Past

Due to a family emergency, the speaker unfortunately had to cancel this talk.

Since their introduction in 2017, Transformers have revolutionized large language models and the broader field of deep learning. Central to this success is the ground-breaking self-attention mechanism. In this presentation, I’ll introduce a mathematical framework that casts this mechanism as a mean-field interacting particle system, revealing a desirable long-time clustering behaviour. This perspective leads to a trove of fascinating questions with unexpected connections to Kuramoto oscillators, sphere packing, Wasserstein gradient flows, and slow dynamics.

Bio: Philippe Rigollet is a Distinguished Professor of Mathematics at MIT, where he serves as Chair of the Applied Math Committee and Director of the Statistics and Data Science Center. His research spans multiple dimensions of mathematical data science, including statistics, machine learning, and optimization, with recent emphasis on optimal transport and its applications. See https://math.mit.edu/~rigollet/ for more information.

This talk is hosted by the AI Reading Group

We investigate pattern formation for a 2D PDE-ODE bulk-cell model, where one or more bulk diffusing species are coupled to nonlinear intracellular
reactions that are confined within a disjoint collection of small compartments. The bulk species are coupled to the spatially segregated
intracellular reactions through Robin conditions across the cell boundaries. For this compartmental-reaction diffusion system, we show that
symmetry-breaking bifurcations leading to stable asymmetric steady-state patterns, as regulated by a membrane binding rate ratio, occur even when
two bulk species have equal bulk diffusivities. This result is in distinct contrast to the usual, and often biologically unrealistic, large
differential diffusivity ratio requirement for Turing pattern formation from a spatially uniform state. Secondly, for the case of one-bulk
diffusing species in R^2, we derive a new memory-dependent ODE integro-differential system that characterizes how intracellular
oscillations in the collection of cells are coupled through the PDE bulk-diffusion field. By using a fast numerical approach relying on the
``sum-of-exponentials'' method to derive a time-marching scheme for this nonlocal system, diffusion induced synchrony is examined for various
spatial arrangements of cells using the Kuramoto order parameter. This theoretical modeling framework, relevant when spatially localized nonlinear
oscillators are coupled through a PDE diffusion field, is distinct from the traditional Kuramoto paradigm for studying oscillator synchronization on
networks or graphs. (Joint work with Merlin Pelz, UBC and UMinnesota).

An explicit first-order drift-randomized Milstein scheme for a regime switching stochastic differential equation is proposed and its bi-stability and rate of strong convergence are investigated for a non-differentiable drift coefficient. Precisely, drift is Lipschitz continuous while diffusion along with its derivative is Lipschitz continuous. Further, we explore the significance of evaluating Brownian trajectories at every switching time of the underlying Markov chain in achieving the convergence rate 1 of the proposed scheme. In this context, possible variants of the scheme, namely modified randomized and reduced randomized schemes, are considered and their convergence rates are shown to be 1/2. Numerical experiments are performed to illustrate the convergence rates of these schemes along with their corresponding non-randomized versions. Further, it is illustrated that the half-order non-randomized reduced and modified schemes outperform the classical Euler scheme.

The BSD conjecture predicts that a rational elliptic curve $E$ has infinitely many points if and only if its $L$-function vanishes at $s=1$.

There are $p$-adic versions of similar phenomena. If $E$ is $p$-ordinary, there is, for example, a $p$-adic analytic analogue $L_p(E,s)$ of the $L$-function, and if $E$ has good reduction, then it has infinitely many rational points iff $L_p(E,1) = 0$. However if $E$ has split multiplicative reduction at $p$ - that is, if $E/\mathbf{Q}_p$ admits a Tate uniformisation $\mathbf{C}_p^{\times}/q^{\mathbf{Z}}$ - then $L_p(E,1) = 0$ for trivial reasons, regardless of $L(E,1)$; it has an 'exceptional zero'. Mazur--Tate--Teitelbaum's exceptional zero conjecture, proved by Greenberg--Stevens in '93, states that in this case the first derivative $L_p'(E,1)$ is much more interesting: it satisfies $L_p'(E,1) = \mathrm{log}(q)/\mathrm{ord}(q) \times L(E,1)/(\mathrm{period})$. In particular, it should vanish iff $L(E,1) = 0$ iff $E(\mathbf{Q})$ is infinite; and even better, it has a beautiful and surprising connection to the Tate period $q$, via the 'L-invariant' $\mathrm{log}(q)/\mathrm{ord}(q)$.

In this talk I will discuss exceptional zero phenomena and L-invariants, and a generalisation of the exceptional zero conjecture to automorphic representations of GL(3). This is joint work in progress with Daniel Barrera and Andrew Graham.

We propose and analyse two structure preserving finite volume schemes to approximate the solutions to a cross-diffusion system with self-consistent electric interactions introduced by Burger, Schlake & Wolfram (2012). This system has been derived thanks to probabilistic arguments and admits a thermodynamically motivated Lyapunov functional that is preserved by suitable two-point flux finite volume approximations. This allows to carry out the mathematical analysis of two schemes to be compared.

This is joint work with Maxime Herda and Annamaria Massimini.

The goal of the talk is to give an overview of the metric theory of currents by Ambrosio-Kirchheim, together with some recent progress in the setting of Banach spaces. Metric currents are a generalization to the metric setting of classical currents. Classical currents are the natural generalization of oriented submanifolds, as distributions play the same role for functions. We present a structure result for 1-metric currents as superposition of 1-rectifiable sets in Banach spaces, which generalizes a previous result by Schioppa. This is based on an approximation result of metric 1-currents with normal 1-currents. This is joint work with D. Bate, J. Takáč, P. Valentine, and P. Wald (Warwick).

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.

The absolute Galois group of ℚₚ determines its field structure: a field K is p-adically closed if and only if its absolute Galois group is isomorphic to that of ℚₚ. This Galois-theoretic characterisation was proved by Koenigsmann in 1995, building on previous work by Arason, Elman, Jacob, Ware, and Pop. Similar results were obtained by Efrat and further developed in his 2006 book.

Our project aims to provide an optimal proof of this characterisation, incorporating improvements and new developments. These include a revised proof strategy; Efrat's construction of valuations via multiplicative stratification; the Galois characterisation of henselianity; systematic use of the standard decomposition; and the function field analogy of Krasner-Kazhdan-Deligne type. Moreover, we replace arguments that use Galois cohomology with elementary ones.

In this talk, I will focus on two key components of the proof: the construction of valuations from rigid elements, and the role of the function field analogy as developed via the non-standard methods of Jahnke-Kartas.

This is joint work with Jochen Koenigsmann and Benedikt Stock.

Right-angled Artin groups (RAAGs) can be viewed as a generalisation of free groups. To what extent, then, do the techniques used to study automorphisms of free groups generalise to the setting of RAAGs? One significant advance in this direction is the construction of 'untwisted Outer space' for RAAGs, a generalisation of the influential Culler-Vogtmann Outer space for free groups. A consequence of this construction is an upper bound on the virtual cohomological dimension of the 'untwisted subgroup' of outer automorphisms of a RAAG. However, this bound is sometimes larger than one expects; I present work showing that, in fact, it can be arbitrarily so, by forming a new complex as a deformation retraction of the untwisted Outer space. In a different direction, another subgroup of interest is that consisting of symmetric automorphisms. Generalising work in the free groups setting from 1989, I present an Outer space for the symmetric automorphism group of a RAAG. A consequence of the proof is a strong finiteness property for many other subgroups of the outer automorphism group.

In stochastic optimal control and conditional generative modelling, a central computational task is to modify a reference diffusion process to maximise a given terminal-time reward. Most existing methods require this reward to be differentiable, using gradients to steer the diffusion towards favourable outcomes. However, in many practical settings, like diffusion bridges, the reward is singular, taking an infinite value if the target is hit and zero otherwise. We introduce a novel framework, based on Malliavin calculus and path-space integration by parts, that enables the development of methods robust to such singular rewards. This allows our approach to handle a broad range of applications, including classification, diffusion bridges, and conditioning without the need for artificial observational noise. We demonstrate that our approach offers stable and reliable training, outperforming existing techniques. 

Understanding the size of the partial sums of the Möbius function is one of the most fundamental problems in analytic number theory. This motivated the 1944 paper of Wintner, where he introduced the concept of a random multiplicative function: a probabilistic model for the Möbius function. In recent years, it has been uncovered that there is an intimate connection between random multiplicative functions and the theory of Gaussian Multiplicative Chaos, an area of probability theory introduced by Kahane in the 1980's. We will survey selected results and discuss recent research on the distribution of partial sums of random multiplicative functions when restricted to integers with a large prime factor.

Quotients of hyperbolic groups (groups that act geometrically on a hyperbolic space) and their generalizations have long been a powerful tool for proving strong algebraic results. In this talk, I will describe the geometry of random quotients of certain of groups, that is, a quotient by a subgroup normally generated by k independent random walks.  I will focus on the class of hierarchically hyperbolic groups (HHGs), a generalization of hyperbolic groups that includes hyperbolic groups, mapping class groups, most CAT(0) cubical groups including right-angled Artin and Coxeter groups, many 3–manifold groups, and various combinations of such groups.  In this context, I will explain why a random quotient of an HHG that does not split as a direct product is again an HHG, definitively showing that the class of HHGs is quite broad.  I will also describe how the result can also be applied to understand the geometry of random quotients of hyperbolic and relatively hyperbolic groups. This is joint work with Giorgio Mangioni, Thomas Ng, and Alexander Rasmussen.

In this talk, we discuss nearly G2 structures, which define positive Einstein metrics, and are, up to scale, critical points of a geometric flow called (modified) Laplacian co-flow. We will discuss a recent joint work with Jason Lotay showing that many of these nearly G2 critical points are unstable for the flow. 

Semidefinite programming (SDP) is a powerful tool in the design of approximation algorithms. After providing a gentle introduction to the basics of this method, I will explore a different facet of SDP and show how it can be used to derive short and elegant proofs of both classical and new estimates related to the MaxCut problem and discrepancy theory in graphs and matrices.

Building on this, I will demonstrate how these results lead to an improved upper bound on the celebrated log-rank conjecture in communication complexity.

I will describe a new method for constructing conformal blocks for the Virasoro vertex algebra with central charge c=1, by "nonabelianization", relating them to conformal blocks for the Heisenberg algebra on a branched double cover. The construction is joint work with Qianyu Hao. Special cases give rise to formulas for tau-functions and solutions of integrable systems of PDE, such as Painleve I and its higher analogues. The talk will be reasonably self-contained (in particular I will explain what a conformal block is).

We are going to study the Ginzburg-Landau energy for two specific geometries, related to the very experiments on fermionic condensates: annuli and strips 

The specific geometry of a strip provides connections between solitons and vortices, called solitonic vortices, which are vortices with a solitonic behaviour in the infinite direction of the strip. Therefore, they are very different from classical vortices which have an algebraic decay at infinity. We show that there exist stationary solutions to the Gross-Pitaevskii equation with k vortices on a transverse line, which bifurcate from the soliton solution as the width of the strip is increased. This is motivated by recent experiments on the instability of solitons by imposing a phase shift in an elongated condensate for bosonic or fermionic atoms.

For annuli, we prescribe a very large degree on the outer boundary and find that either there is a transition from a giant vortex to vortices also in the bulk but tending to the outer boundary.

This is joint work with Ph. Gravejat and E.Sandier for solitonice vortices and Remy Rodiac for annuli.
 

TBC

Temperley-Lieb algebras are certain finite-dimensional algebras coming originally from statistical physics and knot theory. Around 2019, they became one of the first examples of homological stability for algebras (homology is here taken to be certain Tor-groups), when Boyd and Hepworth showed that in low dimensions the homology vanishes. We're now able to give complete calculations of their homology, which has a surprisingly rich structure (and in particular is very far from vanishing). This is joint work in progress with Rachael Boyd, Oscar Randal-Williams, and Robin Sroka. Prerequisites will be minimal: it will be enough to know what Tor is.

Keywords: SDE on the plane, Brownian sheet, path by path uniqueness, space time local time integral, Malliavin calculus

In this talk, we discuss the existence, uniqueness, and regularisation by noise for stochastic differential equations (SDEs) on the plane. These equations can also be interpreted as quasi-linear hyperbolic stochastic partial differential equations (HSPDEs). More specifically, we address path-by-path uniqueness for multidimensional SDEs on the plane, under the assumption that the drift coefficient satisfies a spatial linear growth condition and is componentwise non-decreasing. In the case where the drift is only measurable and uniformly bounded, we show that the corresponding additive HSPDE on the plane admits a unique strong solution that is Malliavin differentiable. Our approach combines tools from Malliavin calculus with variational techniques originally introduced by Davie (2007), which we non-trivially extend to the setting of SDEs on the plane.

This talk is based on a joint works with A. M. Bogso, M. Dieye and F. Proske.