Past

I'll discuss how a certain notion of symmetry captures the computational complexity of approximating homomorphism problems between relational structures, also known as constraint satisfaction problems. I'll present recent results on inapproximability of conflict-free and linearly-ordered hypergraph colourings and solvability of systems of equations.

Let X --> S be a family of algebraic varieties parametrized by an infinitesimal disk S, possibly of mixed characteristic. The Bloch conductor conjecture expresses the difference of the Euler characteristics of the special and generic fibers in algebraic and arithmetic terms. I'll describe a proof of some new cases of this conjecture, including the case of isolated singularities. The latter was a conjecture of Deligne generalizing Milnor's formula on vanishing cycles. 

This is joint work with Massimo Pippi; our methods use derived and non-commutative algebraic geometry. 

Celestial Holography posits the existence of a holographic description of gravitational theories in asymptotically flat space-times. To date, top-down constructions of such dualities involve a combination of twisted holography and twistor theory. The gravitational theory is the closed string B model living in a suitable twistor space, while the dual is a chiral 2d gauge theory living on a stack of D1 branes wrapping a twistor line. I’ll talk about a variant of these models that yields a theory of self-dual Einstein gravity (via the Plebanski equations) in four dimensions. This is based on work in progress with Roland Bittleston, Kevin Costello & Atul Sharma.

In this talk I will give a short introduction to moderately interacting particle systems and the general notion of fluctuations around the mean-field limit. We will see how a central limit theorem can be shown for moderately interacting particles on the whole space for certain types of interaction potentials. The interaction potential approximates singular attractive potentials of sub-Coulomb type and we can show that the fluctuations become asymptotically Gaussians. The methodology is inspired by the classical work of Oelschläger in the 1980s on fluctuations for the porous-medium equation. To allow for attractive potentials we use a new approach of quantitative mean-field convergence in probability in order to include aggregation effects. 

The Farrell--Jones conjecture predicts that the algebraic K-theory of a group ring is isomorphic to a certain equivariant homology theory, and there are also versions for L-theory and Waldhausen's A-theory. In principle, this provides a way to calculate these K-groups, and has many applications. These include classifying manifolds admitting a given fundamental group and a positive resolution of the Borel conjecture.

I will discuss work with Yassine Guerch and Sam Hughes on the Farrell--Jones conjecture for extensions of relatively hyperbolic groups, as well as an application to their automorphism groups in the one-ended case. The methods are from geometric group theory: we go via the theory of JSJ decompositions to produce acylindrical actions on trees.

In this talk, we will make an explicit link between self-dual Yang-Mills instantons on the Taub-NUT space, and G2-instantons on the BGGG space, by displaying the latter space as a fibration by the former. In doing so, we will discuss analysis on non-compact manifolds, circle symmetries, and a new method of constructing solutions to quadratically singular ODE systems. This talk is based on joint work with Matt Turner: https://arxiv.org/pdf/2409.03886. 

Abelian gauge theories in 2+1 dimensions are very interesting QFTs: they are strongly coupled and exhibit non-trivial dynamics. However, they are somewhat more tractable than non-Abelian theories in 3+1 dimensions. In this talk, I will first review the known properties of fermions in 2+1 dimensions and some conjectures about QED_3 with a single Dirac fermion. I will then present the recent proposal from [arXiv:2409.17913] regarding the phase diagram of QED_3 with two fermions. The findings reveal surprising (yet compelling) features: while semiclassical analysis would suggest two trivially gapped phases and a single phase transition, the actual dynamics indicate the presence of two distinct phase transitions separated by a "quantum phase." This intermediate phase exists over a finite range of parameters in the strong coupling regime and is not visible semiclassically. Moreover, these phase transitions are second-order and exhibit symmetry enhancement. The proposal is supported by several non-trivial checks and is consistent with results from numerical bootstrap, lattice simulations, and extrapolations from the large-Nf expansion.

The geodesic cycles (resp. Eisenstein classes) for SL(2,Z) are special classes in the homology (resp. cohomology) of modular curve (for SL(2,Z)) defined by the closed geodesics (resp. Eisenstein series). It is known that the pairing between these geodesic cycles and Eisenstein classes gives the special values of partial zeta functions of real quadratic fields, and this has many applications. In this talk, I would like to report on some recent observations on the size of the homology subgroup generated by geodesic cycles and their applications. This is a joint work with Ryotaro Sakamoto.

 I will first give a short review of the concepts of Asymptotically Flat Spacetimes, IR triangle and Noether's theorems. I will then present what Asymptotic Higher Spin Symmetries are and how they were introduced as a candidate for an approximate symmetry of General Relativity and the S-matrix. Next, I'll move on to the recent developments of establishing these symmetries as Noether symmetries and describing how they are canonically and non-linearly realized on the asymptotic gravitational phase space. I will discuss how the introduction of dual equations of motion encapsulates the non-perturbativity of the analysis. Finally I'll emphasize the relation to twistor, especially with 2407.04028. Based on 2409.12178 and 2410.15219

I want to give a survey about the rational cohomology of SL_n
Z. This includes recent developments of finding Hopf algebras in the
direct sum of all cohomology groups of SL_n Z for all n. I will give a
quick overview about Hopf algebras and what this structure implies for
the cohomology of SL_n Z.

Are you thinking of launching your own start-up or considering joining an early-stage company? Navigating the entrepreneurial landscape can be both exciting and challenging. Join Pete for an interactive exploration of the unwritten rules and hidden insights that can make or break a start-up journey.

Drawing from personal experience, Pete's talk will offer practical wisdom for aspiring founders and team members, revealing the challenges and opportunities of building a new business from the ground up.

Whether you're an aspiring entrepreneur, a potential start-up team member, or simply curious about innovative businesses, you'll gain valuable perspectives on the realities of creating something from scratch.

This isn't a traditional lecture – it will be a lively conversation that invites participants to learn, share, and reflect on the world of start-ups. Come prepared to challenge your assumptions and discover practical insights that aren't found in standard business guides.
 

Speaker: Professor Pete Grindrod

Topological data analysis, and in particular persistent homology, has provided robust results for numerous applications, such as protein structure, cancer detection, and material science. In the field of neuroscience, the applications of TDA are abundant, ranging from the analysis of single cells to the analysis of neuronal networks. The topological representation of branching trees has been successfully used for a variety of classification and clustering problems of neurons and microglia, demonstrating a successful path of applications that go from the space of trees to the space of barcodes. In this talk, I will present some recent results on topological representation of brain cells, with a focus on neurons. I will also describe our solution for solving the inverse TDA problem on neurons: how can we efficiently go from persistence barcodes back to the space of neuronal trees and what can we learn in the process about these spaces. Finally, I will demonstrate how algebraic topology can be used to understand the links between single neurons and networks and start understanding the brain differences between species. The organizational principles that distinguish the human brain from other species have been a long-standing enigma in neuroscience. Human pyramidal cells form highly complex networks, demonstrated by the increased number and simplex dimension compared to mice. This is unexpected because human pyramidal cells are much sparser in the cortex. The number and size of neurons fail to account for this increased network complexity, suggesting that another morphological property is a key determinant of network connectivity. By comparing the topology of dendrites, I will show that human pyramidal cells have much higher perisomatic (basal and oblique) branching density. Therefore greater dendritic complexity, a defining attribute of human L2 and 3 neurons, may provide the human cortex with enhanced computational capacity and cognitive flexibility.

Would you like to meet some of your fellow students, and some graduate students and postdocs, in an informal and relaxed atmosphere, while building your communication skills? In this Friday@2 session, you'll be able to play a selection of board games, meet new people, and practise working together. What better way to spend the final Friday afternoon of term?! We'll play the games in the south Mezzanine area of the Andrew Wiles Building.

In "Higher Operations in Perturbation Theory", Gaiotto, Kulp, and Wu discussed Feynman integrals that control certain deformations in quantum field theory. The corresponding integrands are differential forms in Schwinger parameters. Specifically, the integrand $\alpha$ is associated to a single topological direction of the theory.
I will show how the combinatorial properties of graph polynomials lead to a relatively simple, explicit formula for $\alpha$, that can be evaluated quickly with a computer. This is interesting for two reasons. Firstly, knowing the explicit formula leads to an elementary proof of the fact that $\alpha$ squares to zero, which asserts the absence of quantum corrections in topological field theories of two (or more) dimensions, known as Kontsevich's formality theorem. Secondly, the underlying constructions and proofs are not intrinsically limited to topological theories. In this sense, they serve as a particularly instructive example for simplifications that can occur in Feynman integrals with numerators.

Advances in spatial profiling technologies are providing insights into how molecular programs are influenced by local signalling and environmental cues. However, cell fate specification and tissue patterning involve the interplay of biochemical and mechanical feedback. Here, we propose a new computational framework that enables the joint statistical analysis of transcriptional and mechanical signals in the context of spatial transcriptomics. To illustrate the application and utility of the approach, we use spatial transcriptomics data from the developing mouse embryo to infer the forces acting on individual cells, and use these results to identify mechanical, morphometric, and gene expression signatures that are predictive of tissue compartment boundaries. In addition, we use geoadditive structural equation modelling to identify gene modules that predict the mechanical behaviour of cells in an unbiased manner. This computational framework is easily generalized to other spatial profiling contexts, providing a generic scheme for exploring the interplay of biomolecular and mechanical cues in tissues.

Model-theoretic dividing lines have long been a source of tameness for various areas of mathematics, with combinatorics jumping on the bandwagon over the last decade or so. Szemerédi’s regularity lemma saw improvements in the realm of NIP, which were further refined in the subrealms of stability and distality. We show how relations satisfying the distal regularity lemma enjoy improved bounds for Zarankiewicz’s problem. We then pivot to arithmetic regularity lemmas as pioneered by Green, for which NIP and stability also imply improvements. Unsettled by the absence of distality in this picture, we discuss the role of distality in additive combinatorics, appealing to our result connecting distality with arithmetic tameness.

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.

The study of polynomials whose coefficients lie in a given set $S$ (the most notable examples being $S=\{0,1\}$ or $\{-1,1\}$) has a long history leading to many interesting results and open problems. We begin with a brief general overview of this topic and then focus on the following old problem of Littlewood. Let $A$ be a set of positive integers, let $f_A(x)=\sum_{n\in A}\cos(nx)$ and define $Z(f_A)$ to be the number of zeros of $f_A$ in $[0,2\pi]$. The problem is to estimate the quantity $Z(N)$ which is defined to be the minimum of $Z(f_A)$ over all sets $A$ of size $N$. We discuss recent progress showing that $Z(N)\geqslant (\log \log N)^{1-o(1)}$ which provides an exponential improvement over the previous lower bound. 

A closely related question due to Borwein, Erd\'elyi and Littmann asks about the minimum number of zeros of a cosine polynomial with $\pm 1$-coefficients. Until recently it was unknown whether this even tends to infinity with the degree $N$. We also discuss work confirming this conjecture.

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. 

The reduced twisted C*-algebra A of an étale groupoid G has a canonical abelian subalgebra D: functions on G's unit space. When G has no non-trivial abelian subgroupoids (i.e., G is principal), then D is in fact maximal abelian. Remarkable work by Kumjian shows that the tuple (A,D) allows us to reconstruct the underlying groupoid G and its twist uniquely; this uses that D is not only masa but even what is called a C*-diagonal. In this talk, I show that twisted C*-algebras of non-principal groupoids can also have such C*-diagonal subalgebras, arising from non-trivial abelian subgroupoids, and I will discuss the reconstructed principal twisted groupoid of Kumjian for such pairs of algebras.

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.

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.

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.

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.

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.

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.

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. 

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).

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.

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. 

In ’95 Lusztig gave a local Langlands correspondence for unramified representations of inner to split adjoint groups combining many deep results from type theory and geometric representation theory. In this talk, I will present a gentle reformulation of his construction revealing some interesting new structures, and with a view toward proving functoriality results in this framework. 

This seminar is organised jointly with the Junior Algebra and Representation Theory Seminar - all are very welcome!