Past

We give a purely topological formula for the square class of the central value of the L-function of a symplectic representation on a curve. We also formulate a topological analogue of the statement, in which the central value of the L-function is replaced by Reidemeister torsion of 3-manifolds. This is related to the theory of epsilon factors in number theory and Meyer’s signature formula in topology among other topics. We will present some of these ideas and sketch aspects of the proof. This is joint work with Akshay Venkatesh.

Bilevel optimization has been part of machine learning for over 4 decades now, although perhaps not always in an obvious way. The interconnection between the two topics started appearing more clearly in publications since about 20 years now, and in the last 10 years, the number of machine learning applications of bilevel optimization has literally exploded. This rise of bilevel optimization in machine learning has been highly positive, as it has come with many innovations in the theoretical and numerical perspectives in understanding and solving the problem, especially with the rebirth of the implicit function approach, which seemed to have been abandoned at some point.
Overall, machine learning has set the bar very high for the whole field of bilevel optimization with regards to the development of numerical methods and the associated convergence analysis theory, as well as the introduction of efficient tools to speed up components such as derivative calculations among other things. However, it remains unclear how the techniques from the machine learning—based bilevel optimization literature can be extended to other applications of bilevel programming. 
For instance, many machine learning loss functions and the special problem structures enable the fulfillment of some qualification conditions that will fail for multiple other applications of bilevel optimization. In this talk, we will provide an overview of machine learning applications of bilevel optimization while giving a flavour of corresponding solution algorithms and their limitations. 
Furthermore, we will discuss possible paths for algorithms that can tackle more complicated machine learning applications of bilevel optimization, while also highlighting lessons that can be learned for more general bilevel programs.

applied mathematics.

Cancelled

We will try to solve the isomorphism problem amongst Coxeter groups by looking at finite quotients.  Some success is found in the classes of affine and right-angled Coxeter groups.  Based on joint work with Samuel Corson, Philip Moeller, and Olga Varghese.

The ground state of N noninteracting Fermions in a rotating harmonic trap enjoys a one-to-one mapping to the complex Ginibre ensemble. This setup is equivalent to electrons in a magnetic field described by Landau levels. The mean, variance and higher order cumulants of the number of particles in a circular domain can be computed exactly for finite N and in three different large-N limits. In the bulk and at the edge of the spectrum the result is universal for a large class of rotationally invariant potentials. In the bulk the variance and entanglement entropy are proportional and satisfy an area law. The same universality can be proven for the quaternionic Ginibre ensemble and its corresponding generalisation. For the real Ginibre ensemble we determine the large-N limit at the origin and conjecture its universality in the bulk and at the edge.

Associated to every shift space, the Cuntz-Krieger algebra (C-K algebra for abbreviation) is an invariant of conjugacy defined and developed by K. Matsumoto, S. Eilers, T. Carlsen, and many of their collaborators in the last decade. In particular, Carlsen defined the C-K algebra to be the full groupoid C*-algebra of the “cover”, which is a topological system consisting of a surjective local homeomorphism on a zero-dimensional space induced by the shift space. 

In 2022, K. Brix proved that the C-K algebra of the Sturmian shift has finite nuclear dimension, where the Sturmian shift is the (unique) minimal shift space with the smallest complexity function: p_X(n)=n+1. In recent results (joint with Z. He), we show that for any minimal shift space with finitely many left special elements, its C-K algebra always have finite nuclear dimension. In fact, this can be further applied to the class of aperiodic shift spaces with non-superlinear growth complexity. 

I will survey some recent results on rigidity and automorphisms of von Neumann algebras of groups with Kazhdan property (T) obtained in a series of joint papers with I. Chifan, A. Ioana, and B. Sun. Specifically, we show that certain groups, constructed via a group-theoretic version of Dehn filling in 3-manifolds, satisfy several conjectures proposed by A. Connes, V. Jones, and S. Popa. Previously, no nontrivial examples of groups satisfying these conjectures were known. At the core of our approach is the new notion of a wreath-like product of groups, which seems to be of independent interest.

Fourth-order elliptic problems arise in a variety of applications from thin plates to phase separation to liquid crystals. A conforming Galerkin discretization requires a finite dimensional subspace of H2, which in turn means that conforming finite element subspaces are C1-continuous. In contrast to standard H1-conforming C0-elements, C1-elements, particularly those of high order, are less understood from a theoretical perspective and are not implemented in many existing finite element codes. In this talk, we address the implementation of the elements. In particular, we present algorithms that compute C1-finite element approximations to fourth-order elliptic problems and which only require elements with at most C0-continuity. The algorithms are suitable for use in almost all standard finite element packages. Iterative methods and preconditioners for the subproblems in the algorithm will also be presented.

In their study of GL(N)-GL(m) Howe duality, Cautis-Kamnitzer-Morrison observed that the GL(N) Reshetikhin-Turaev link invariant can be computed in terms of quantum gl(m). This idea inspired Cautis and Lauda-Queffelec-Rose to give a construction of GL(N) link homology in terms of Khovanov-Lauda's categorified quantum gl(m). There is a Spin(2n+1)-Spin(m) Howe duality, and a quantum analogue that was first studied by Wenzl. In the first half of the talk, I will explain how to use this duality to compute the Spin(2n+1) link polynomial, and present calculations which suggest that the Spin(2n+1) link invariant is obtained from the GL(2n) link invariant by folding. In the second part of the talk, I will introduce the parallel categorified constructions and explain how to use them to define Spin(2n+1) link homology.

This is based on joint work in progress with Ben Elias and David Rose.

Population structure substantially affects evolutionary dynamics. Networks that promote the spreading of fitter mutants are called amplifiers of selection, and those that suppress the spreading of fitter mutants are called suppressors of selection. It has been discovered that most networks are amplifiers under the so-called birth-death updating combined with uniform initialization, which is a common condition. We discuss constant-selection evolutionary dynamics with binary node states (which is equivalent to the biased voter model with two opinions in statistical physics research community) on higher-order networks, i.e., hypergraphs, temporal networks, and multilayer networks. In contrast to the case of conventional networks, we show that a vast majority of these higher-order networks are suppressors of selection, which we show by random-walk and Martingale analyses as well as by numerical simulations. Our results suggest that the modeling framework for structured populations in addition to the specific network structure is an important determinant of evolutionary dynamics.
 

I shall review some aspects of the relationship between scale and conformal invariance in 2-dimensional sigma models.  Then, I shall explain how such an investigation is related to the Perelman's ideas of proving the Poincare' conjecture.  Using this, I shall demonstrate that scale invariant sigma models  with B-field coupling and  compact target space  are conformally invariant. Several examples will also be presented that elucidate the results.  The talk is based on the arXiv paper 2404.19526.

Scaling limits of random developments of a path into a matrix Lie Group have recently been used to construct signature-based kernels on path space, while mitigating some of the dimensionality challenges that come with using signatures directly. General linear group developments have been shown to be connected to the ordinary signature kernel (Muça Cirone et al.), while unitary developments have been used to construct a path characteristic function distance (Lou et al.). By leveraging the tools of random matrix theory and free probability theory, we are able to provide a unified treatment of the limits in both settings under general assumptions on the vector fields. For unitary developments, we show that the limiting kernel is given by the contraction of a signature against the monomials of freely independent semicircular random variables. Using the Schwinger-Dyson equations, we show that this kernel can be obtained by solving a novel quadratic functional equation. 

This is joint work with Thomas Cass.

Introducing an inhomogeneous shift allows for generalisations of many multiplicative results in diophantine approximation. In this talk, we discuss an inhomogeneous version of Gallagher's theorem, established by Chow and Technau, which describes the rates for which we can approximate a typical product of fractional parts. We will sketch the methods used to prove an earlier version of this result due to Chow, using continued fraction expansions and geometry of numbers to analyse the structure of Bohr sets and bound sums of reciprocals of fractional parts.

We will discuss the problem of finding hyperbolic manifolds fibring over the circle; and show a method to construct and analyse maps from particular hyperbolic manifolds to S^1, which relies on Bestvina-Brady Morse theory. 
This technique can be used to build and detect fibrations, algebraic fibrations, and Morse functions with minimal number of critical points, which are interesting in the even dimensional case. 
After an introduction to the problem, and presentation of the main results, we will use the remaining time to focus on some easy 3-dimensional examples, in order to explicitly show the construction at work.
 

In this talk we study wave propagation in random media using multiscale analysis.
We show that the wavefield can be described by a stochastic partial differential equation.
We can then address the following physical conjecture: for large propagation distances, the wavefield has Gaussian statistics, mean zero, and second-order moments determined by radiative transfer theory.
The results for the first two moments can be proved under general circumstances.
The Gaussian conjecture for the statistical distribution of the wavefield can be proved in some propagation regimes, but it turns out to be wrong in other regimes.

Low rank structure is pervasive in real-world datasets.

This talk shows how to accelerate the solution of fundamental computational problems, including eigenvalue decomposition, linear system solves, composite convex optimization, and stochastic optimization (including deep learning), by exploiting this low rank structure.

We present a simple method based on randomized numerical linear algebra for efficiently computing approximate top eigende compositions, which can be used to replace large matrices (such as Hessians and constraint matrices) with low rank surrogates that are faster to apply and invert.

The resulting solvers for linear systems (NystromPCG), composite convex optimization (NysADMM), and stochastic optimization (SketchySGD and PROMISE) demonstrate strong theoretical and numerical support, outperforming state-of-the-art methods in terms of speed and robustness to hyperparameters.

In our first ever joint event with WISOx (Oxford Women in Statistics), we will be having a panel discussion about the hidden labour of minorities, such as extra committee work, editorial work, etc. 

We will be joined by panellists Helen Byrne (Maths) and Gesine Reinert (Stats). 

In this talk I will discuss how phenotypic heterogeneity affects emergent pattern formation in living active matter with chemical communication between cells. In doing so, I will explore how the emergent dynamics of multicellular communities are qualitatively different in comparison to the dynamics of isolated or non-interacting cells. I will focus on two specific projects. First, I will show how genetic regulation of chemical communication affects motility-induced phase separation in cell populations. Second, I will demonstrate how chemotaxis along self-generated signal gradients affects cell populations undergoing 3D morphogenesis.