation theory. I will focus on new rigidity results in low regularity, surveying important techniques, geometrically motivated examples, and open problems.

Let H be a planar graph. By a classical result of Robertson and Seymour, there is a function f(k) such that for all k and all graphs G, either G contains k vertex-disjoint subgraphs each containing H as a minor, or there is a subset X of at most f(k) vertices such that G−X has no H-minor. We prove that this remains true with f(k)=ck log k for some constant c depending on H. This bound is best possible, up to the value of c, and improves upon a recent bound of Chekuri and Chuzhoy. The proof is constructive and yields the first polynomial-time O(log 𝖮𝖯𝖳)-approximation algorithm for packing subgraphs containing an H-minor.

This is joint work with Wouter Cames van Batenburg, Gwenaël Joret, and Jean-Florent Raymond.

This series of lecture will introduce the study of groups acting on the circle and the line, the moduli spaces of such actions, and the role of these spaces in questions of geometric topology, dynamics, and foliation theory. I will focus on new rigidity results in low regularity, surveying important techniques, geometrically motivated examples, and open problems.

Let X be a compact Kahler manifold of dimension n > 0. Let G be a group of zero entropy automorphisms of X.

Let G_0 be the set of elements in G which are isotopic to the identity. We prove that after replacing G by a suitable finite-index subgroup,

G/G_0 is a unipotent group of derived length at most n-1. This is a corollary of an optimal upper bound of length involving the Kodaira dimension.

We also study the algebro-geometric structure of X when it admits a group action with maximal derived length n-1.

This is a joint work with Dinh and Oguiso.

Given a group G and a manifold M, can one describe all the actions of G on M? This is a basic, natural question in geometric topology, but also a very difficult one -- even in the case where M is 1-dimensional, and G is a familiar, finitely generated group.

This talk will introduce the theory of groups acting on 1-manifolds, through the study of orderable groups. I will describe some connections between this theory and themes in topology and dynamics (like rigidity and foliation theory ), some current open problems, and indicate new approaches coming from recent joint work with C. Rivas.