Past

We investigate a stochastic differential game in which a major player has a private information (the knowledge of a random variable), which she discloses through her control to a population of small players playing in a Nash Mean Field Game equilibrium. The major player’s cost depends on the distribution of the population, while the cost of the population depends on the random variable known by the major player. We show that the game has a relaxed solution and that the optimal control of the major player is approximatively optimal in games with a large but finite number of small players. Joint work with Pierre Cardaliaguet and Catherine Rainer.

In this talk we will provide an overview of a number of mathematical models of cancer growth and development - gene regulatory networks, the immune response to cancer, avascular solid tumour growth, tumour-induced angiogenesis, cancer invasion and metastasis. In the talk we will also discuss (the art of) mathematical modelling itself giving illustrations and analogies from works of art. 

We are now able to solve many partial differential equation problems that were well beyond reach when I started in academia. Some of this success is due to computer hardware but much is due to algorithmic advances. 

I will give a personal perspective of the development of computational methodology in this area over my career thus far. 

Perturbation theory is one of the main tools in the modern physicist's toolbox to solve problems. Indeed, it can often the only approach we have to computing any quantity of interest in a physical theory. However, perturbative contributions can actually grow as we increase the order. Thus, many perturbative series in physics are asymptotic, with 0 radius of convergence. In this talk, I will describe resurgence, which gives us a way of treat such series, by adding non-perturbative effects in a systematic manner.

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.

It is well known that adding even small amounts of  long chain polymers (e.g. few parts per million) to Newtonian solvents can drastically change the flow behaviour by introducing elasticity. In particular,  two decades ago, experiments in curved geometries  demonstrated  that polymer flows can be  chaotic even at vanishingly small Reynolds numbers. The situation in `straight’ flows  such as pressure-driven flow down a channel is less clear  and hence an area of current focus. I will discuss recent progress.

The widespread need to solve large-scale linear systems has sparked a growing interest in randomized techniques. One such class of techniques is known as iterative random sketching methods (e.g., Randomized Block Kaczmarz and Randomized Block Coordinate Descent). These methods "sketch" the linear system to generate iterative, easy-to-compute updates to a solution. By working with sketches, these methods can often enable more efficient memory operations, potentially leading to faster performance for large-scale problems. Unfortunately, tracking the progress of these methods still requires computing the full residual of the linear system, an operation that undermines the benefits of the solvers. In practice, this cost is mitigated by occasionally computing the full residual, typically after an epoch. However, this approach sacrifices real-time progress tracking, resulting in wasted computations. In this talk, we use statistical techniques to develop a progress estimation procedure that provides inexpensive, accurate real-time progress estimates at the cost of a small amount of uncertainty that we effectively control.

Jochen Koenigsmann’s Habilitation introduced a classification of fields elementarily characterised by their absolute Galois groups, including two conjecturally empty families. The emptiness of one of these families would follow from a Galois cohomological conjecture concerning radically closed fields formulated by Koenigsmann. A promising approach to resolving this conjecture involves the use of local-global principles in Galois cohomology. This talk examines the conceptual foundations of this method, highlights its relevance to Koenigsmann’s classification, and evaluates existing local-global principles with regard to their applicability to this conjecture.

In this talk, I will introduce fusion categories as categorical versions of finite rings. We will discuss some examples which may already be familiar, like the category of representations of a finite group and the category of vector spaces graded over a finite group. Then, we will define Tambara-Yamagami categories, which are a certain type of fusion categories which have one simple object which is non-invertible. I will provide the classification results of Tambara and Yamagami on these categories and give some small examples. Time permitting, I will discuss current work in progress on how to generalize Tambara-Yamagami fusion categories to locally compact groups. 

This talk will not assume familiarity with category theory further than the definition of a category and a functor.

According to a conjecture by Landau-Pekar (1948) and by Spohn (1986), the effective mass of the Fröhlich Polaron should diverge in the strong coupling limit like a quartic power of the coupling constant. In a recent joint with R. Bazaes, M. Sellke and S.R.S. Varadhan, we prove this conjecture.

Selberg’s CLT informs us that the logarithm of the Riemann zeta function evaluated on the critical line behaves as a complex Gaussian. It is natural, therefore, to study how far this Gaussianity persists. This talk will present conditional and unconditional results on atypically large values, and concerns work joint with Louis-Pierre Arguin and Asher Roberts.

A trace on a group is a positive-definite conjugation-invariant function on it. These traces correspond to tracial states on the group's maximal  C*-algebra. In the past couple of decades, the study of traces has led to exciting connections to the rigidity, stability, and dynamics of groups. In this talk, I will explain these connections and focus on the topological structure of the space of traces of some groups. We will see the different behaviours of these spaces for free groups vs. higher-rank lattices, and how our strategy for the free group can be used to answer a question of Musat and Rørdam regarding free products of matrix algebras. This is based on joint works with Arie Levit, Joav Orovitz, and Itamar Vigdorovich.

Consider a connected graph and choose a subset of its vertices. From this simple setup, Iyama and Wemyss define a collection of real hyperplanes known as an intersection arrangement, going on to classify all tilings of the affine plane that arise in this way. These "local" generalisations of Coxeter combinatorics also admit a nice wall-crossing structure via Dynkin involutions and longest Weyl elements. In this talk I give an analogous classification in the hyperbolic setting using the data of an "overextended" ADE diagram with three distinguished vertices. I then discuss ongoing work applying intersection arrangements to parametrise notions of stability conditions for preprojective algebras.

In 1975, Bollobás, Erdős, and Szemerédi asked the following Zarankiewicz-type problem. What is the smallest $\tau$ such that an $n \times n \times n$ tripartite graph with minimum degree $n + \tau$ must contain $K_{t, t, t}$? They further conjectured that $\tau = O(n^{1/2})$ when $t = 2$.

I will discuss our proof that $\tau = O(n^{1 - 1/t})$ (confirming their conjecture) and an infinite family of extremal examples. The bound $O(n^{1 - 1/t})$ is best possible whenever the Kővári-Sós-Turán bound $\operatorname{ex}(n, K_{t, t}) = O(n^{2 - 1/t})$ is (which is widely-conjectured to be the case).

This is joint work with Francesco Di Braccio (LSE).

Topically, my goal is to provide a fun and instructive introduction to graph cumulants: a hierarchical set of subgraph statistics that extend the classical cumulants (mean, (co)variance, skew, kurtosis, etc) to relational data.  

Intuitively, graph cumulants quantify the propensity (if positive) or aversion (if negative) for the appearance of any particular subgraph in a larger network.  

Concretely, they are derived from the “bare” subgraph densities via a Möbius inversion over the poset of edge partitions.  

Practically, they offer a systematic way to measure similarity between graph distributions, with a notable increase in statistical power compared to subgraph densities.  

Algebraically, they share the defining properties of cumulants, providing clever shortcuts for certain computations.  

Generally, their definition extends naturally to networks with additional features, such as edge weights, directed edges, and node attributes.  

Finally, I will discuss how this entire procedure of “cumulantification” suggests a promising framework for a motif-centric statistical analysis of general structured data, including temporal and higher-order networks, leaving ample room for exploration. 

Realizing chiral global symmetries on a finite lattice is a long-standing challenge in lattice gauge theory, with potential implications for non-perturbative regularization of the Standard Model. One of the simplest examples of such a symmetry is the axial U(1) symmetry of the 1+1d massless Dirac fermion field theory: it acts by equal and opposite phase rotations on the left- and right-moving Weyl components of the Dirac field. This field theory also has a vector U(1) symmetry which acts identically on left- and right-movers. The two U(1) symmetries exhibit a mixed anomaly, known as the chiral anomaly. In this talk, we will discuss how both symmetries are realized as ordinary U(1) symmetries of an "ultra-local" lattice Hamiltonian, on a finite-dimensional Hilbert space. Intriguingly, the anomaly of the Abelian U(1) symmetries in the infrared (IR) field theory is matched on the lattice by a non-Abelian Lie algebra. The lattice symmetry forces the low-energy phase to be gapless, closely paralleling the effects of the anomaly in the field theory.

I will introduce the class of causally-null-compactifiable spacetimes that can be canonically converted into compact timed-metric spaces using the cosmological time function of Andersson-Galloway-Howard and the null distance of Sormani-Vega. This class of space-times includes future developments of compact initial data sets and regions exhausting asymptotically flat space-times. I will discuss various intrinsic notions of distance between such space-times and show that some of them are definite in the sense that they are equal to zero if and only if there is a time-oriented Lorentzian isometry between the space-times. These definite distances allow us to define notions of convergence of space-times to limit space-times that are not necessarily smooth. This is joint work with Christina Sormani.

TBC

Global well-posedness of 3D Navier-Stokes equations (NSEs) is one of the biggest open problems in modern mathematics. A long-standing conjecture in stochastic fluid dynamics suggests that physically motivated noise can prevent (potential) blow-up of solutions of the 3D NSEs. This phenomenon is often referred to as `regularization by noise'. In this talk, I will review recent developments on the topic and discuss the solution to this problem in the case of the 3D NSEs with small hyperviscosity, for which the global well-posedness in the deterministic setting remains as open as for the 3D NSEs. An extension of our techniques to the case without hyperviscosity poses new challenges at the intersection of harmonic and stochastic analysis, which, if time permits, will be discussed at the end of the talk.

This talk reports on two projects. The first work (in progress), joint  with Amanda Hirschi, constructs (genus 0) open Gromov-Witten invariants for any Lagrangian submanifold using a global Kuranishi chart construction. As an application we show open Gromov-Witten invariants are invariant under Lagrangian cobordisms. I will then describe how open Gromov-Witten invariants fit into mirror symmetry, which brings me to the second project: obtaining open Gromov-Witten invariants from the Fukaya category.

We begin with a short overview of Random Matrix Theory (RMT), focusing on the Marchenko-Pastur (MP) spectral approach. 

Next, we present recent analytical and numerical results on accelerating the training of Deep Neural Networks (DNNs) via MP-based pruning ([1]). Furthermore, we show that combining this pruning with L2 regularization allows one to drastically decrease randomness in the weight layers and, hence, simplify the loss landscape. Moreover, we show that the DNN’s weights become deterministic at any local minima of the loss function. 
 

Finally, we discuss our most recent results (in progress) on the generalization of the MP law to the input-output Jacobian matrix of the DNN. Here, our focus is on the existence of fixed points. The numerical examples are done for several types of DNNs: fully connected, CNNs and ViTs. These works are done jointly with PSU PhD students M. Kiyashko, Y. Shmalo, L. Zelong and with E. Afanasiev and V. Slavin (Kharkiv, Ukraine). 

[1] Berlyand, Leonid, et al. "Enhancing accuracy in deep learning using random matrix theory." Journal of Machine Learning. (2024).

Measurement-Based Quantum Computation (MBQC) is a model of quantum computation driven by measurements instead of unitary gates.   In 2D it is capable of supporting universal quantum computations.   Interestingly, while all measurements are local, the computational output involves non local observables.   We will use the simpler case of 1D MBQC to illustrate how these features can be captured by ideas from gauge theory and extended TQFT. We will also explain  MBQC from the perspective of the extended Hilbert space construction in gauge theories, in which the entanglement edge modes play the role of the logical qubit.

How can it be that so many clever, competent and capable people can feel that they are just one step away from being exposed as a complete fraud? Despite evidence that they are performing well they can still have that lurking fear that at any moment someone is going to tap them on the shoulder and say "We need to have a chat". If you've ever felt like this, or you feel like this right now, then this Friday@2 session might be of interest to you. We'll explore what "Imposter Feelings" are, why we get them and steps you can start to take to help yourself and others. This event is likely to be of interest to undergraduates and MSc students at all stages.