C2

The Ising model is a well-known statistical physics model, defined on a two-dimensional lattice. It is interesting because it exhibits a "phase transition" at a certain critical temperature. Recent mathematical research has revealed an intriguing geometry in the model, involving discrete holomorphic functions, spinors, spin structures, and the Dirac equation. I will try to outline some of these ideas.

I will introduce the general idea of p-adic Hodge theory from the view point of a beginner. Also, I will give a sketch of the proof of the crystalline comparison theorem in the case of good reduction using 'almost mathematics'.

We will give a sketch overview of Scholze's theory of perfectoid spaces and the tilting equivalence, starting from Huber's geometric approach to valuation theory. Applications to weight-monodromy and p-adic Hodge theory we will only hint at, preferring instead to focus on examples which illustrate the philosophy of tilting equivalence.
 

In this talk I will describe two instances of Langlands functoriality concerning the group $\mathrm{Sp}_{2n}$. I will then very briefly explain how this enables one to attach Galois representations to automorphic representations of (inner forms of) $\mathrm{Sp}_{2n}$. 

Let G be a transitive permutation group. If G is finite, then a classical theorem of Jordan implies the existence of fixed-point-free elements, which we call derangements. This result has some interesting and unexpected applications, and it leads to several natural problems on the abundance and order of derangements that have been the focus of recent research. In this talk, I will discuss some of these related problems, and I will report on recent joint work with Hung Tong-Viet on primitive permutation groups with extremal derangement properties.

Quiver varieties and their quantizations feature prominently in
geometric representation theory. Multiplicative quiver varieties are
group-like versions of ordinary quiver varieties whose quantizations
involve quantum groups and $q$-difference operators. In this talk, we will
define and give examples of representations of quivers, ordinary quiver
varieties, and multiplicative quiver varieties. No previous knowledge of
quivers will be assumed. If time permits, we will describe some phenomena
that occur when quantizing multiplicative quiver varieties at a root of
unity, and work-in-progress with Nicholas Cooney.

Abstract: Joint work with Syahida Che Dzul-Kifli

Let $f:X\to X$ be a continuous function on a compact metric space forming a discrete dynamical system. There are many definitions that try to capture what it means for the function $f$ to be chaotic. Devaney’s definition, perhaps the most frequently cited, asks for the function $f$ to be topologically transitive, have a dense set of periodic points and is sensitive to initial conditions.  Bank’s et al show that sensitive dependence follows from the other two conditions and Velleman and Berglund show that a transitive interval map has a dense set of periodic points.  Li and Yorke (who coined the term chaos) show that for interval maps, period three implies chaos, i.e. that the existence of a period three point (indeed of any point with period having an odd factor) is chaotic in the sense that it has an uncountable scrambled set.

The existence of a period three point is In this talk we examine the relationship between transitivity and dense periodic points and look for simple conditions that imply chaos in interval maps. Our results are entirely elementary, calling on little more than the intermediate value theorem.