Events

We construct new families of groups with property (T) and infinitely many alternating group quotients. One of those consists of subgroups of Aut(Fp[x1,…,xn]) generated by a suitable set of tame automorphisms. Finite quotients are constructed using the natural action of Aut(F_p[x_1,…,x_n]) on the n-dimensional affine spaces over finite extensions of F_p. As a consequence, we obtain explicit presentations of Gromov hyperbolic groups with property (T) and infinitely many alternating group quotients. Our construction also yields an explicit infinite family of expander Cayley graphs of degree 4 for alternating groups of degree p^7−1 for any odd prime p.

Given a non-smooth vector field in a Sobolev space and a smooth hypersurface in the Euclidean space, we can show that there exists a one-parameter family of hypersurfaces whose velocity is equal to the mean curvature plus the given vector field in a strong PDE sense for a short time, and more generalized sense time-globally. In some sense, there is a competition between the surface tension (regularizing effect) and non-smooth flow field (roughening effect), and apparently, there cannot be uniqueness in the usual sense for the solution despite the availability of a good regularity theorem. I plan to describe what I know about the problem, rough idea of the proof as well as some possible way to look into the uniqueness question suggested from the construction.

Hypoelliptic differential operators play a central role in various fields, from stochastic analysis to contact and sub-riemannian geometry. A computable criterion of hypoellipticity was proposed by Helffer and Nourrigat in 1979. In this lecture we will give an overview of hypoellipticity and present the proof of the Helffer-Nourrigat conjecture. This is joint work with Omar Mohsen and Robert Yuncken.

It is well known that the empirical failure of Kolmogorov theoretical prediction (K41) on the local structures of incompressible turbulent flows fails because of the intermittent nature of the velocity field. Several physical intermittency models have been thus proposed to reconcile K41 to experiments and most of them builds on the observable phenomenon that the nontrivial energy dissipation (zero-th law of turbulence) is not space filling, i.e. it is lower dimensional. In this talk I will discuss a recent work, obtained together with Philip Isett, where we propose a machinery to quantitatively translate dimensionality assumptions on the dissipation into deviations from K41 prediction. The approach is rather geometrical and if time permits I will describe how it connects to much more abstract/general settings.

Based on the Chernoff approximation, we provide an approximation result for convex monotone semigroups and, in particular, for dynamic convex risk measures. Starting with a generating family $(I(t))_{t\geq 0}$ of operators on the space of bounded continuous functions, the semigroup is constructed as $S(t)f:=\lim_{n\to\infty}I(\frac{t}{n})^n f$ and is uniquely determined by the time derivative $I’(0)f$ for smooth functions $f$. Moreover, we identify explicit conditions for the generating family that are transferred to the semigroup and can easily be verified in applications. It turns out that there is a structural connection between Chernoff approximations for semigroups and LLN and CLT type results for convex risk measures and convex expectations. Other applications are scaling limits of discrete-time models to continuous models under model uncertainty.

A CM point in the moduli space of complex elliptic curves is a point corresponding to an elliptic curve with complex multiplication. A classical result of William Duke (1988), complemented by Laurent Clozel and Emmanuel Ullmo (2004), states that CM points become uniformly distributed in the moduli space, with respect to the hyperbolic measure, when the discriminant of the underlying ring of endomorphisms grows. Since CM points are algebraic, it is possible to study p-adic analogues of this phenomenon. In this talk I will present a description of the p-adic asymptotic distribution of CM points in the moduli space of p-adic elliptic curves. In contrast to the complex case, there are infinitely many measures describing the p-adic asymptotic distribution of CM points. This is joint work with Ricardo Menares (Pontificia Universidad Católica de Chile) and Juan Rivera-Letelier (University of Rochester).

We consider the notions of (i) critical points, (ii) second-order points, (iii) local minima, and (iv) strict local minima for multivariate polynomials. For each type of point, and as a function of the degree of the polynomial, we study the complexity of deciding (1) if a given point is of that type, and (2) if a polynomial has a point of that type. Our results characterize the complexity of these two questions for all degrees left open by prior literature. Our main contributions reveal that many of these questions turn out to be tractable for cubic polynomials. By contrast, we show that unless P=NP, there cannot be a polynomial-time algorithm that finds a point within Euclidean distance $c^n$ (for any constant $c\geq 0$) of a local minimizer of an $n$-variate quadratic polynomial over a polytope. This result (with $c=0$) answers a question of Pardalos and Vavasis that appeared on a list of seven open problems in complexity theory for numerical optimization in 1992. Based on joint work with Jeffrey Zhang (CMU).