Thu, 29 Feb 2024

11:00 - 12:00
C3

Coherent group actions

Martin Bays
(University of Oxford)
Abstract

I will discuss aspects of some work in progress with Tingxiang Zou, in which we continue the investigation of pseudofinite sets coarsely respecting structures of algebraic geometry, focusing on algebraic group actions. Using a version of Balog-Szemerédi-Gowers-Tao for group actions, we find quite weak hypotheses which rule out non-abelian group actions, and we are applying this to obtain new Elekes-Szabó results in which the general position hypothesis is fully weakened in one co-ordinate.

Wed, 28 Feb 2024

16:00 - 17:00
L6

Revisiting property (T)

Ismael Morales
(University of Oxford)
Abstract

Property (T) was introduced by Kazhdan in the sixties to show that lattices in higher rank semisimple Lie groups are finitely generated. We will discuss some classical examples of groups that satisfy this property, with a particular focus on SL(3, R).

Wed, 28 Feb 2024
15:00
Lecture room 5

Mathematics of magic angles for twisted bilayer graphene.

Prof Maciej Zworski
(University of California, Berkeley)
Further Information

This is a joint seminar with Random Matrix Theory and Oxford Centre for Nonlinear Partial Differential Equations.

Abstract

Magic angles refer to a remarkable theoretical (Bistritzer--MacDonald, 2011) and experimental (Jarillo-Herrero et al 2018) discovery, that two sheets of graphene twisted by a certain (magic) angle display unusual electronic properties such as superconductivity.

 

Mathematically, this is related to having flat bands of nontrivial topology for the corresponding periodic Hamiltonian and their existence be shown for the chiral model of twisted bilayer graphene (Tarnopolsky-Kruchkov-Vishwanath, 2019). A spectral characterization of magic angles (Becker--Embree--Wittsten--Z, 2021, Galkowski--Z, 2023) also produces complex values and the distribution of their reciprocals looks remarkably like a distribution of scattering resonances for a two dimensional problem, with the real magic angles corresponding to anti-bound states. I will review various results on that distribution as well as on the properties of the associated eigenstates.

 

The talk is based on joint works with S Becker, M Embree, J Galkowski, M Hitrik, T Humbert and J Wittsten

Wed, 28 Feb 2024
12:00
L6

Non-regular spacetime geometry, curvature and analysis

Clemens Saemann
(Mathematical Institute, University of Oxford)
Abstract

I present an approach to Lorentzian geometry and General Relativity that does neither rely on smoothness nor
on manifolds, thereby leaving the framework of classical differential geometry. This opens up the possibility to study
curvature (bounds) for spacetimes of low regularity or even more general spaces. An analogous shift in perspective
proved extremely fruitful in the Riemannian case (Alexandrov- and CAT(k)-spaces). After introducing the basics of our
approach, we report on recent progress in developing a Sobolev calculus for time functions on such non-smooth
Lorentzian spaces. This seminar talk can also be viewed as a primer and advertisement for my mini course in
May: Current topics in Lorentzian geometric analysis: Non-regular spacetimes

Tue, 27 Feb 2024
16:00
L6

Dynamics in interlacing arrays, conditioned walks and the Aztec diamond

Theodoros Assiotis
(University of Edinburgh)
Abstract

I will discuss certain dynamics of interacting particles in interlacing arrays with inhomogeneous, in space and time, jump probabilities and their relations to conditioned random walks and random tilings of the Aztec diamond.

Tue, 27 Feb 2024

16:00 - 17:00
C2

Simplicity of crossed products by FC-hypercentral groups

Shirly Geffen
(Munster, DE)
Abstract

Results from a few years ago of Kennedy and Schafhauser attempt to characterize the simplicity of reduced crossed products, under an assumption which they call vanishing obstruction. 

However, this is a strong condition that often fails, even in cases of finite groups acting on finite dimensional C*-algebras. In this work, we give complete C*-dynamical characterization, of when the crossed product is simple, in the setting of FC-hypercentral groups. 

This is a large class of amenable groups that, in the finitely-generated setting, is known to coincide with the set of groups with polynomial growth.

Tue, 27 Feb 2024

15:30 - 16:30
Online

Discrepancy of graphs

István Tomon
(Umea University)
Further Information

Part of the Oxford Discrete Maths and Probability Seminar, held via Zoom. Please see the seminar website for details.

Abstract

The positive discrepancy of a graph $G$ is the maximum surplus of edges in an induced subgraph of $G$ compared to its expected size. This quantity is closely related to other well studied parameters, such as the minimum bisection and the spectral gap. I will talk about the extremal behavior of the positive discrepancy among graphs with given number of vertices and average degree, uncovering a surprising pattern. This leads to an almost complete solution of a problem of Alon on the minimum bisection and let's us extend the Alon-Boppana bound on the second eigenvalue to dense graphs.

Joint work with Eero Räty and Benny Sudakov.

Tue, 27 Feb 2024
15:00
L6

Sublinear rigidity of lattices in semisimple Lie groups

Ido Grayevsky
Abstract

I will talk about the coarse geometry of lattices in real semisimple Lie groups. One great result from the 1990s is the quasi-isometric rigidity of these lattices: any group that is quasi-isometric to such a lattice must be, up to some minor adjustments, isomorphic to lattice in the same Lie group. I present a partial generalization of this result to the setting of Sublinear Bilipschitz Equivalences (SBE): these are maps that generalize quasi-isometries in some 'sublinear' fashion.

Tue, 27 Feb 2024

14:00 - 15:00
Online

Geodesics networks in the directed landscape

Duncan Dauvergne
(University of Toronto)
Further Information

Part of the Oxford Discrete Maths and Probability Seminar, held via Zoom. Please see the seminar website for details.

Abstract

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.

Tue, 27 Feb 2024

14:00 - 15:00
L5

Modular Reduction of Nilpotent Orbits

Jay Taylor
(University of Manchester)
Abstract

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

We may assume that our group 𝐺𝕜 = 𝐺 × 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 ⨂ 𝕜, as a module for 𝐺𝕜, is 𝔤𝕜 or 𝔤𝕜* respectively.

It is known that each 𝐺 -orbit 𝒪 ⊆ 𝒩(𝑉) contains a representative ξ ∈ 𝑉 in the ℤ-form. Reducing ξ one gets an element ξ𝕜 ∈ 𝑉𝕜 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).

Tue, 27 Feb 2024
12:30
L4

Page curves and replica wormholes from chaotic dynamics

Andrew Rolph
(Vrije U., Brussels)
Abstract

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.

Tue, 27 Feb 2024
11:00
L5

Deep Transfer Learning for Adaptive Model Predictive Control

Harrison Waldon
(Oxford Man Institute)
Abstract

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.

Mon, 26 Feb 2024
16:00
L2

The Metaplectic Representation is Faithful

Christopher Chang, Simeon Hellsten, Mario Marcos Losada, and Sergiu Novac.
(University of Oxford)
Abstract

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.

Mon, 26 Feb 2024
15:30
L4

Morava K-theory of infinite groups and Euler characteristic

Irakli Patchkoria
(University of Aberdeen)
Abstract

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.

Mon, 26 Feb 2024
15:30
Lecture room 5

McKean-Vlasov S(P)Des with additive noise

Professor Michela Ottobre
(Heriot Watt University)
Abstract

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.  

Mon, 26 Feb 2024
14:15
L4

Hessian geometry of $G_2$-moduli spaces

Thibault Langlais
(Oxford)
Abstract

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.

Mon, 26 Feb 2024

14:00 - 15:00
Lecture Room 3

Fantastic Sparse Neural Networks and Where to Find Them

Dr Shiwei Liu
(Maths Institute University of Oxford)
Abstract

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).

Fri, 23 Feb 2024
16:00
L1

Demystifying careers for mathematicians in the Civil Service

Sarah Livermore (Department for Business and Trade)
Abstract

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.

Fri, 23 Feb 2024
14:30
C6

Flat from anti de Sitter - a Carrollian perspective

Prof Marios Petropoulos
(Ecole Polytechnique, Paris)
Abstract

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.  

Fri, 23 Feb 2024

12:00 - 13:00
Quillen Room

Homotopy type of SL2 quotients of simple simply connected complex Lie groups

Dylan Johnston
(University of Warwick)
Abstract
We say an element X in a Lie algebra g is nilpotent if ad(X) is a nilpotent operator. It is known that G_{ad}-orbits of nilpotent elements of a complex semisimple Lie algebra g are in 1-1 correspondence with Lie algebra homomorphisms sl2 -> g, which are in turn in 1-1 correspondence with Lie group homomorphisms SL2 -> G.
Thus, we may denote the homogeneous space obtained by quotienting G by the image of a Lie group homomorphism SL2 -> G by X_v, where v is a nilpotent element in the corresponding G_{ad}-orbit.
In this talk we introduce some algebraic tools that one can use to attempt to classify the homogeneous spaces, X_v, up to homotopy equivalence.
Thu, 22 Feb 2024
18:00
The Auditorium, Citigroup Centre, London, E14 5LB

Frontiers in Quantitative Finance: Statistical Predictions of Trading Strategies in Electronic Markets

Prof Samuel N Cohen
Abstract

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.

Thu, 22 Feb 2024

17:00 - 18:00

Sets that are very large and very small

Asaf Karagila (Leeds)
Abstract
We can compare the relative sizes of sets by using injections or (partial) surjections, but without the axiom of choice we cannot prove that every two sets can be compared. We can use the ordinals to define a notion of size which allows us to determine whether a set is "large" or "small" relative to another. The first is defined by the Hartogs number, which is the least ordinal which does not inject into the set; the second is the Lindenbaum number of a set, which is the first ordinal which is not an image of the set. In this talk we will discuss some basic properties of these numbers and some basic historical results. 

 
In a new work with Calliope Ryan-Smith we showed that given any pair of (infinite) cardinals, we can onstruct a symmetric extension in which there is a set whose Hartogs is the smaller and the Lindenbaum is the larger. Moreover, using the techniques of iterated symmetric extensions, we can realise all possible pairs in a single model.

 
This work appears on arXiv: https://arxiv.org/abs/2309.11409
Thu, 22 Feb 2024
16:00
Lecture Room 4

Tangent spaces of Schubert varieties

Rong Zhou
(University of Cambridge)
Abstract

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.