Past

We consider quadratic deformations of the q-symmetric algebras A_q given by x_i x_j = q_{ij} x_j x_i, for q_{ij} in C*.   We explicitly describe the Hochschild cohomology and compute the weights of the torus action (dilating the x_i variables). We describe new families of filtered deformations of A_q, which are Koszul and Calabi—Yau algebras. This also applies to abelian category deformations of coh(P^n), and for n=3 we give examples having no homogeneous coordinate ring.  We then focus on the case where n is even and the deformations are obtainable from deformation quantisation of toric log symplectic structures on P^n.  In this case we construct formally universal families of quadratic algebras deforming A_q, obtained by tensoring filtered deformations and Feigin—Odesskii elliptic algebras. The universality is a consequence of a beautiful combinatorial classification of deformations via "smoothing diagrams", a collection of disjoint cycles and segments in the complete graph on n vertices, viewed as the dual complex for the coordinate hyperplanes in P^{n-1}.  Already for n=5 there are 40 of these, mostly entirely new. Our proof also applies to deformations of Poisson structures, recovering the P^n case of our previous results on general log symplectic varieties with normal crossings divisors, which motivated this project.  This is joint work with Mykola Matviichuk and Brent Pym.

A cluster graph is a disjoint union of complete graphs. We consider the random $G(n,p)$ graph on $n$ vertices with connection probability $p$, conditioned on the rare event of being a cluster graph. There are three main motivations for our study.

1. For $p = 1/2$, each random cluster graph occurs with the same probability, resulting in the uniform distribution over set partitions. Interpreting such a partition as a graph adds additional structural information.
2. To study how the law of a well-studied object like $G(n,p)$ changes when conditioned on a rare event; an evidence of this fact is that the conditioned random graph overcomes a phase transition at $p=1/2$ (not present in the dense $G(n,p)$ model).
3. The original motivation was an application to community detection. Taking a random cluster graph as a model for a prior distribution of a partition into communities leads to significantly better community-detection performance.

This is joint work with Martijn Gösgens, Lukas Lüchtrath, Elena Magnanini and Élie de Panafieu.

One of the major insights gained from holographic duality is the relation between the physics of black holes and quantum chaotic systems. This relation is made precise in the duality between two dimensional JT gravity and random matrix theory.  In this work, we generalize this to a duality between AdS3 gravity and a random ensemble of approximate CFT's.  The latter is described by a combined tensor and matrix model, describing the OPE coefficients and spectrum of a theory that approximately satisfies the bootstrap constraints.   We show that the Feynman diagrams of the random ensemble produce a sum over 3 manifolds that agrees with the partition function of 3d gravity.  A crucial element of this dictionary is the Virasoro TQFT, which defines the bulk gravitational path integral via the cutting and sewing relations of topological field theory.  Time permitting, we will explain why this TQFT has gravitational edge modes degrees of freedom whose entanglement gives rise to gravitational entropy.

In this talk, we present a graph discretized approximation scheme for diffusions with drift and killing on a complete Riemannian manifold M. More precisely, for a given Schrödinger operator with drift on M having the form A = — Δ — b + V , we introduce a family of discrete time random walks in the  ow generated by the drift b with killing on a sequence of proximity graphs, which are constructed by partitions cutting M into small pieces. As a main result, we prove that the drifted Schrodinger semigroup {e^—tA}_t≥0 is approximated by discrete semigroups generated by the family of random walks with a suitable scale change. This result gives a  nite dimensional summation approximation of a Feynman-Kac type functional integral over M. Furthermore, when M is compact, we also obtain a quantitative error estimate of the convergence.
This talk is based on a joint work with Satoshi Ishiwata (Yamagata University), and the full paper can be found on https://doi.org/10.1007/s00208-024-02809-9.

Course Overview
The course gives an introduction to a topic of current interest in Lorentzian geometic analysis and mathematical General Relativity: an approach to nonregular spacetimes based on a “metric” point of view.
 

Learning Outcomes
Becoming acquainted with Lorentzian length spaces, sectional and Ricci curvature bounds for non-regular Lorentzian spaces and the appropriate techniques.
 

Course Synopsis
Lecture 1a: Review of Lorentzian geometry, spaces of constant curvature, causality theory, singularity theorems.
Lecture 1b: Introduction to Lorentzian length spaces, timelike sectional curvature bounds.

Lecture 2a: Optimal transport, timelike Ricci curvature bounds
Lecture 2b: Sobolev calculus for time functions. Literature: [O’N83, KS18, CM20].
 

Reading group: Depending on student’s interest one could discuss the papers [GKS19, AGKS21, ABS22].

References
[ABS22] L. Aké Hau, S. Burgos, and D. A. Solis. Causal completions as Lorentzian pre-length spaces. General Relativity and Gravitation, 54(9), 2022. doi:10.1007/s10714-022-02980-x.
[AGKS21] S. B. Alexander, M. Graf, M. Kunzinger, and C. Sämann. Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems. Comm. Anal. Geom., to appear, 2021. doi:10.48550/arXiv.1909.09575. arXiv:1909.09575 [math.MG].
[CM20] F. Cavalletti and A. Mondino. Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications. Cambridge Journal of Mathematics, to appear, arXiv:2004.08934 [math.MG], 2020. doi:10.48550/arXiv.2004.08934.
[GKS19] J. D. E. Grant, M. Kunzinger, and C. Sämann. Inextendibility of spacetimes and Lorentzian length spaces. Ann. Global Anal. Geom., 55(1):133–147, 2019. doi:10.1007/s10455-018-9637-x.
[KS18] M. Kunzinger and C. Sämann. Lorentzian length spaces. Ann. Glob. Anal. Geom., 54(3):399–447, 2018. doi:10.1007/s10455-018-9633-1.
[O’N83] B. O’Neill. Semi-Riemannian geometry with applications to relativity, volume 103 of Pure and Applied Mathematics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983.

Should you be interested in taking part in the course, please send an email to @email  by 10 May 2024. 

Functoriality is a key feature in Langlands’ conjectured relationship between automorphic representations and Galois representations; it predicts that certain Galois representations are automorphic, i.e. should come from automorphic representations. We discuss the idea of $p$-adic propagation of automorphy, which seeks to establish the automorphy of everything in a “neighborhood” given the automorphy of something in that neighborhood. The “neighborhoods” that we consider will be the irreducible components of a $p$-adic analytic space called the eigenvariety, which parameterizes $p$-adic automorphic representations. This technique was introduced by Newton and Thorne in their proof of symmetric power functoriality, and can be adapted to investigate similar problems.

We consider the general framework of distributionally robust optimization under a martingale restriction. We provide explicit expressions for model risk sensitivities in this context by considering deviations in the Wasserstein distance and the corresponding adapted one. We also extend the dual formulation to this context.

The quadratic Euler characteristic of an algebraic variety is a (virtual) symmetric bilinear form which refines the topological Euler characteristic and contains interesting arithmetic information when the base field is not algebraically closed. For smooth projective varieties, it has a quite concrete expression in terms of the cup product and Serre duality for Hodge cohomology. However, for singular varieties, it is defined abstractly (using either cut and paste relations or motivic homotopy theory) and is still rather mysterious. I will first introduce this invariant and place it in the broader context of quadratic enumerative geometry. I will then explain some progress on concrete computations, first for symmetric powers (joint with Lenny Taelman) and second for conductor formulas for hypersurface singularities (older results with Marc Levine and Vasudevan Srinivas on the one hand, and joint work in progress with Ran Azouri, Niels Feld, Yonathan Harpaz and Tasos Moulinos on the other).

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.

What makes a good talk? This year, graduate students and postdocs will give a series talks on how to give talks! There may even be a small prize for the audience’s favourite.

If you’d like to have a go at informing, entertaining, or just have an axe to grind about a particularly bad talk you had to sit through, we’d love to hear from you (you can email Ric Wade or ask any of the organizers).
 

The role of tissue stiffness in controlling cell behaviours ranging from proliferation to signalling and activation is by now well accepted. A key focus of experimental studies into mechanotransduction are focal adhesions, localised patches of strong adhesion, where cell signalling has been established to occur. However, these adhesion sites themselves alter the mechanical equilibrium of the system determining the force balance and work done. To explore this I have developed an active matter continuum description of cellular contractility and will discuss recent results on the specific role of spatial positioning of adhesions in mechanotransduction. I show using energy arguments why the experimentally observed arrangements of focal adhesions develop and the implications this has for stiffness sensing and cellular contractility control. I will also show how adhesions play distinct roles in single cells and tissue layers respectively drawing on recent experimental work with Dr JR Davis (Manchester University) and Dr Nic Tapon (Crick Institute) with applications to epithelial layers and organoids.

The orbit method is a fundamental tool to study a finite dimensional solvable Lie algebra g. It relates the annihilators of simple U(g)-module to the coadjoint orbits of the adjoint group on g^* . In my talk, I will extend this story to the Witt algebra – a simple (non-solvable) infinite dimensional Lie algebra which is important in physics and representation theory. I will construct an induced module from an element of W^* and show that its annihilator is a primitive ideal. I will also construct an algebra homomorphism that allows one to relate the orbit method for W to that of a finite dimensional solvable algebra.

The chiralization in the title denotes a certain procedure which turns cluster X-varieties into q-W algebras. Many important notions from cluster and q-W worlds, such as mutations, global functions, screening operators, R-matrices, etc emerge naturally in this context. In particular, we discover new bosonizations of q-W algebras and establish connections between previously known bosonizations. If time permits, I will discuss potential applications of our approach to the study of 3d topological theories and local systems with affine gauge groups. This talk is based on a joint project with J. Shiraishi, J.E. Bourgine, B. Feigin, A. Shapiro, and G. Schrader.

There have been a lot of interests in understanding the behaviour of random multiplicative functions, which are probabilistic models for deterministic arithmetic functions such as the Möbius function and Dirichlet characters. Despite recent advances, the limiting distributions of partial sums of random multiplicative functions remain mysterious even at the conjectural level. In this talk, I shall discuss the so-called $L^2$ regime of twisted sums and provide a precise answer to the distributional problem. This is based on ongoing work with Ofir Gorodetsky.

In this seminar we introduce a portfolio optimisation framework, in which the use of rough path signatures (Lyons, 1998) provides a novel method of incorporating path-dependencies in the joint signal-asset dynamics, naturally extending traditional factor models, while keeping the resulting formulas lightweight, tractable and easily interpretable. Specifically, we achieve this by representing a trading strategy as a linear functional applied to the signature of a path (which we refer to as “Signature Trading” or “Sig-Trading”). This allows the modeller to efficiently encode the evolution of past time-series observations into the optimisation problem. In particular, we derive a concise formulation of the dynamic mean-variance criterion alongside an explicit solution in our setting, which naturally incorporates a drawdown control in the optimal strategy over a finite time horizon. Secondly, we draw parallels between classical portfolio stategies and Sig-Trading strategies and explain how the latter leads to a pathwise extension of the classical setting via the “Signature Efficient Frontier”. Finally, we give explicit examples when trading under an exogenous signal as well as examples for momentum and pair-trading strategies, demonstrated both on synthetic and market data. Our framework combines the best of both worlds between classical theory (whose appeal lies in clear and concise formulae) and between modern, flexible data-driven methods (usually represented by ML approaches) that can handle more realistic datasets. The advantage of the added flexibility of the latter is that one can bypass common issues such as the accumulation of heteroskedastic and asymmetric residuals during the optimisation phase. Overall, Sig-Trading combines the flexibility of data-driven methods without compromising on the clarity of the classical theory and our presented results provide a compelling toolbox that yields superior results for a large class of trading strategies.

This is based on works with Blanka Horvath and Magnus Wiese.

In this talk, we discuss continuous in time dynamics for the problem of approaching the set of zeros of a single-valued monotone and continuous operator V . Such problems are motivated by minimax convexconcave and, in particular, by convex optimization problems with linear constraints. The central role is played by a second-order dynamical system that combines a vanishing damping term with the time derivative of V along the trajectory, which can be seen as an analogous of the Hessian-driven damping in case the operator is originating from a potential. We show that these methods exhibit fast convergence rates for kV (z(t))k as t ! +1, where z( ) denotes the generated trajectory, and for the restricted gap function, and that z( ) converges to a zero of the operator V . For the corresponding implicit and explicit discrete time models with Nesterov’s momentum, we prove that they share the asymptotic features of the continuous dynamics.

Extensions to variational inequalities and fixed-point problems are also addressed. The theoretical results are illustrated by numerical experiments on bilinear games and the training of generative adversarial networks.

What do fiber polymers and ice sheets have in common? They both flow with a directionally dependent - anisotropic - viscosity. This behaviour occurs in other geophysical flows, such as the Earth's mantle, where a material's microstructure affects its large-scale flow. In ice, the alignment of crystal orientations can cause the viscosity to vary by an order of magnitude, consequently having a strong impact on the flow of ice sheets and glaciers. However, the effect of anisotropy on large-scale flow is not well understood, due to a lack of understanding of a) the best physical approximations to model crystal orientations, and b) how crystal orientations affect rheology. In this work, we aim to address both these questions by linking rheology to crystal orientation predictions, and testing a range of models against observations from the Greenland ice sheet. The results show assuming all grains experience approximately the same stress provides realistic predictions, and we suggest a set of equations and parameters which can be used in large-scale models of ice sheets. 

I will present a recent work with G. Kocharyan, where we show the undecidability of the following two problems: given a finitely generated subgroup G of GL(n,Q), a) determine whether G has a non-identity element whose (i,j) entry is equal to zero, and b) determine whether the stabilizer of a given vector in G is non-trivial. Undecidability of problem b) answers a question of Dixon from 1985. The proofs reduce to the undecidability of the word problem for finitely presented groups.

I'll talk about the Morse local-to-global property and try to convince you that is a good property. There are three reasons. Firstly, it is satisfied by many examples of interest. Secondly, it allows to prove many theorems. Thirdly, it sits nicely in the larger program of classifying groups up to quasi-isometry and it has connections with open questions.

A $C^*$-diagonal is a certain commutative subalgebra of a $C^∗$ -algebra with a rich structure. Renault and Kumjian showed that finding a $C^*$ -diagonal of a $C^∗$-algebra is equivalent to realizing the $C^*$-algebra via a groupoid. This establishes a close connection between $C^∗$-diagonals and dynamics and allows one to relate the geometric properties of groupoids to the properties of $C^∗$ -diagonals. 

In this talk, I will explore the unique pure state extension property of an Abelian $C^*$-subalgebra of a 1-dim NCCW complex, the approximation of morphisms between two 1-dim NCCW complexes by $C^*$-diagonal preserving morphisms, and the existence of $C^*$-diagonal in inductive limits of certain 1-dim NCCW complexes.

Coboundary expansion is a notion introduced by Linial and Meshulam, and by Gromov that combines combinatorics topology and linear algebra. Kaufman and Lubotzky observed its relation to "Property testing", and in recent years it has found several applications in theoretical computer science, including for error correcting codes (both classical and quantum), for PCP agreement tests, and even for studying polarization in social networks.

In the talk I will introduce this notion and some of its applications. No prior knowledge is assumed, of course.