L3

Highly oscillatory integrals arise in a range of wave-based problems. For example, they may occur when a basis for a boundary element has been enriched with oscillatory functions, or as part of a localised approximation to various short-wavelength phenomena. A range of contemporary methods exist for the efficient evaluation of such integrals. These methods have been shown to be very effective for model integrals, but may require expertise and manual intervention for
integrals with higher complexity, and can be unstable in practice.

The PathFinder toolbox aims to develop robust and fully automated numerical software for a large class of oscillatory integrals. In this talk I will introduce the method of numerical steepest descent (the technique upon which PathFinder is based) with a few simple examples, which are also intended to highlight potential causes for numerical instability or manual intervention. I will then explain the novel approaches that PathFinder uses to avoid these. Finally I will present some numerical examples, demonstrating how to use the toolbox, convergence results, and an application to the parabolic wave equation.

Supergravity backgrounds are an essential ingredient in string theory or field theories via AdS/CFT. The simplest example of a 4d, N=1 background is the product of four-dimensional Minkowski space with a seven-dimensional manifold with G_2 holonomy in M-theory. For more complicated backgrounds where we allow non-zero fluxes, the supersymmetry conditions can be rephrased in terms of G-structure data. The geometry of these backgrounds is often complicated and their general features are not well understood.

In this talk, I will define the analogue of G_2 geometry for generic 4d, N=1 backgrounds with flux in both type II and eleven-dimensional supergravity. The geometry is characterised by a G-structure in 'exceptional generalised geometry' that includes G_2 structures and Hitchin's generalised geometry as subcases. Supersymmetry is then equivalent to integrability of the structures, which appears as an involutivity condition and a moment map for diffeomorphisms and gauge transformations. I will show how this works in a few simple examples and discuss how this can be used to understand general properties of supersymmetric backgrounds.

 

I will describe recent advances in the study of quantum gravity where one can explicitly show in examples that superpositions of states with fixed topology can change the topology of spacetime. These effects lead to paradoxes that are resolved in effective field theory by the introduction of code subspaces. I will also talk about more typical states and issues related on how to decide if a black hole horizon is smooth or not.

A virtue of the special geometry underlying the string theory moduli space of  Calabi--Yau manifolds is the existence of a canonical choice of moduli space coordinates. In heterotic theories, as much as we would desire it, there is no obvious choice of coordinates and so we should be covariant. I will discuss some issues in doing this.

One of the key unsolved challenges at the interface of physical and life sciences is to formulate comprehensive computational modeling of cells of higher organisms that is based on microscopic molecular principles of chemistry and physics. Towards addressing this problem, we have developed a unique reactive mechanochemical force-field and software, called MEDYAN (Mechanochemical Dynamics of Active Networks: http://medyan.org).  MEDYAN integrates dynamics of multiple mutually interacting phases: 1) a spatially resolved solution phase is treated using a reaction-diffusion master equation; 2) a polymeric gel phase is both chemically reactive and also undergoes complex mechanical deformations; 3) flexible membrane boundaries interact mechanically and chemically with both solution and gel phases.  In this talk, I will first outline our recent progress in simulating multi-micron scale cytosolic/cytoskeletal dynamics at 1000 seconds timescale, and also highlight the outstanding challenges in bringing about the capability for routine molecular modeling of eukaryotic cells. I will also report on MEDYAN’s applications, in particular, on developing a theory of contractility of actomyosin networks and also characterizing dissipation in cytoskeletal dynamics. With regard to the latter, we devised a new algorithm for quantifying dissipation in cytoskeletal dynamics, finding that simulation trajectories of entropy production provide deep insights into structural evolution and self-organization of actin networks, uncovering earthquake-like processes of gradual stress accumulation followed by sudden rupture and subsequent network remodeling.
 

Abstract:  As is well known, every rational representation of a finite group $G$ can be realized over $\mathbb{Z}$, that is, the corresponding $\mathbb{Q}G$-module $V$ admits a $\mathbb{Z}$-form. Although $\mathbb{Z}$-forms are usually far from being unique, the famous Jordan--Zassenhaus Theorem shows that there are only finitely many $\mathbb{Z}$-forms of any given $\mathbb{Q}G$-module, up to isomorphism. Determining the precise number of these isomorphism classes or even explicit representatives is, however, a hard task in general. In this talk we shall be concerned with the case where $G$ is the symmetric group $\mathfrak{S}_n$ and $V$ is a simple $\mathbb{Q}\mathfrak{S}_n$-module labelled by a hook partition. Building on work of Plesken and Craig we shall present some results as well as open problems concerning the construction of the
integral forms of these modules. This is joint work with Tommy Hofmann from Kaiserslautern.

Many singular stochastic PDEs are expected to be universal objects that govern a wide range of microscopic models in different universality classes. Two notable examples are KPZ and \Phi^4_3. In these cases, one usually finds a parameter in the system, and tunes according to the space-time scale in such a way that the system rescales to the SPDE in the large-scale limit. We justify this belief for a large class of continuous microscopic growth models (for KPZ) and phase co-existence models (for Phi^4_3), allowing microscopic nonlinear mechanisms far beyond polynomials. Aside from the framework of regularity structures, the main new ingredient is a moment bound for general nonlinear functionals of Gaussians. This essentially allows one to reduce the problem of a general function to that of a polynomial. Based on a joint work with Martin Hairer, and another joint work in progress with Chenjie Fan and Jiawei Li.