Title: Elliptic curves and modularity

Abstract: The goal of this talk is to give you a glimpse of the Langlands program, a central topic at the intersection of algebraic number theory, algebraic geometry and representation theory. I will focus on a celebrated instance of the Langlands correspondence, namely the modularity of elliptic curves. In the first part of the talk, I will give an explicit example, discuss the different meanings of modularity for rational elliptic curves, and mention applications. In the second part of the talk, I will discuss what is known about the modularity of elliptic curves over more general number fields.

In recent years, methods and concepts of algebraic geometry, particularly those of real and computational algebraic geometry, have been used in many applied domains. In this talk, aimed at a broad audience, I will review applications to molecular biology. The goal is to analyze standard models in systems biology to predict dynamic behavior in regions of parameter space without the need for simulations. I will also mention some challenges in the field of real algebraic geometry that arise from these applications.

Interacting particle systems provide flexible and powerful models that are useful in many application areas such as sociology (agents), molecular dynamics (proteins) etc. However, particle systems with large numbers of particles are very complex and difficult to handle, both analytically and computationally. Therefore, a common strategy is to derive effective equations that describe the time evolution of the empirical particle density, the so-called Dean-Kawasaki equation.

Our aim is to derive and study continuum models for the mesoscopic behavior of particle systems. In particular, we are interested in finite size effects. We will introduce nonlinear and non-Gaussian models that approximate the Dean-Kawasaki equation, in the special case of non-interacting particles. We want to study the well-posedness of these nonlinear SPDE models and to control the weak error of the SPDE approximation.  This is the joint work with H. Kremp (TU Wien) and N. Perkowski (FU Berlin).

This is a report of joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams. Based on random matrix theory, Conrey-Farmer-Keating-Rubinstein-Snaith have conjectured precise asymptotics for moments of families of quadratic L-functions over number fields. There is an extremely similar function field analogue, worked out by Andrade-Keating. I will explain that one can relate this problem to understanding the homology of the braid group with symplectic coefficients. With Bergström-Diaconu-Westerland we compute the stable homology groups of the braid groups with these coefficients, together with their structure as Galois representations. We moreover show that the answer matches the number-theoretic predictions. With Miller-Patzt-Randal-Williams we prove an improved range for homological stability with these coefficients. Together, these results imply the conjectured asymptotics for all moments in the function field case, for all sufficiently large (but fixed) q.

About 20 years ago, Dror Bar-Natan described an elegant generalisation
of Khovanov homology to tangles with any number of endpoints, by
considering certain quotients of two-dimensional relative cobordism
categories.  I claim that these categories are in general not
well-understood (not by me in any case).  However, if we restrict to
tangles with four endpoints, things simplify and Bar-Natan's category
turns out to be closely related to the wrapped Fukaya category of the
four-punctured sphere.  This relationship gives rise to a symplectic
interpretation of Khovanov homology that is useful both for doing
calculations and for proving theorems.  I will discuss joint work in
progress with Artem Kotelskiy and Liam Watson where we investigate what
happens when we fill in one of the punctures.
 

Given two mathematical objects, the most basic question is whether they are the same. We will discuss this question for triangulations of three-manifolds. In practice there is fast software to answer this question and theoretically the problem is known to be decidable. However, our understanding is limited and known theoretical algorithms could have extremely long run-times. I will describe a programme to show that the 3-manifold homeomorphism problem is in the complexity class NP, and discuss the important sub-case of Seifert fibered spaces. 

Many researchers are developing stable and accurate numerical methods for flow problems, which roughly belong to upwind methods or characteristics(-based) methods. 
The Lagrange-Galerkin method proposed and analyzed in, e.g., [O. Pironneau. NM, 1982] and [E. S\"uli. NM, 1988] is the finite element method combined with the idea of the method of characteristics; hence, it belongs to the characteristics(-based) methods. The advantages are the CFL-free robustness for convection-dominated problems and the symmetry of the resulting coefficient matrix. In this talk, we introduce stabilized Lagrange-Galerkin schemes of second order in time for viscous and viscoelastic flow problems, which employ the cheapest conforming P1-element with the help of pressure-stabilization [F. Brezzi and J. Pitk\"aranta. Vieweg+Teubner, 1984] for all the unknown functions, i.e., velocity, pressure, and conformation tensor, reducing the number of DOFs. 
Focusing on the recent developments of discretizations of the (non-conservative and conservative) material derivatives and the upper-convected time derivative, we present theoretical and numerical results.