Past events

For a compact Lie group $G$, its massless Coulomb branch algebra is the $G$-equivariant Borel-Moore homology of its based loop space. This algebra is the same as the algebra of regular functions on the BFM space. In this talk, we will explain how this algebra acts on the equivariant symplectic cohomology of Hamiltonian $G$-manifolds when the symplectic manifolds are open and convex. This is a generalization of the closed case where symplectic cohomology is replaced with quantum cohomology. Following Teleman, we also explain how it relates to the Coulomb branch algebra of cotangent-type representations. This is joint work with Eduardo González and Dan Pomerleano.

Data that have an intrinsic network structure can be found in various contexts, including social networks, biological systems (e.g., protein-protein interactions, neuronal networks), information networks (computer networks, wireless sensor networks),  economic networks, etc. As the amount of graphical data that is generated is increasingly large, compressing such data for storage, transmission, or efficient processing has become a topic of interest. 

In this talk, I will give an information theoretic perspective on graph compression. The focus will be on compression limits and their scaling with the size of the graph. For lossless compression, the Shannon entropy gives the fundamental lower limit on the expected length of any compressed representation. 
I will discuss the entropy of some common random graph models, with a particular emphasis on our results on the random geometric graph model. 
Then, I will talk about the problem of compressing a graph with side information, i.e., when an additional correlated graph is available at the decoder. Turning to lossy compression, where one accepts a certain amount of distortion between the original and reconstructed graphs, I will present theoretical limits to lossy compression that we obtained for the Erdős–Rényi and stochastic block models by using rate-distortion theory.

The conference ‘Beyond the Pipeline: Women and Non-binary People in Mathematics Day’ will be held at the University of Oxford on the 17th February 2024. This is a joint event between the Mathematrix and the Mirzakhani societies of the University of Oxford. It is kindly funded by the London Mathematical Society and the Mathematical Institute at the University of Oxford, with additional funding from industry sponsors. 

The metaphor of the 'leaky pipeline' for the decreasing number of women and other gender minorities in Mathematics is problematic and outdated. It conceals the real reasons that women and non-binary people choose to leave Mathematics. This conference, 'Beyond the Pipeline', aims to encourage women and non-binary people to pursue careers in Mathematics, to promote women and non-binary role models, and to create a community of like-minded people. 

Speakers: 

• Brigitte Stenhouse, The Open University
• Mura Yakerson, The University of Oxford
• Vandita Patel, The University of Manchester
• Melanie Rupflin, The University of Oxford
• Christl Donnelly, The University of Oxford

The conference will also include: 

• A panel discussion on careers in and out of academia
• Talks by early-career speakers
• Poster presentations
• 1:1 bookable appointments with our industry sponsors (Cisco, Jane Street, ING, and Optiver)
• Careers stands with our sponsors and the IMA

More information can be found on our website https://www.oxwomeninmaths2024.co.uk/.

This conference is open to everyone regardless of their gender identity. Registration is via the following google form https://forms.gle/cDGaeJCPbBFEPfDB6 and will close when we have reached capacity. We have limited travel funding to support travel to Oxford from within the UK and you can apply for this on the registration form. The deadline for those applying to give a talk and for those applying for travel funding is the 27th January.

If you have any questions email us at @email. 

Conferences and networking are important parts of academic life, particularly early in your academic career.  But how do you make the most out of conferences?  And what are the does and don'ts of networking?  Learn about the answers to these questions and more in this panel discussion by postdocs from across the Mathematical Institute.

The ability of biological matter to move and deform itself is facilitated by microscopic out-of-equilibrium processes that convert chemical energy into mechanical work. In many cases, this mechano-chemical activity takes place on effectively two-dimensional domains formed by, for example, multicellular structures like epithelial tissues or the outer surface of eukaryotic cells, the so-called actomyosin cortex.
We will show in the first part of the talk, that the large-scale dynamics and self-organisation of such structures can be captured by the theory of active fluids. Specifically, using a minimal model of active isotropic fluids, we can rationalize the emergence of asymmetric epithelial tissue flows in the flower beetle during early development, and explain cell rotations in the context of active chiral flows and left-right symmetry breaking that occurs as the model organism C. elegans sets up its body plan.
To develop a more general understanding of such processes, specifically the role of geometry, curvature and interactions with the environment, we introduce in the second part a theory of active fluid surfaces and discuss analytical and numerical tools to solve the corresponding momentum balance equations of curved and deforming surfaces. By considering mechanical interactions with the environment and the fully self-organized shape dynamics of active surfaces, these tools reveal novel mechanisms of symmetry breaking and pattern formation in active matter.

Perverse equivalences, introduced by Chuang and Rouquier, are derived equivalences with a particularly nice combinatorial description. This generalised an earlier construction, with which they proved Broué’s abelian defect group conjecture for blocks of the symmetric groups. Perverse equivalences are of much wider significance in the representation theory of finite dimensional symmetric algebras. Grant has shown that periodic algebras admit perverse autoequivalences. In a similar vein, I will present some perverse equivalences arising from certain periodic modules, with an application to the setting of the symmetric groups.

I am going to discuss main results of my paper "Physics over a finite field and Wick rotation", arxiv 2306.15698. It introduces a structure over a pseudo-finite field which might be of interest in Foundations of Physics. The main theorem establishes an analogue of the polar co-ordinate system in the pseudo-finite field. A stability classification status of the structure is an open question.

Often, one tries to understand the behaviour of non-commutative random variables or of von Neumann algebras through matricial approximations. In some cases, such as when appealing to the determinant conjecture or investigating the soficity of a group, it is important to find approximations by matrices with good algebraic conditions on their entries (e.g., being integers). On the other hand, the most common tool for generating asymptotic independence -- conjugating with random unitaries -- often destroys such delicate structure.

 I will speak on recent joint work with de Santiago, Hayes, Jekel, Kunnawalkam Elayavalli, and Nelson, where we investigate graph products (an interpolation between free and tensor products) and conjugation of matrix models by large structured random permutations. We show that with careful control of how the permutation matrices are chosen, we can achieve asymptotic graph independence with amalgamation over the diagonal matrices. We are able to use this fine structure to prove that strong $1$-boundedness for a large class of graph product von Neumann algebras follows from the vanishing of the corresponding first $L^2$-Betti number. The main idea here is to show that a version of the determinant conjecture holds as long as the individual algebras have generators with approximations by matrices with entries in the ring of integers of some finite extension of Q satisfying some conditions strongly reminiscent of soficity for groups.

A plethora of stochastic models used in particular in mathematical finance, but also population genetics and physics, stems from the class of affine and polynomial processes. The history of these processes is on the one hand closely connected with the important concept of tractability, that is a substantial reduction of computational efforts due to special structural features, and on the other hand with a unifying framework for a large number of probabilistic models. One early instance in the literature where this unifying affine and polynomial point of view can be applied is Lévy's stochastic area formula. Starting from this example,  we present a guided tour through the main properties and recent results, which lead to signature stochastic differential equations (SDEs). They constitute a large class of stochastic processes, here driven by Brownian motions, whose characteristics are entire or real-analytic functions of their own signature, i.e. of iterated integrals of the process with itself, and allow therefore for a generic path dependence. We show that their prolongation with the corresponding signature is an affine and polynomial process taking values in subsets of group-like elements of the extended tensor algebra. Signature SDEs are thus a class of stochastic processes, which is universal within Itô processes with path-dependent characteristics and which allows - due to the affine theory - for a relatively explicit characterization of the Fourier-Laplace transform and hence the full law on path space.

We show the primes have level of distribution 66/107 using triply well-factorable weights. This gives the highest level of distribution for primes in any setting, improving on the prior record level 3/5 of Maynard. We also extend this level to 5/8, assuming Selberg's eigenvalue conjecture. As a result, we obtain new upper bounds for twin primes and for Goldbach representations of even numbers $a$. For the Goldbach problem, this is the first use of a level of distribution beyond the 'square-root barrier', and leads to the greatest improvement on the problem since Bombieri--Davenport from 1966.

We will discuss the physics of M-theory compactifications onto G2-orbifolds of the type that can be desingularised via the method of Joyce and Karigiannis i.e. orbifolds where one has a singular locus of A1 singularities that admits a nowhere-vanishing (Z2-twisted) harmonic 1-form. Interestingly, there are topologically distinct desingularisations of such orbifolds which we show can be physically interpreted as different branches of the 4d vacuum moduli space of the arising gauge theories: Coulomb and Higgs branches. The results suggest generalisations of the results of Joyce and Karigiannis to G2-orbifolds with more diverse ADE singularities and higher order twists. As a bonus, we also get an isomorphism between the moduli space of flat connections on flat compact 3-manifolds and the moduli space of Ricci flat metrics on the G2-orbifolds. We will briefly discuss this. Based on 2309.12869 and 2312.12311.

As machine learning algorithms get integrated into the decision-making process of companies and organizations, insurance products are being developed to protect their providers from liability risk. Algorithmic liability differs from human liability since it is based on data-driven models compared to multiple heterogeneous decision-makers and its performance is known a priori for a given set of data. Traditional actuarial tools for human liability do not consider these properties, primarily focusing on the distribution of historical claims. We propose, for the first time, a quantitative framework to estimate the risk exposure of insurance contracts for machine-driven liability, introducing the concept of algorithmic insurance. Our work provides ML model developers and insurance providers with a comprehensive risk evaluation approach for this new class of products. Thus, we set the foundations of a niche area of research at the intersection of the literature in operations, risk management, and actuarial science. Specifically, we present an optimization formulation to estimate the risk exposure of a binary classification model given a pre-defined range of premiums. Our approach outlines how properties of the model, such as discrimination performance, interpretability, and generalizability, can influence the insurance contract evaluation. To showcase a practical implementation of the proposed framework, we present a case study of medical malpractice in the context of breast cancer detection. Our analysis focuses on measuring the effect of the model parameters on the expected financial loss and identifying the aspects of algorithmic performance that predominantly affect the risk of the contract.

Paper Reference: Bertsimas, D. and Orfanoudaki, A., 2021. Pricing algorithmic insurance. arXiv preprint arXiv:2106.00839.

Paper link: https://arxiv.org/pdf/2106.00839.pdf

Graph braid groups are similar to braid groups, except that they are defined as ‘braids’ on a graph, rather than the real plane. We can think of graph braid groups in terms of the discrete configuration space of a graph, which is a CW-complex. One can compute a presentation of a graph braid group using Morse theory. In this talk I will give a few examples on how to compute these presentations in terms of generating circuits of the graph. I will then go through a detailed example of a graph that gives a one-ended braid group.

We introduce a new class of groups with Thompson-like group properties. In the surface case, the asymptotic mapping class group contains mapping class groups of finite type surfaces with boundary. In dimension three, it contains automorphism groups of all finite rank free groups. I will explain how asymptotic mapping class groups act on a CAT(0) cube complex which allows us to show that they are of type F_infinity. 

This is joint work with Javier Aramayona, Kai-Uwe Bux, Jonas Flechsig and Xaolei Wu.

Brown-Erdős-Sós initiated the study of the maximum number of edges in an $n$-vertex $r$-graph such that no $k$ edges span at most $s$ vertices. If $s=rk-2k+2$ then this function is quadratic in $n$ and its asymptotic was previously known for $k=2,3,4$. I will present joint work with Stefan Glock, Jaehoon Kim, Lyuben Lichev and Shumin Sun where we resolve the cases $k=5,6,7$.

Porta and Yue Yu's model of derived analytic geometry takes as its category of basic, or affine, objects the category opposite to simplicial algebras over the entire functional calculus Lawvere theory. This is analogous to Lurie's approach to derived algebraic geometry where the Lawvere theory is the one governing simplicial commutative rings, and Spivak's derived smooth geometry, using the Lawvere theory of C-infinity-rings. Although there have been numerous important applications including GAGA, base-change, and Riemann-Hilbert theorems, these methods are still missing some crucial ingredients. For example, they do not naturally beget a good definition of quasi-coherent sheaves satisfying descent. On the other hand, the Toen-Vezzosi-Deligne approach of geometry relative to a symmetric monoidal category naturally provides a definition of a category of quasi-coherent sheaves, and in two such approaches to analytic geometry using the categories of bornological and condensed abelian groups respectively, these categories do satisfy descent.  In this talk I will explain how to compare the Porta and Yue Yu model of derived analytic geometry with the bornological one. More generally we give conditions on a Lawvere theory such that its simplicial algebras embed fully faithfully into commutative bornological algebras. Time permitting I will show how the Grothendieck topologies on both sides match up, allowing us to extend the embedding to stacks.

This is based on joint work with Oren Ben-Bassat and Kobi Kremnitzer, and follows work of Kremnitzer and Dennis Borisov.

The S-Matrix in flat space is a naturally holographic observable. S-Matrix elements thus contain valuable information about the putative dual CFT. In this talk, I will first introduce some basic aspects of Celestial Holography and then explain how these can be inferred directly from scattering amplitudes. I will then focus on how the singularity structure of amplitudes interplays with traditional CFT structures particularly in the context of the operator product expansion (OPE) of the dual CFT. I will conclude with some discussion about the role played by supersymmetry in simplifying the putative dual CFT.
 

First we will show that the continuum counterpart of the discrete individual-based mechanical model that describes the dynamics of two contiguous cell populations is given by a free-boundary problem for the cell densities.  Then, in addition to interactions, we will consider the microscopic movement of cells and derive a fractional cross-diffusion system as the many-particle limit of a multi-species system of moderately interacting particles.

First we will show that the continuum counterpart of the discrete individual-based mechanical model that describes the dynamics of two contiguous cell populations is given by a free-boundary problem for the cell densities.  Then, in addition to interactions, we will consider the microscopic movement of cells and derive a fractional cross-diffusion system as the many-particle limit of a multi-species system of moderately interacting particles.

Let $E$ be an elliptic curve over a number field $K$ and $n \geq 2$ an integer. We recall that elements of the $n$-Selmer group of $E/K$ can be explicitly written in terms of certain equations for $n$-coverings of $E/K$. Writing the elements in this way is called conducting an explicit $n$-descent. One of the applications of explicit $n$-descent is in finding generators of large height for $E(K)$ and from this point of view one would like to be able to take $n$ as large as possible. General algorithms for explicit $n$-descent exist but become computationally challenging already for $n \geq 5$. In this talk we discuss combining $n$- and $(n+1)$-descents to $n(n+1)$-descent and the role that invariant theory plays in this procedure.