In this talk I present a recent result about the free-boundary problem for 2D current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity near the onset of the instability, Hunter and Thoo have introduced an asymptotic quadratically nonlinear integro-differential equation for the amplitude of small perturbations of the planar discontinuity. We study such amplitude equation and prove its nonlinear well-posedness under a stability condition given in terms of a longitudinal strain of the fluid along the discontinuity. This is a joint work with A.Morando and P.Trebeschi.

Expansion, Random Walks and Sieving in $SL_2 (\mathbb{F}_p[t])$

We pose the question of how to characterize "generic" elements of finitely generated groups. We set the scene by discussing recent results for linear groups in characteristic zero. To conclude we describe some new work in positive characteristic.

 This is joint work with Angus Macintyre. We prove a general model-completeness theorem for Henselian valued fields
stating that a Henselian valued field of characteristic zero with value group a Z-group and with finite ramification is model-complete in the language of rings provided that its residue field is model-complete. We apply this to extensions of p-adic fields showing that any finite or infinite extension of p-adics with finite ramification is model-complete in the language of rings.

The talk will discuss the mean value theorem and Wooley's breakthrough with his "efficent congruencing" method.

 Let $X$ be a smooth cubic hypersurface of dimension $m$ defined over a global field $K$. A conjecture of Colliot-Thelene(02) states that $X$ satisfies the Hasse Principle and Weak approximation as long as $m\geq 3$. We use a global version of Hardy-Littlewood circle method along with the theory of global $L$-functions to establish this for $m\geq 6$, in the case $K=\mathbb{F}_q(t)$, where $\text{char}(\mathbb{F}_{q})> 3$.

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.

I will present a joint work with Leonid Kovalev and Jani Onninen. The proofs are  based on topological arguments (degree theory)  and the properties  of planar  quasiconformal mappings. These new ideas  apply well to inner variational equations of conformally invariant energy integrals; in particular, to the Hopf-Laplace equation for the Dirichlet integral.

A meromorphic connection on a complex curve can be interpreted as a representation of a simple Lie algebroid.  By integrating this Lie algebroid to a Lie groupoid, one obtains a complex surface on which the parallel transport of the connection is globally well-defined and holomorphic, despite the apparent singularities of the corresponding differential equations.  I will describe these groupoids and explain how they can be used to illuminate various aspects of the classical theory of singular ODEs, such as the resummation of divergent series solutions.  (This talk is based on joint work with Marco Gualtieri and Songhao Li.)

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.