Past

We study the distribution of eigenvalues of kernel random matrices where each element is the empirical covariance between the feature map evaluations of a random fully-connected neural network. We show that, under mild assumptions on the non-linear activation function, namely Lipschitz continuity and measurability, the limiting spectral distribution can be written as successive free multiplicative convolutions between the Marchenko-Pastur law and a nonrandom measure specific to the neural network. The latter has no known analytical expression but can be simulated empirically, separately from the random matrices of interest.

Motivated by the desire to construct large independent sets in random graphs, Karp and Sipser modified the usual greedy construction to yield an algorithm that outputs an independent set with a large cardinal called the Karp-Sipser core. When run on the Erdős-Rényi $G(n,c/n)$ random graph, this algorithm is optimal as long as $c < e$. We will present the proof of a physics conjecture of Bauer and Golinelli (2002) stating that at criticality, the size of the Karp-Sipser core is of order $n^{3/5}$. Along the way we shall highlight the similarities and differences with the usual greedy algorithm and the $k$-core algorithm.
Based on a joint work with Nicolas Curien and Thomas Budzinski.

The Dehn function of a finitely presented group provides a quantitative measure for the difficulty of detecting if a word in its generators represents the trivial element of the group. By work of Gersten, Holt and Riley the Dehn function of a nilpotent group of class $c$ is bounded above by $n^{c+1}$. However, we are still far from determining the precise Dehn functions of all nilpotent groups. In this talk, I will explain recent results that allow us to determine the Dehn functions of large classes of nilpotent groups arising as central products. As a consequence, for every $k>2$, we obtain many pairs of finitely presented $k$-nilpotent groups with bilipschitz asymptotic cones, but with different Dehn functions. This shows that Dehn functions can distinguish between nilpotent groups with the same asymptotic cone, making them interesting in the context of the conjectural quasi-isometry classification of nilpotent groups.  This talk is based on joint works with García-Mejía, Pallier and Tessera.

A $k$-block in a graph is a set of $k$ vertices every two of which are joined by $k$ vertex disjoint paths. By a result of Weissauer, graphs with no $k$-blocks admit tree-decompositions with especially useful structure. While several constructions show that it is probably very difficult to characterize induced subgraph obstructions for bounded tree width, a lot can be said about graphs with no $k$-blocks. On the other hand, forbidding induced subgraphs places significant restrictions on the structure of a $k$-block in a graphs. We will discuss this phenomenon and its consequences in the study of tree-decompositions in classes of graphs defined by forbidden induced subgraphs.

The resolutions of Slodowy slices S͂_e are symplectic varieties that contain the Springer fiber (G/B)_e as a Lagrangian subvariety. In joint work with R. Bezrukavnikov, M. McBreen, and Z. Yun, we construct analogues of these spaces for homogeneous affine Springer fibers. We further understand the categories of microlocal sheaves in these symplectic spaces supported on the affine Springer fiber as some categories of coherent sheaves.

In this talk I will mostly focus on the case of the homogeneous element ts for s a regular semisimple element and will discuss some relations of these categories with the small quantum group providing a categorification of joint work with R.Bezrukavnikov, P. Shan and E. Vasserot.

We shall consider a crossing equation of the Euclidean conformal group in terms of conformal partial waves and in particular, a position independent representation of this equation. We shall briefly discuss the relevance of this equation to the problem of cosmological bootstrap. Thereafter, we shall sketch the derivation of the Biedenharn-Eliiot identity (a pentagon identity) for the 6j symbols of the conformal group and show how this provides us with an exact solution to said crossing equation. For the conformal group (which is non-compact), this involves some careful bookkeeping of the spinning representations. Finally, we shall discuss some consistency checks on the result obtained, and some open questions. 

Plateau's problem asks whether every boundary curve in 3-space is spanned by an area minimizing surface. Various interpretations of this problem have been solved using eg. geometric measure theory. Froehlich and Struwe proposed a PDE approach, in which the desired surface is produced using smooth sections of a twisted line bundle over the complement of the boundary curve. The idea is to consider sections of this bundle which minimize an analogue of the Allen--Cahn functional (a classical model for phase transition phenomena) and show that these concentrate energy on a solution of Plateau's problem. After some background on the link between phase transition models and minimal surfaces, I will describe new work with Marco Guaraco in which we produce smooth solutions of Plateau's problem using this approach. 

In 2019 Masser and Zannier proved that "most" abelian varieties over the algebraic numbers are not isogenous to the jacobian of any curve (where "most" refers to an ordering by some suitable height function). We will see how this result fits in the general Zilber-Pink Conjecture picture and we discuss some (rather concrete) analogous problems in a power of the modular curve Y(1).

Given two mathematical objects, the most basic question is whether they are the same. We will discuss this question for triangulations of three-manifolds. In practice there is fast software to answer this question and theoretically the problem is known to be decidable. However, our understanding is limited and known theoretical algorithms could have extremely long run-times. I will describe a programme to show that the 3-manifold homeomorphism problem is in the complexity class NP, and discuss the important sub-case of Seifert fibered spaces. 

Neural SDEs are continuous-time generative models for sequential data. State-of-the-art performance for irregular time series generation has been previously obtained by training these models adversarially as GANs. However, as typical for GAN architectures, training is notoriously unstable, often suffers from mode collapse, and requires specialised techniques such as weight clipping and gradient penalty to mitigate these issues. In this talk, I will introduce a novel class of scoring rules on path space based on signature kernels and use them as an objective for training Neural SDEs non-adversarially. The strict properness of such kernel scores and the consistency of the corresponding estimators, provide existence and uniqueness guarantees for the minimiser. With this formulation, evaluating the generator-discriminator pair amounts to solving a system of linear path-dependent PDEs which allows for memory-efficient adjoint-based backpropagation. Moreover, because the proposed kernel scores are well-defined for paths with values in infinite-dimensional spaces of functions, this framework can be easily extended to generate spatiotemporal data. This procedure permits conditioning on a rich variety of market conditions and significantly outperforms alternative ways of training Neural SDEs on a variety of tasks including the simulation of rough volatility models, the conditional probabilistic forecasts of real-world forex pairs where the conditioning variable is an observed past trajectory, and the mesh-free generation of limit order book dynamics.

We construct vertex algebras associated to divisors $S$ in toric Calabi-Yau threefolds $Y$, satisfying conjectures of Gaiotto-Rapcak and Feigin-Gukov, and in particular such that the characters of these algebras are given by a local analogue of the Vafa-Witten partition function of the underlying reduced subvariety $S^{red}$. These results are part of a broader program to establish a dictionary between the enumerative geometry of coherent sheaves on surfaces and Calabi-Yau threefolds, and the representation theory of vertex algebras and affine Yangian-type quantum groups.

How do you make the most of graduate supervisions?  Whether you are a first year graduate wanting to learn about how to manage meetings with your supervisor, or a later year DPhil student, postdoc or faculty member willing to share their experiences and give advice, please come along to this informal discussion led by DPhil students for the first Fridays@4 session of the term.  You can also continue the conversation and learn more about graduate student life at Oxford at Happy Hour afterwards.

Various constructions relevant to practical problems such as clustering and graph classification give rise to multiparameter persistence modules (MPPM), that is, linear representations of non-totally ordered sets. Much of the mathematical interest in multiparameter persistence comes from the fact that there exists no tractable classification of MPPM up to isomorphism, meaning that there is a lot of room for devising invariants of MPPM that strike a good balance between discriminating power and complexity of their computation. However, there is no consensus on what type of information we want these invariants to provide us with, and, in particular, there seems to be no good notion of “global” or “high persistence” features of MPPM.

With the goal of substantiating these claims, as well as making them more precise, I will start with an overview of some of the known invariants of MPPM, including joint works with Bauer and Oudot. I will then describe recent work of Bjerkevik, which contains relevant open questions and which will help us make sense of the notion of global feature in multiparameter persistence.

Many natural and social phenomena involve individual agents coming together to create group dynamics, whether the agents are drivers in a traffic jam, cells in a developing tissue, or locusts in a swarm. Here I will focus on two examples of such emergent behavior in biology, specifically cell interactions during pattern formation in zebrafish skin and gametophyte development in ferns. Different modeling approaches provide complementary insights into these systems and face different challenges. For example, vertex-based models describe cell shape, while more efficient agent-based models treat cells as particles. Continuum models, which track the evolution of cell densities, are more amenable to analysis, but it is often difficult to relate their few parameters to specific cell interactions. In this talk, I will overview our models of cell behavior in biological patterns and discuss our ongoing work on quantitatively relating different types of models using topological data analysis and data-driven techniques.

We will kick off the start of the academic year and the Junior Algebra and Representation Theory seminar (JART) with a fun social event in the common room. Come catch up with your fellow students about what happened over the summer, meet the new students and play some board games. We'll go for lunch together afterwards.

This is a report of joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams. Based on random matrix theory, Conrey-Farmer-Keating-Rubinstein-Snaith have conjectured precise asymptotics for moments of families of quadratic L-functions over number fields. There is an extremely similar function field analogue, worked out by Andrade-Keating. I will explain that one can relate this problem to understanding the homology of the braid group with symplectic coefficients. With Bergström-Diaconu-Westerland we compute the stable homology groups of the braid groups with these coefficients, together with their structure as Galois representations. We moreover show that the answer matches the number-theoretic predictions. With Miller-Patzt-Randal-Williams we prove an improved range for homological stability with these coefficients. Together, these results imply the conjectured asymptotics for all moments in the function field case, for all sufficiently large (but fixed) q.

A Path Shadowing Monte-Carlo method provides prediction of future paths given any generative model.

At a given date, it averages future quantities over generated price paths whose past history matches, or “shadows”, the actual (observed) history.

We test our approach using paths generated from a maximum entropy model of financial prices,

based on the recently introduced “Scattering Spectra” which are multi-scale analogues of the standard skewness and kurtosis.

This model promotes diversity of generated paths while reproducing the main statistical properties of financial prices, including stylized facts on volatility roughness.

Our method yields state-of-the-art predictions for future realized volatility. It also allows one to determine conditional option smiles for the S&P500.

These smiles depend only on the distribution of the price process, and are shown to outperform both the current version of the Path Dependent Volatility model and the option market itself.

This work considers weighted and preconditioned GMRES. The objective is to provide a way of choosing the preconditioner and the inner product, also called weight, that ensure fast convergence. The main focus of the article is on Hermitian preconditioning (even for non-Hermitian problems).

It is indeed proposed to choose a Hermitian preconditioner H, and to apply GMRES in the inner product induced by H. If moreover, the problem matrix A is positive definite, then a new convergence bound is proved that depends only on how well H preconditions the Hermitian part of A, and on a measure of how non-Hermitian A is. In particular, if a scalable preconditioner is known for the Hermitian part of A, then the proposed method is also scalable. I will also illustrate this result numerically.

Over the years, I've often taught a first course in asymptotics and perturbation methods, even though I don't know much about the subject. In this talk, I'll discuss a textbook example of a singularly perturbed nonlinear boundary-value problem that has revealed delightful new surprises, every time I teach it. These include a pitchfork bifurcation in the number of solutions as one varies the small parameter, and transcendentally small terms in the solutions' initial conditions that can be calculated by elementary means.

If some structure, mathematical or otherwise, is giving you grief, then often the first thing to do is to attempt to break the offending object down into (finitely many) simpler pieces.

In group theory, when we speak of questions of *accessibility* we are referring to the ability to achieve precisely this. The idea of an 'accessible group' was first coined by Terry Wall in the 70s, and since then has left quite a mark on our field (and others). In this talk I will introduce the toolbox required to study accessibility, and walk you and your groups through some reasons to be accessible.