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Let G be a reductive group defined over a local field of characteristic 0 (real or p-adic). By Harish-Chandra’s regularity theorem, the character Θ_π of an irreducible representation π of G is given by a locally integrable function f_π on G. It turns out that f_π has even better integrability properties, namely, it is locally L^{1+r}-integrable for some r>0. This gives rise to a new singularity invariant of representations \e_π by considering the largest such r.

We explore \e_π, show it is bounded below only in terms of the group G, and calculate it in the case of a p-adic GL(n). To do so, we relate \e_π to the integrability of Fourier transforms of nilpotent orbital integrals appearing in the local character expansion of Θ_π. As a main technical tool, we use explicit resolutions of singularities of certain hyperplane arrangements. We obtain bounds on the multiplicities of K-types in irreducible representations of G for a p-adic G and a compact open subgroup K.

Based on a joint work with Itay Glazer and Julia Gordon.

Gaussian processes provide a powerful probabilistic kernel learning framework, which allows high-quality nonparametric learning via methods such as Gaussian process regression. Nevertheless, its learning phase requires unrealistic massive computations for large datasets. In this talk, we present a quadrature-based approach for scaling up Gaussian process regression via a low-rank approximation of the kernel matrix. The low-rank structure is utilized to achieve effective hyperparameter learning, training, and prediction. Our Gauss-Legendre features method is inspired by the well-known random Fourier features approach, which also builds low-rank approximations via numerical integration. However, our method is capable of generating high-quality kernel approximation using a number of features that is poly-logarithmic in the number of training points, while similar guarantees will require an amount that is at the very least linear in the number of training points when using random Fourier features. The utility of our method for learning with low-dimensional datasets is demonstrated using numerical experiments.

The Bethe-Gauge Correspondence (BGC) of Nekrasov and Shatashvili, linking quantum integrable spin chains to two-dimensional supersymmetric gauge theories with N=2 supersymmetry, stands out as a significant instance of the deep connection between supersymmetric gauge theories and integrable models. In this talk, I will delve into this correspondence and its origins for superspin chains. To achieve this, I will first elucidate the Bethe Side and its corresponding Gauge Side of the BGC. Subsequently, it becomes evident that the BGC can be naturally realized within String Theory. I will initially outline the brane configuration for the realization of the Gauge Side. Through the use of string dualities, this brane configuration will be mapped to another, embodying the Bethe Side of the correspondence. The 4D Chern-Simons theory plays a crucial role in this latter duality frame, elucidating the integrability of the Bethe Side. Lastly, I will elaborate on computing the main object of interest for integrable superspin chains—the R-matrix—from the BGC. While this provides a rather comprehensive picture of the correspondence, some important questions remain for further clarification. I will summarize some of the most interesting ones at the end of the talk.

In a recent preprint, together with Bailleul and Chevyrev we introduced a class of random fields which try to model the basic properties of quantum fields. I will try to explain the basic ideas and some of the many open problems.

To read the preprint, please click here.

Busenberg and Travis suggested in 1983 a population system that exhibits complete segregation of the species. This system can be rigorously derived from interacting particle systems in a mean-field-type limit. It consists of parabolic cross-diffusion equations with an indefinite diffusion matrix. It is known that this system can be formulated in terms of so-called entropy variables such that the transformed equations possess a positive semidefinite diffusion matrix. We consider in this talk the case of incomplete diffusion, which means that the diffusion matrix has zero eigenvalues, and the problem is not parabolic in the sense of Petrovskii. 

We show that the cross-diffusion equations can be written as a normal form of symmetric hyperbolic-parabolic type beyond the Kawashima-Shizuta theory. Using results for symmetric hyperbolic systems, we prove the existence of a unique local classical solution. As solutions may become discontinuous in finite time, only global solutions with very low regularity can be expected. We prove the existence of global dissipative measure-valued solutions satisfying a weak-strong uniqueness property. The proof is based on entropy methods and a finite-volume approximation with a mesh-dependent artificial diffusion. 

Many congruences between modular forms (or at least their q-expansions) can be explained by the theory of $p$-adic families of modular forms. In this talk, I will discuss properties of eigenvarieties, a geometric interpretation of the idea of $p$-adic families. In particular, focusing initially on the well-understood case of (elliptic) modular forms, before delving into the considerably murkier world of Bianchi modular forms. In this second case, this work gives numerical verification of a couple of conjectures, including BSD by work of Loeffler and Zerbes.

We propose a novel generative model for time series based on Schrödinger bridge (SB) approach. This consists in the entropic interpolation via optimal transport between a reference probability measure on path space and a target measure consistent with the joint data distribution of the time series. The solution is characterized by a stochastic differential equation on finite horizon with a path-dependent drift function, hence respecting  the temporal dynamics of the time series distribution. We  estimate the drift function from data samples by nonparametric, e.g. kernel regression methods,  and the simulation of the SB diffusion  yields new synthetic data samples of the time series. The performance of our generative model is evaluated through a series of numerical experiments.  First, we test with autoregressive models, a GARCH Model, and the example of fractional Brownian motion,  and measure the accuracy of our algorithm with marginal, temporal dependencies metrics, and predictive scores. Next, we use our SB generated synthetic samples for the application to deep hedging on real-data sets. 

Let $(M, g)$ be a Riemannian manifold equipped with a calibration $k$-form $\alpha$. In earlier work with Cheng and Madnick (AJM 2021), we studied the analytic properties of a special class of $k$-harmonic maps into $M$ satisfying a first order nonlinear PDE, whose images (away from a critical set) are $\alpha$-calibrated submanifolds of $M$. We call these maps Smith immersions, as they were originally introduced in an unpublished preprint of Aaron Smith. They have nice properties related to conformal geometry, and are higher-dimensional analogues of the $J$-holomorphic map equation. In new joint work (arXiv:2311.14074) with my PhD student Anton Iliashenko, we have obtained analogous results for maps out of $M$. Slightly more precisely, we define a special class of $k$-harmonic maps out of $M$, satisfying a first order nonlinear PDE, whose fibres (away from a critical set) are $\alpha$-calibrated submanifolds of $M$. We call these maps Smith submersions. I will give an introduction to both of these sets of equations, and discuss many future questions.

Mathematical methods are developed to characterize the asymptotics of recurrent neural networks (RNN) as the number of hidden units, data samples in the sequence, hidden state updates, and training steps simultaneously grow to infinity. In the case of an RNN with a simplified weight matrix, we prove the convergence of the RNN to the solution of an infinite-dimensional ODE coupled with the fixed point of a random algebraic equation. 
The analysis requires addressing several challenges which are unique to RNNs. In typical mean-field applications (e.g., feedforward neural networks), discrete updates are of magnitude O(1/N ) and the number of updates is O(N). Therefore, the system can be represented as an Euler approximation of an appropriate ODE/PDE, which it will converge to as N → ∞. However, the RNN hidden layer updates are O(1). Therefore, RNNs cannot be represented as a discretization of an ODE/PDE and standard mean-field techniques cannot be applied. Instead, we develop a fixed point analysis for the evolution of the RNN memory state, with convergence estimates in terms of the number of update steps and the number of hidden units. The RNN hidden layer is studied as a function in a Sobolev space, whose evolution is governed by the data sequence (a Markov chain), the parameter updates, and its dependence on the RNN hidden layer at the previous time step. Due to the strong correlation between updates, a Poisson equation must be used to bound the fluctuations of the RNN around its limit equation. These mathematical methods allow us to prove a neural tangent kernel (NTK) limit for RNNs trained on data sequences as the number of data samples and size of the neural network grow to infinity.

Say we want to view the function-Rips multifiltration as an estimator. Then, what is the target? And what kind of consistency, bias, or convergence rate, should we expect? In this talk I will present on-going joint work with Ethan André (Ecole Normale Supérieure) that aims at laying the algebro-topological ground to start answering these questions.

Motile cells inside living tissues often encounter junctions, where their path branches into several alternative directions of migration. We present a theoretical model of cellular polarization for cells migrating along one-dimensional lines, exhibiting diverse migration modes. When arriving at a symmetric Y-junction and extending protrusions along the different paths that emanate from the junction. The model predicts the spontaneous emergence of deterministic oscillations between competing protrusions, whereby the cellular polarization and growth alternates between the competing protrusions. These predicted oscillations are found experimentally for two different cell types, noncancerous endothelial and cancerous glioma cells, migrating on patterned network of thin adhesive lanes with junctions. Finally we present an analysis of the migration modes of multicellular "trains" along one-dimensional tracks.

The Junior Algebra and Representation Theory Seminar will kick-off the start of Hilary term with a social event in the common room. Come to catch up with your fellow students and maybe play a board game or two. Afterwards we'll have lunch together.

We introduce a methodology based on Multireference Alignment (MRA) for lead-lag detection in multivariate time series, and demonstrate its applicability in developing trading strategies. Specifically designed for low signal-to-noise ratio (SNR) scenarios, our approach estimates denoised latent signals from a set of time series. We also investigate the impact of clustering the time series on the recovery of latent signals. We demonstrate that our lead-lag detection module outperforms commonly employed cross-correlation-based methods. Furthermore, we devise a cross-sectional trading strategy that capitalizes on the lead-lag relationships uncovered by our approach and attains significant economic benefits. Promising backtesting results on daily equity returns illustrate the potential of our method in quantitative finance and suggest avenues for future research.

In ring theory, Morita equivalence is an invariant for many properties, generalising the isomorphism of commutative rings. A strong Morita equivalence for selfadjoint operator algebras was introduced by Rieffel in the 60s, and works as a correspondence between their representations. In the past 30 years, there has been an interest to develop a similar theory for nonselfadjoint operator algebras and operator spaces with much success. Taking motivation from recent work of Connes and van Suijlekom, we will present a Morita theory for operator systems. We will give equivalent characterizations of Morita equivalence via Morita contexts, bihomomoprhisms and stable isomorphisms, while we will highlight properties that are preserved in this context. Time permitted we will provide applications to rigid systems, function systems and non-commutative graphs. This is joint work with George Eleftherakis and Ivan Todorov.

We present a general framework for preconditioning Hermitian positive definite linear systems based on the Bregman log determinant divergence. This divergence provides a measure of discrepancy between a preconditioner and a target matrix, giving rise to

the study of preconditioners given as the sum of a Hermitian positive definite matrix plus a low-rank correction. We describe under which conditions the preconditioner minimises the $\ell^2$ condition number of the preconditioned matrix, and obtain the low-rank 

correction via a truncated singular value decomposition (TSVD). Numerical results from variational data assimilation (4D-VAR) support our theoretical results.

We also apply the framework to approximate factorisation preconditioners with a low-rank correction (e.g. incomplete Cholesky plus low-rank). In such cases, the approximate factorisation error is typically indefinite, and the low-rank correction described by the Bregman divergence is generally different from one obtained as a TSVD. We compare these two truncations in terms of convergence of the preconditioned conjugate gradient method (PCG), and show numerous examples where PCG converges to a small tolerance using the proposed preconditioner, whereas PCG using a TSVD-based preconditioner fails. We also consider matrices arising from interior point methods for linear programming that do not admit such an incomplete factorisation by default, and present a robust incomplete Cholesky preconditioner based on the proposed methodology.

The talk is based on papers with Martin S. Andersen (DTU).

During the last fifteen years, there has been a paradigm shift in the continuum modelling of granular materials; most notably with the development of rheological models, such as the μ(I)-rheology (where μ is the friction and I is the inertial number), but also with significant advances in theories for particle segregation. This talk details theoretical and numerical frameworks (based on OpenFOAM®) which unify these disconnected endeavours. Coupling the segregation with the flow, and vice versa, is not only vital for a complete theory of granular materials, but is also beneficial for developing numerical methods to handle evolving free surfaces. This general approach is based on the partially regularized incompressible μ(I)-rheology, which is coupled to a theory for gravity/shear-driven segregation (Gray & Ancey, J. Fluid Mech., vol. 678, 2011, pp. 353–588). These advection–diffusion–segregation equations describe the evolving concentrations of the constituents, which then couple back to the variable viscosity in the incompressible Navier–Stokes equations. A novel feature of this approach is that any number of differently sized phases may be included, which may have disparate frictional properties. The model is used to simulate the complex particle-size segregation patterns that form in a partially filled triangular rotating drum. There are many other applications of the theory to industrial granular flows, which are the second most common material used after fluids. The same processes also occur in geophysical flows, such as snow avalanches, debris flows and dense pyroclastic flows. Depth-averaged models, that go beyond the μ(I)-rheology, will also be derived to capture spontaneous self-channelization and levee formation, as well as complex segregation-induced flow fingering effects, which enhance the run-out distance of these hazardous flows.

The Ivanov conjecture is equivalent to the statement that every covering map of surfaces has the so-called Putman-Wieland property. I will discuss my recent work with Vlad Marković, where we prove it for asymptotically all coverings as the degree grows. I will give some overview of our main tool: spectral geometry, which is related to objects like the heat kernel of a hyperbolic surface, or Cheeger connectivity constant.

The grazing limit of the Boltzmann equation to Landau equation is well-known and has been justified by using cutoff near the grazing angle with some suitable scaling. In this talk, we will present a new approach by applying a natural scaling on the Boltzmann equation. The proof is based on an improved well-posedness theory for the Boltzmann equation without angular cutoff in the regime with an optimal range of parameters so that the grazing limit can be justified directly that includes the Coulomb potential. With this new understanding, the scaled Boltzmann operator in fact can be decomposed into two parts. The first one converges to the Landau operator when the parameter of deviation angle tends to its singular value and the second one vanishes in the limit. Hence, the scaling and limiting process exactly capture the grazing collisions. The talk is based on a recent joint work with Yu-Long Zhou.

A central question in infinite group theory is to determine how much global information about a group is encoded in its set of finite quotients. In this talk, we will discuss this problem in the case of algebraically fibered groups, which naturally generalise fundamental groups of compact manifolds that fiber over the circle. The study of such groups exploits the relationships between the geometry of the classifying space, the dynamics of the monodromy map, and the algebra of the group, and as such draws from all of these areas.