Past

The directed landscape is a random directed metric on the plane that is the scaling limit for models in the KPZ universality class (i.e. last passage percolation on $\mathbb{Z}^2$, TASEP). In this metric, typical pairs of points are connected by a unique geodesic.  However, certain exceptional pairs are connected by more exotic geodesic networks. The goal of this talk is to describe a full classification for these exceptional pairs.

Suppose 𝐺_𝕜 is a connected reductive algebraic 𝕜-group where 𝕜 is an algebraically closed field. If 𝑉_𝕜 is a 𝐺_𝕜-module then, using geometric invariant theory, Kempf has defined the nullcone 𝒩(𝑉_𝕜) of 𝑉_𝕜. For the Lie algebra 𝔤_𝕜 = Lie(𝐺_𝕜), viewed as a 𝐺_𝕜-module via the adjoint action, we have 𝒩(𝔤_𝕜) is precisely the set of nilpotent elements.

We may assume that our group 𝐺_𝕜 = 𝐺 ×_ℤ 𝕜 is obtained by base-change from a suitable ℤ-form 𝐺. Suppose 𝑉 is 𝔤 = Lie(G) or its dual 𝔤* = Hom(𝔤, ℤ) which are both modules for 𝐺, that are free of finite rank as ℤ-modules. Then 𝑉 ⨂_ℤ 𝕜, as a module for 𝐺_𝕜, is 𝔤_𝕜 or 𝔤_𝕜* respectively.

It is known that each 𝐺_ℂ -orbit 𝒪 ⊆ 𝒩(𝑉_ℂ) contains a representative ξ ∈ 𝑉 in the ℤ-form. Reducing ξ one gets an element ξ_𝕜 ∈ 𝑉_𝕜 for any algebraically closed 𝕜. In this talk, we will explain two ways in which we might want ξ to have “good reduction” and how one can find elements with these properties. We will also discuss the relationship to Lusztig’s special orbits.

This is on-going joint work with Adam Thomas (Warwick).

What is the bare minimum needed to get a unitarity-consistent black hole radiation entropy curve? In this talk, I will show how to capture both Hawking's non-unitary entropy curve, and density matrix-connecting contributions that restore unitarity, in a toy quantum system with chaotic dynamics. The motivation is to find the simplest possible dynamical model, dropping all superfluous details, that captures this aspect of gravitational physics. In the model, the Hamiltonian obeys random matrix statistics within microcanonical windows, the entropy of the averaged state gives the non-unitary curve, the averaged entropy gives the unitary curve, and the difference comes from matrix index contractions in the Haar averaging that connect the density matrices in a replica wormhole-like manner.

This paper presents the (Adaptive) Iterative Linear Quadratic Regulator Deep Galerkin Method (AIR-DGM), a novel approach for solving optimal control (OC) problems in dynamic and uncertain environments. Traditional OC methods face challenges in scalability and adaptability due to the curse-of-dimensionality and reliance on accurate models. Model Predictive Control (MPC) addresses these issues but is limited to open-loop controls. With (A)ILQR-DGM, we combine deep learning with OC to compute closed-loop control policies that adapt to changing dynamics. Our methodology is split into two phases; offline and online. In the offline phase, ILQR-DGM computes globally optimal control by minimizing a variational formulation of the Hamilton-Jacobi-Bellman (HJB) equation. To improve performance over DGM (Sirignano & Spiliopoulos, 2018), ILQR-DGM uses the ILQR method (Todorov & Li, 2005) to initialize the value function and policy networks. In the online phase, AIR-DGM solves continuously updated OC problems based on noisy observations of the environment. We provide results based on HJB stability theory to show that AIR-DGM leverages Transfer Learning (TL) to adapt the optimal policy. We test (A)ILQR-DGM in various setups and demonstrate its superior performance over traditional methods, especially in scenarios with misspecified priors and changing dynamics.

Iwasawa algebras are completed group rings that arise in number theory, so there is interest in understanding their prime ideals. For some special Iwasawa algebras, it is conjectured that every non-zero such ideal has finite codimension and in order to show this it is enough to establish the faithfulness of the modules arising from the completion of highest weight modules. In this talk we will look at methods for doing this and apply them to the specific case of the metaplectic representation for the symplectic group.

Given an infinite discrete group G with a finite model for the classifying space for proper actions, one can define the Euler characteristic of G and the orbifold Euler characteristic of G. In this talk we will discuss higher chromatic analogues of these invariants in the sense of stable homotopy theory. We will study the Morava K-theory of G and associated Euler characteristic, and give a character formula for the Lubin-Tate theory of G. This will generalise the results of Hopkins-Kuhn-Ravenel from finite to infinite groups and the K-theoretic results of Adem, Lück and Oliver from chromatic level one to higher chromatic levels. At the end we will mention explicit computations for some arithmetic groups and mapping class groups in terms of class numbers and special values of zeta functions. This is all joint with Wolfgang Lück and Stefan Schwede.

Many systems in the applied sciences are made of a large number of particles. One is often not interested in the detailed behaviour of each particle but rather in the collective behaviour of the group. An established methodology in statistical mechanics and kinetic theory allows one to study the limit as the number of particles in the system N tends to infinity and to obtain a (low dimensional) PDE for the evolution of the density of the particles. The limiting PDE is a non-linear equation, where the non-linearity has a specific structure and is called a McKean-Vlasov nonlinearity. Even if the particles evolve according to a stochastic differential equation, the limiting equation is deterministic, as long as the particles are subject to independent sources of noise. If the particles are subject to the same noise (common noise) then the limit is given by a Stochastic Partial Differential Equation (SPDE). In the latter case the limiting SPDE is substantially the McKean-Vlasov PDE + noise; noise is furthermore multiplicative and has gradient structure.  One may then ask the question about whether it is possible to obtain McKean-Vlasov SPDEs with additive noise from particle systems. We will explain how to address this question, by studying limits of weighted particle systems.  

This is a joint work with L. Angeli, J. Barre,  D. Crisan, M. Kolodziejzik.  

The moduli space of torsion-free $G_2$-structures on a compact $7$-manifold $M$ is a smooth manifold, locally diffeomorphic to an open subset of $H^3(M)$. It is endowed with a natural metric which arises as the Hessian of a potential, the properties of which are still poorly understood. In this talk, we will review what is known of the geometry of $G_2$-moduli spaces and present new formulae for the fourth derivative of the potential and the curvatures of the associated metric. We explain some interesting consequences for the simplest examples of $G_2$-manifolds, when the universal cover of $M$ is $\mathbb{R}^7$ or $\mathbb{R}^3 \times K3$. If time permits, we also make some comments on the general case.

Sparse neural networks, where a substantial portion of the components are eliminated, have widely shown their versatility in model compression, robustness improvement, and overfitting mitigation. However, traditional methods for obtaining such sparse networks usually involve a fully pre-trained, dense model. As foundation models become prevailing, the cost of this pre-training step can be prohibitive. On the other hand, training intrinsic sparse neural networks from scratch usually leads to inferior performance compared to their dense counterpart. 

In this talk, I will present a series of approaches to obtain such fantastic sparse neural networks by training from scratch without the need for any dense pre-training steps, including dynamic sparse training, static sparse with random pruning, and only masking no training. First, I will introduce the concept of in-time over-parameterization (ITOP) (ICML2021) which enables training sparse neural networks from scratch (commonly known as sparse training) to attain the full accuracy of dense models. By dynamically exploring new sparse topologies during training, we avoid the costly necessity of pre-training and re-training, requiring only a single training run to obtain strong sparse neural networks. Secondly, ITOP involves additional overhead due to the frequent change in sparse topology. Our following work (ICLR2022) demonstrates that even a naïve, static sparse network produced by random pruning can be trained to achieve dense model performance as long as our model is relatively larger. Moreover, I will further discuss that we can continue to push the extreme of training efficiency by only learning masks at initialization without any weight updates, addressing the over-smoothing challenge in building deep graph neural networks (LoG2022).

Sarah Livermore has worked in the Civil Service for over 10 years, using the maths skills gained in her physics degrees (MPhys, DPhil) whilst studying at Oxford. In this session she’ll discuss some of the roles available to people with a STEM background in the Civil Service, a ‘day in the life’ of a civil servant, typical career paths and how to apply.

In recent years, the theme of asymptotically flat spacetimes has come back to the fore, fueled by the discovery of gravitational waves and the growing interest in what flat holography could be. In this quest, the standard tools pertaining to asymptotically anti-de Sitter spacetimes have been insufficiently exploited. I will show how Ricci-flat spacetimes are generally reached as a limit of Einstein geometries and how they are in fact constructed by means of data defined on the conformal Carrollian boundary that is null infinity. These data, infinite in number, are obtained as the coefficients of the Laurent expansion of the energy-momentum tensor in powers of the cosmological constant. This approach puts this tensor back at the heart of the analysis, and at the same time reveals the versatile role of the boundary Cotton tensor. Both appear in the infinite hierarchy of flux-balance equations governing the gravitational dynamics.  

We build statistical models to describe how market participants choose the direction, price, and volume of orders. Our dataset, which spans sixteen weeks for four shares traded in Euronext Amsterdam, contains all messages sent to the exchange and includes algorithm identification and member identification. We obtain reliable out-of-sample predictions and report the top features that predict direction, price, and volume of orders sent to the exchange. The coefficients from the fitted models are used to cluster trading behaviour and we find that algorithms registered as Liquidity Providers exhibit the widest range of trading behaviour among dealing capacities. In particular, for the most liquid share in our study, we identify three types of behaviour that we call (i) directional trading, (ii) opportunistic trading, and (iii) market making, and we find that around one third of Liquidity Providers behave as market markers.

This is based on work with Álvaro Cartea, Saad Labyad, Leandro Sánchez-Betancourt and Leon van Veldhuijzen. View the working paper here.
 

Attendance is free of charge but requires prior online registration. To register please click here.

Schubert varieties in (twisted) affine Grassmannians and their singularities are of interest to arithmetic geometers because they model the étale local structure of the special fiber of Shimura varieties. In this talk, I will discuss a proof of a conjecture of Haines-Richarz classifying the smooth locus of Schubert varieties, generalizing a classical result of Evens-Mirkovic. The main input is to obtain a lower bound for the tangent space at a point of the Schubert variety which arises from considering certain smooth curves passing through it. In the second part of the talk, I will explain how in many cases, we can prove this bound is actually sharp, and discuss some applications to Shimura varieties. This is based on joint work with Pappas and Kisin-Pappas.

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.

A novel framework for hierarchical low-rank matrices is proposed that combines an adaptive hierarchical partitioning of the matrix with low-rank approximation. One typical application is the approximation of discretized functions on rectangular domains; the flexibility of the format makes it possible to deal with functions that feature singularities in small, localized regions. To deal with time evolution and relocation of singularities, the partitioning can be dynamically adjusted based on features of the underlying data. Our format can be leveraged to efficiently solve linear systems with Kronecker product structure, as they arise from discretized partial differential equations (PDEs). For this purpose, these linear systems are rephrased as linear matrix equations and a recursive solver is derived from low-rank updates of such equations. 
We demonstrate the effectiveness of our framework for stationary and time-dependent, linear and nonlinear PDEs, including the Burgers' and Allen–Cahn equations.

This is a joint work with Daniel Kressner and Stefano Massei.

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For many systems (certainly those in biology, medicine and pharmacology) the mathematical models that are generated invariably include state variables that cannot be directly measured and associated model parameters, many of which may be unknown, and which also cannot be measured.  For such systems there is also often limited access for inputs or perturbations. These limitations can cause immense problems when investigating the existence of hidden pathways or attempting to estimate unknown parameters and this can severely hinder model validation. It is therefore highly desirable to have a formal approach to determine what additional inputs and/or measurements are necessary in order to reduce or remove these limitations and permit the derivation of models that can be used for practical purposes with greater confidence.

Structural identifiability arises in the inverse problem of inferring from the known, or assumed, properties of a biomedical or biological system a suitable model structure and estimates for the corresponding rate constants and other model parameters.  Structural identifiability analysis considers the uniqueness of the unknown model parameters from the input-output structure corresponding to proposed experiments to collect data for parameter estimation (under an assumption of the availability of continuous, noise-free observations).  This is an important, but often overlooked, theoretical prerequisite to experiment design, system identification and parameter estimation, since estimates for unidentifiable parameters are effectively meaningless.  If parameter estimates are to be used to inform about intervention or inhibition strategies, or other critical decisions, then it is essential that the parameters be uniquely identifiable. 

Numerous techniques for performing a structural identifiability analysis on linear parametric models exist and this is a well-understood topic.  In comparison, there are relatively few techniques available for nonlinear systems (the Taylor series approach, similarity transformation-based approaches, differential algebra techniques and the more recent observable normal form approach and symmetries approaches) and significant (symbolic) computational problems can arise, even for relatively simple models in applying these techniques.

In this talk an introduction to structural identifiability analysis will be provided demonstrating the application of the techniques available to both linear and nonlinear parameterised systems and to models of (nonlinear mixed effects) population nature.

It was shown by Balasubramanya that any acylindrically hyperbolic group (a natural generalisation of a hyperbolic group) must act acylindrically and non-elementarily on some quasi-tree. It is therefore sensible to ask to what extent this is true for trees, i.e. given an acylindrically hyperbolic group, does it admit a non-elementary acylindrical action on some simplicial tree? In this talk I will introduce the concepts of acylindrically hyperbolic and acylindrically arboreal groups and discuss some particularly interesting examples of acylindrically hyperbolic groups which do and do not act acylindrically on trees.

We review progress made by our group on soft matter at interfaces and related physics from the nano- to macroscopic lengthscales. Specifically, to capture nanoscale properties very close to interfaces and to establish a link to the macroscale behaviour, we employ elements from the statistical mechanics of classical fluids, namely density-functional theory (DFT). We formulate a new and general dynamic DFT that carefully and systematically accounts for the fundamental elements of any classical fluid and soft matter system, a crucial step towards the accurate and predictive modelling of physically relevant systems. In a certain limit, our DDFT reduces to a non-local Navier-Stokes-like equation that we refer to as hydrodynamic DDFT: an inherently multiscale model, bridging the micro- to the macroscale, and retaining the relevant fundamental microscopic information (fluid temperature, fluid-fluid and wall-fluid interactions) at the macroscopic level.

Work analysing the moving contact line in both equilibrium and dynamics will be presented. This has been a longstanding problem for fluid dynamics with a major challenge being its multiscale nature, whereby nanoscale phenomena manifest themselves at the macroscale. A key property captured by DFT at equilibrium, is the fluid layering on the wall-fluid interface, amplified as the contact angle decreases. DFT also allows us to unravel novel phase transitions of fluids in confinement. In dynamics, hydrodynamic DDFT allows us to benchmark existing phenomenological models and reproduce some of their key ingredients. But its multiscale nature also allows us to unravel the underlying physics of moving contact lines, not possible with any of the previous approaches, and indeed show that the physics is much more intricate than the previous models suggest.

We will close with recent efforts on machine learning and DFT. In particular, the development of a novel data-driven physics-informed framework for the solution of the inverse problem of statistical mechanics: given experimental data on the collective motion of a classical many-body system, obtain the state functions, such as free-energy functionals.

email Hannah Hughes for further information.

We discuss Connes’ quantized calculus on quantum tori and Euclidean spaces, as applications of the recent development of noncommutative analysis.
This talk is based on a joint work in progress with Xiao Xiong and Kai Zeng.
 

Abstract: It’s a classical result by Huber that the number of closed geodesics of length bounded by L on a closed hyperbolic surface S is asymptotic to exp(L)/L as L grows. This result has been generalized in many directions, for example by counting certain subsets of closed geodesics. One such result is the asymptotic growth of those that are homologically trivial, proved independently by both by Phillips-Sarnak and Katsura-Sunada. A homologically trivial curve can be written as a product of commutators, and in this talk we will look at those that can be written as a product of g commutators (in a sense, those that bound a genus g subsurface) and obtain their asymptotic growth. As a special case, our methods give a geometric proof of Huber’s classical theorem. This is joint work with Juan Souto. 

An $n$-vertex graph $G$ is a $C$-expander if $|N(X)|\geq C|X|$ for every $X\subseteq V(G)$ with $|X|< n/2C$ and there is an edge between every two disjoint sets of at least $n/2C$ vertices.

We show that there is some constant $C>0$ for which every $C$-expander is Hamiltonian. In particular, this implies the well known conjecture of Krivelevich and Sudakov from 2003 on Hamilton cycles in $(n,d,\lambda)$-graphs. This completes a long line of research on the Hamiltonicity of sparse graphs, and has many applications.

Joint work with R. Montgomery, D. Munhá Correia, A. Pokrovskiy and B. Sudakov.

High-order tensor methods for solving both convex and nonconvex optimization problems have recently generated significant research interest, due in part to the natural way in which higher derivatives can be incorporated into adaptive regularization frameworks, leading to algorithms with optimal global rates of convergence and local rates that are faster than Newton's method. On each iteration, to find the next solution approximation, these methods require the unconstrained local minimization of a (potentially nonconvex) multivariate polynomial of degree higher than two, constructed using third-order (or higher) derivative information, and regularized by an appropriate power of the change in the iterates. Developing efficient techniques for the solution of such subproblems is currently, an ongoing topic of research,  and this talk addresses this question for the case of the third-order tensor subproblem.

In particular, we propose the CQR algorithmic framework, for minimizing a nonconvex Cubic multivariate polynomial with  Quartic Regularisation, by sequentially minimizing a sequence of local quadratic models that also incorporate both simple cubic and quartic terms. The role of the cubic term is to crudely approximate local tensor information, while the quartic one provides model regularization and controls progress. We provide necessary and sufficient optimality conditions that fully characterise the global minimizers of these cubic-quartic models. We then turn these conditions into secular equations that can be solved using nonlinear eigenvalue techniques. We show, using our optimality characterisations, that a CQR algorithmic variant has the optimal-order evaluation complexity of $O(\epsilon^{-3/2})$ when applied to minimizing our quartically-regularised cubic subproblem, which can be further improved in special cases.  We propose practical CQR variants that judiciously use local tensor information to construct the local cubic-quartic models. We test these variants numerically and observe them to be competitive with ARC and other subproblem solvers on typical instances and even superior on ill-conditioned subproblems with special structure.

Iwasawa algebras are completions of group algebras for p-adic Lie groups, and have applications for studying the representations of these groups. It is an ongoing project to study the prime ideals, and more generally the two-sided ideals, of these algebras.

In the case of Iwasawa algebras corresponding to a simple Lie algebra with a Chevalley basis, we aim to prove that all non-zero two-sided ideals have finite codimension. To prove this, it is sufficient to show faithfulness of modules arising from highest-weight modules for the corresponding Lie algebra.

I have proved two main results in this direction: firstly, I proved the faithfulness of generalised Verma modules over the Iwasawa algebra. Secondly, I proved the faithfulness of all infinite-dimensional highest-weight modules in the case where the Lie algebra has type A. In this talk, I will outline the methods I used to prove these cases.

We discuss timelike surfaces of finite size in general relativity and the initial boundary value problem. We consider obstructions with the standard Dirichlet problem, and conformal version with improved properties. The ensuing dynamical features are discussed with general cosmological constant.

I will give an overview of a recent method introduced by P. Duch to solve some subcritical singular SPDEs, in particular the stochastic quantisation equation for scalar fields. 

In this seminar, I will talk about a notion of entropy of arithmetic functions and some properties of this entropy.  This notion was introduced to study Sarnak's Moebius Disjointness Conjecture.

I will speak about a central question in higher algebra (aka brave new algebra), namely which rings or schemes admit 'higher models', that is lifts to the sphere spectrum. This question is in some sense very classical, but there are many open questions. These questions are closely related to questions about higher versions of prismatic cohomology and delta ring, asked e.g. by Scholze and Lurie. Concretely we will consider the case of group algebras and explain how to understand maps between lifts of group algebras to the sphere spectrum. The results we present are joint with Carmeli and Yuan and on the prismatic side with Antieau and Krause.

Rough path theory provides a framework for the study of nonlinear systems driven by highly oscillatory (deterministic) signals. The corresponding analysis is inherently distinct from that of classical stochastic calculus, and neither theory alone is able to satisfactorily handle hybrid systems driven by both rough and stochastic noise. The introduction of the stochastic sewing lemma (Khoa Lê, 2020) has paved the way for a theory which can efficiently handle such hybrid systems. In this talk, we will discuss how this can be done in a general setting which allows for jump discontinuities in both sources of noise.

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.