Abstract: The dynamics of complex biological systems is often driven by multiscale, rare but important events. In this talk, I will first introduce the numerical methods for computing transition states, in particular, the Optimization-based Shrinking Dimer (OSD) method we recently proposed. Then I will give two applications of rare events and transition states in biology: boundary sharpening in zebrafish hindbrain and neuroblast delamination in Drosophila. The joint work with Qiang Du (Columbia), Qing Nie (UC Irvine), Yan Yan (HKUST).

Many consequences of the Proper Forcing Axiom (PFA) factor through the stationarity of the class of guessing models. Such consequences include the Tree Property at $\omega_2$, absence of (weak) Kurepa Trees on $\omega_1$, and failure of square principles.  On the other hand, stationarity of guessing models does not decide the value of the continuum, even when one requires that the guessing models are also indestructible in some sense.  I will give an introduction to the topic and discuss some recent results due to John Krueger and me.

In this talk, I will discuss a simplified Ericksen-Leslie system modeling
the hydrodynamics of nematic liquid crystals, that is coupling between Navier-Stokes equations and harmonic map heat flows. I will describe some existence results of global weak solutions in dimensions two and three, and a finite time singularity result in dimension three. This is based on some joint works with Tao Huang, Junyu Lin, Fanghua Lin, and Chun Lin.

The Langlands program is an intricate network of conjectures, which are meant to connect different areas of mathematics, such as number theory, harmonic analysis and representation theory. One striking consequence of the Langlands program is the Ramanujan conjecture, which is a statement purely within harmonic analysis, about the growth rate of Fourier coefficients of modular forms. It turns out to be intimately connected to the Weil conjectures, a statement about the cohomology of projective, smooth varieties defined over finite fields. 

I will explain this connection and then move towards a mod p analogue of these ideas. More precisely, I will explain a strategy for understanding torsion occurring in the cohomology of locally symmetric spaces and how to detect which degrees torsion will contribute to. The main theorem is joint work with Peter Scholze and relies on a p-adic version of Hodge theory and on recent developments in p-adic geometry.

This talk will concern two topics.  The first topic is the applications of Riemann--Hilbert (RH) problems.   RH problems provide a powerful and rigorous tool to study many problems in pure and applied mathematics.  Important problems in integrable systems and random matrix theory have been solved with the aid of RH problems. RH problems can also be approached numerically with applications to the numerical solution of PDEs and the sampling of random matrix ensembles.  The resulting methods are seen to have accuracy and complexity advantages over previously existing methods.  The second topic is recent progress on the statistical analysis of numerical algorithms.  In particular, with appropriate randomness, the fluctuations of the iteration count of numerous numerical algorithms have been demonstrated to be universal, see Pfrang, Deift and Menon (2014).  I will discuss simple algorithms where universality is provable and the wide persistence of this phenomenon.