Past

Scattered topological spaces and their C*-analogs, known as scattered
C*-algebras, have been studied since the 70's and admit a number of
interesting characterizations. In this talk, I will define nowhere
scattered C*-algebras as, informally, those C*-algebras that are very
far from being scattered. I will then characterize this property in
various ways, such as the absence of nonzero elementary ideal-quotients,
topological properties of the spectrum, and divisibility properties in
the Cuntz semigroup. Further, I will also show that these divisibility
properties can be strengthened in the real rank zero or the stable rank
one case.

The talk is based on joint work with Hannes Thiel.

In many contexts a correspondence has been found between the classical compact groups and certain boundary conditions -- $U(n)$ corresponding to periodic, $USp(2n)$ corresponding to Dirichlet, $SO(2n)$ corresponding to Neumann and $SO(2n+1)$ corresponding to Zaremba. In this talk, I will try to elucidate this correspondence in Lie theoretic terms and in the process relate random matrix theory to Yang-Mills theory, free fermions and modular forms.

An $r$-uniform tight cycle of length $k>r$ is a hypergraph with vertices $v_1,\ldots,v_k$ and edges $\{v_i,v_{i+1},…,v_{i+r-1}\}$ (for all $i$), with the indices taken modulo $k$. Sós, and independently Verstraëte, asked the following question: how many edges can there be in an $n$-vertex $r$-uniform hypergraph if it contains no tight cycles of any length? In this talk I will review some known results, and present recent progress on this problem.

A metapopulation model, composed of subpopulations and pairwise connections, is a particle-network framework for epidemic dynamics study. Individuals are well-mixed within each subpopulation and migrate from one subpopulation to another, obeying a given mobility rule. While different mobility rules in metapopulation models have been studied, few efforts have been made to compare the effects of simple (i.e., unbiased) random walks and more complex mobility rules. In this talk, we study susceptible-infectious-susceptible (SIS) dynamics in a metapopulation model, in which individuals obey a second-order parametric random-walk mobility rule called the node2vec. We transform the node2vec mobility rule to a first-order Markov chain whose state space is composed of the directed edges and then derive the epidemic threshold. We find that the epidemic threshold is larger for various networks when individuals avoid frequent backtracking or visiting a neighbor of the previously visited subpopulation than when individuals obey the simple random walk. The amount of change in the epidemic threshold induced by the node2vec mobility is generally not as significant as, but is sometimes comparable with, the one induced by the change in the diffusion rate for individuals.

arXiv links: https://arxiv.org/abs/2006.04904 and https://arxiv.org/abs/2106.08080

Optimal execution of large positions over a given trading period is a fundamental decision-making problem for financial services. In this talk we explore reinforcement learning methods, in particular policy gradient methods, for finding the optimal policy in the optimal liquidation problem. We show results for the case where we assume a linear quadratic regulator (LQR) model for the underlying dynamics and where we apply the method to the data directly. The empirical evidence suggests that the policy gradient method can learn the global optimal solution for a larger class of stochastic systems containing the LQR framework, and that it is more robust with respect to model misspecification when compared to a model-based approach.

In the first part of this talk, we will (briefly) derive the original calculation by Hawking in 1974 to determine the radiation given off by a black hole, giving the result in the form of an integral of a classical solution to the linear wave equation.
In the second part of the talk, we will take this integral as a starting point, and rigorously calculate the radiation given off by a forming spherically symmetric, charged black hole. We will then show that for late times in its formation, the radiation given off approaches the limit predicted by Hawking, including the extremal case. We will also calculate a bound on the rate at which this limit is approached.

Nonuniform Ellipticity is a classical topic in PDE, and regularity of solutions to nonuniformly elliptic and parabolic equations has been studied at length. I will present some recent results in this direction, including the solution to the longstanding issue of the validity of Schauder estimates in the nonuniformly elliptic case obtained in collaboration with Cristiana De Filippis. 

Free-by-cyclic groups are easy to define – all you need is an automorphism of F_n. Their properties (for example hyperbolicity, or relative hyperbolicity) depend on this defining automorphism, but not always transparently. I will introduce these groups and some of their properties, and connect some to properties of the defining automorphism. I'll then discuss some ideas and techniques we can use to understand their automorphisms, including finding useful actions on trees and relationships with certain subgroups of Out(F_n). (This is joint work with Armando Martino.)

I will present a simple abstract quantitative method for proving the hydrodynamic limit of interacting particle systems on a lattice, both in the hyperbolic and parabolic scaling. In the latter case, the convergence rate is uniform in time. This "consistency-stability" approach combines a modulated Wasserstein-distance estimate comparing the law of the stochastic process to the local Gibbs measure, together with stability estimates à la Kruzhkov in weak distance, and consistency estimates exploiting the regularity of the limit solution. It avoids the use of “block estimates” and is self-contained. We apply it to the simple exclusion process, the zero range process, and the Ginzburg-Landau process with Kawasaki dynamics. This is a joint work with Daniel Marahrens and Angeliki Menegaki (IHES).

Examples of 2CY categories include the category of coherent sheaves on a K3 surface, the category of Higgs bundles, and the category of modules over preprojective algebras or fundamental group algebras of compact Riemann surfaces.  Let p:M->N be the morphism from the stack of semistable objects in a 2CY category to the coarse moduli space.  I'll explain, using cohomological DT theory, formality in 2CY categories, and structure theorems for good moduli stacks, how to prove a version of the BBDG decomposition theorem for the exceptional direct image of the constant sheaf along p, even though none of the usual conditions for the decomposition theorem apply: p isn't projective or representable, M isn't smooth, the constant mixed Hodge module complex Q_M isn't pure...  As an application, I'll explain how this allows us to extend nonabelian Hodge theory to Betti/Dolbeault stacks.

A remarkable feature of chaos in many-body quantum systems is the existence of a bound on the quantum Lyapunov exponent. An important question is to understand what is special about maximally chaotic systems which saturate this bound. Here I will discuss a proposal for a `hydrodynamic' origin of chaos in such systems, and discuss hallmarks of maximally chaotic systems. In particular I will discuss how in maximally chaotic systems there is a suppression of exponential growth in commutator squares of generic few-body operators. This suppression appears to indicate that the nature of operator scrambling in maximally chaotic systems is fundamentally different to scrambling in non-maximally chaotic systems.

There are various aspects of the AdS/CFT correspondence that are rather mysterious. For example, how does the gravitational theory know about a discrete boundary spectrum or how does it know moments of the partition function factorize, given the existence of connected (wormhole) geometries? In this talk I will discuss some recent efforts with Andreas Blommaert and Luca Iliesiu on these two puzzles in two dimensional dilaton gravities. These gravity theories are simple enough that we can understand and propose a resolution to the discreteness and factorization puzzles. I will show that a tiny but universal bilocal spacetime interaction in the bulk is enough to ensure factorization, whereas modifying the dilaton potential with tiny corrections gives a discrete boundary spectrum. We will discuss the meaning of these corrections and how they could be related to resolutions of the same puzzles in higher dimensions. 

This event will take place on Teams. A link will be available 30 minutes before the session begins.

Sebastien Racaniere is a Staff Research Engineer at DeepMind. His current main interest is in the use of symmetries in Machine Learning. This offers diverse applications, for example in Neuroscience or Theoretical Physics (in particular Lattice Quantum Chromodynamics). Past interests, still in Machine Learning, include Reinforcement Learning (i.e. learning from rewards), generative models (i.e. learn to sample from probability distributions), and optimisation (i.e. how to find 'good' minima of functions)

The morphogenesis of branched tissues has been a subject of long-standing interest and debate. Although much is known about the signaling pathways that control cell fate decisions, it remains unclear how macroscopic features of branched organs, including their size, network topology and spatial patterning, are encoded. Based on large-scale reconstructions of the mouse mammary gland and kidney, we show that statistical features of the developing branched epithelium can be explained quantitatively by a local self-organizing principle based on a branching and annihilating random walk (BARW). In this model, renewing tip-localized progenitors drive a serial process of ductal elongation and stochastic tip bifurcation that terminates when active tips encounter maturing ducts. Finally, based on reconstructions of the developing mouse salivary gland, we propose a generalisation of BARW model in which tips arrested through steric interaction with proximate ducts reactivate their branching programme as constraints become alleviated through the expansion of the underlying matrix. This inflationary branching-arresting random walk model presents a general paradigm for branching morphogenesis when the ductal epithelium grows cooperatively with the matrix into which it expands.

Let $F$ be a finite field or a $p$-adic field. One method of constructing irreducible representations of $G = GL_2(F)$ is to consider spaces on which $G$ naturally acts and look at the representations arising from invariants of these spaces, such as the action of $G$ on cohomology groups. In this talk, I will discuss how this goes for abstract representations of $G$ (when $F$ is finite), and smooth representations of $G$ (when $F$ is $p$-adic). The first space is an affine algebraic variety, and the second a tower of rigid spaces. I will then mention some recent results about how this tower allows us to construct new interesting $p$-adic representations of $G$, before explaining how trying to adapt these methods leads naturally to considerations about certain geometric properties of these spaces.

The Persistent Homology Transform, and the Euler Characteristic Transform are topological analogs of the Radon transform that can be used in statsistical shape analysis. In this talk I will consider an interesting variant called the Extended Persistent Homology Transform (XPHT) which replaces the normal persistent homology with extended persistent homology. We are particularly interested in the application of the XPHT to binary images. This paper outlines an algorithm for efficient calculation of the XPHT exploting relationships between the PHT of the boundary curves to the XPHT of the foreground.

The use of heuristics to assess the convergence and compress the output of Markov chain Monte Carlo can be sub-optimal in terms of the empirical approximations that are produced. Here we consider the problem of retrospectively selecting a subset of states, of fixed cardinality, from the sample path such that the approximation provided by their empirical distribution is close to optimal. A novel method is proposed, based on greedy minimisation of a kernel Stein discrepancy, that is suitable for problems where heavy compression is required. Theoretical results guarantee consistency of the method and its effectiveness is demonstrated in the challenging context of parameter inference for ordinary differential equations. Software is available in the Stein Thinning package in Python, R and MATLAB.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome

Climate- and weather prediction centres such as the Met Office rely on efficient numerical methods for simulating large scale atmospheric flow. One computational bottleneck in many models is the repeated solution of a large sparse system of linear equations. Preconditioning this system is particularly challenging for state-of-the-art discretisations, such as (mimetic) finite elements or Discontinuous Galerkin (DG) methods. In this talk I will present recent work on developing efficient multigrid preconditioners for practically relevant modelling codes. As reported in a REF2021 Industrial Impact Case Study, multigrid has already led to runtime savings of around 10%-15% for operational global forecasts with the Unified Model. Multigrid also shows superior performance in the Met Office next-generation LFRic model, which is based on a non-trivial finite element discretisation.

The experimental evidence of the existence of a feedback between growth and stress in tumors poses challenging questions. First, the rheological properties (the constitutive equations) of aggregates of malignant cells are to identified. Secondly, the feedback law (the "growth law") that relates stress and mitotic and apoptotic rate should be understood. We address these questions on the basis of a theoretical analysis of in vitro experiments that involve the growth of tumor spheroids. We show that solid tumors exhibit several mechanical features of a poroelastic material, where the cellular component behaves like an elastic solid. When the solid component of the spheroid is loaded at the boundary, the cellular aggregate grows up to an asymptotic volume that depends on the exerted compression.
Residual stress shows up when solid tumors are radially cut, highlighting a peculiar tensional pattern.
The features of the mechanobiological system can be explained in terms of a feedback of mechanics on the cell proliferation rate as modulated by the availability of nutrient, that is radially damped by the balance between diffusion and consumption. The volumetric growth profiles and the pattern of residual stress can be theoretically reproduced assuming a dependence of the target stress on the concentration of nutrient which is specific of the malignant tissue.

The Teichmüller space for a closed surface of genus g is the space of marked complex/hyperbolic structures on the surface. Teichmüller space also identifies with the space of Fuchsian representations of the fundamental group into PSL(2,R) (mod conjugation). Higher Teichmüller theory concerns special representations of surface (or hyperbolic) groups into higher rank Lie groups of non-compact type.