Past

A pair of elliptic curves is said to be $N$-congruent if their mod $N$ Galois representations are isomorphic. We will discuss a construction of the moduli spaces of $N$-congruent elliptic curves, due to Kani--Schanz, and describe how this can be exploited to compute explicit equations. Finally we will outline a proof that there exist infinitely many pairs of elliptic curves with isomorphic mod $12$ Galois representations, building on previous work of Chen and Fisher (in the case where the underlying isomorphism of torsion subgroups respects the Weil pairing).

The strong cosmic censorship conjecture is a fundamental open problem in classical general relativity, first put forth by Roger Penrose in the early 70s. This is essentially the question of whether general relativity is a deterministic theory. Perhaps the most exciting arena where the validity of the conjecture is challenged is the interior of rotating black holes, and there has been a lot of work in the past 50 years in identifying mechanisms ensuring that at least some formulation of the conjecture be true. It turns out that when a nonzero cosmological constant Λ is added to the Einstein equations, these underlying mechanisms change in an unexpected way, and the validity of the conjecture depends on a detailed understanding of subtle aspects of black hole scattering theory, surprisingly involving, in the case of negative Λ, some number theory. Does strong cosmic censorship survive the challenge of non-zero Λ? This talk will try to address this Question!

In this talk I explain the fertile relationship between the foundations of inference and learning and combinatorial geometry.

My presentation contains several powerful examples where famous theorems in discrete geometry answered natural  questions from machine learning and statistical inference:

In this tasting tour I will include the problem of deciding the existence of Maximum likelihood estimator in multiclass logistic regression, the variability of behavior of k-means algorithms with distinct random initializations and the shapes of the clusters, and the estimation of the number of samples in chance-constrained optimization models. These obviously only scratch the surface of what one could do with extra free time. Along the way we will see fascinating connections to the coupon collector problem, topological data analysis, measures of separability of data, and to the computation of Tukey centerpoints of data clouds (a high-dimensional generalization of median). All new theorems are joint work with subsets of the following wonderful folks: T. Hogan, D. Oliveros, E. Jaramillo-Rodriguez, and A. Torres-Hernandez.

Two relevant papers published/ to appear are

https://arxiv.org/abs/1907.09698https://arxiv.org/abs/1907.09698

https://arxiv.org/abs/2205.05743https://arxiv.org/abs/2205.05743

Fluids sculpt many of the shapes we see in the world around us. We present a new mathematical model describing the shape evolution of a body that dissolves or melts under gravitationally stable buoyancy-driven convection, driven by thermal or solutal transfer at the solid-fluid interface. For high Schmidt number, the system is reduced to a single integro-differential equation for the shape evolution. Focusing on the particular case of a cone, we derive complete predictions for the underlying self-similar shapes, intrinsic scales and descent rates. We will present the results of new laboratory experiments, which show an excellent match to the theory. By analysing all initial power-law shapes, we uncover a surprising result that the tips of melting or dissolving bodies can either sharpen or blunt with time subject to a critical condition.

Let $X$ denote an open subset of $\mathbb{C}^d$, and $\mathcal{O}$ its sheaf of holomorphic functions. In the 1970’s, Ishimura studied the morphisms of sheaves $P\colon\mathcal{O}\to\mathcal{O}$ of $\mathbb{C}$-vector spaces which are continuous, that is the maps $P(U)\colon\mathcal{O}(U)\to\mathcal{O}(U)$ on the sections are continuous. In this talk, we explain his result, and explore its analogues in the non-Archimedean world.

Modern Data Assimilation (DA) can be traced back to the sixties and owes a lot to earlier developments in linear filtering theory. Since then, DA has evolved independently of Filtering Theory. To-date it is a massively important area of research due to its many applications in meteorology, ocean prediction, hydrology, oil reservoir exploration, etc. The field has been largely driven by practitioners, however in recent years an increasing body of theoretical work has been devoted to it. In this talk, In my talk, I will advocate the interpretation of DA through the language of stochastic filtering. This interpretation allows us to make use of advanced particle filters to produce rigorously validated DA methodologies. I will present a particle filter that incorporates three additional add-on procedures: nudging, tempering and jittering. The particle filter is tested on a two-layer quasi-geostrophic model with O(10^6) degrees of freedom out of which only a minute fraction are noisily observed.

We study the global linear convergence of policy gradient (PG) methods for finite-horizon exploratory linear-quadratic control (LQC) problems. The setting includes stochastic LQC problems with indefinite costs and allows additional entropy regularisers in the objective. We consider a continuous-time Gaussian policy whose mean is linear in the state variable and whose covariance is state-independent. Contrary to discrete-time problems, the cost is noncoercive in the policy and not all descent directions lead to bounded iterates. We propose geometry-aware gradient descents for the mean and covariance of the policy using the Fisher geometry and the Bures-Wasserstein geometry, respectively. The policy iterates are shown to obey an a-priori bound, and converge globally to the optimal policy with a linear rate. We further propose a novel PG method with discrete-time policies. The algorithm leverages the continuous-time analysis, and achieves a robust linear convergence across different action frequencies. A numerical experiment confirms the convergence and robustness of the proposed algorithm.

This is joint work with Yufei Zhang and Christoph Reisinger.

I will talk about work in progress with Ana Caraiani aimed at proving Ihara’s lemma for quaternionic Shimura varieties, generalising the strategy of Manning-Shotton for Shimura curves. As an arithmetic motivation, in the first part of the talk I will recall an approach to the Birch and Swinnerton-Dyer conjecture based on congruences between modular forms, relying crucially on Ihara’s lemma.

TBA

The Camassa--Holm (CH) equation is a nonlocal equation that manifests supercritical behaviour in ``wave-breaking" and non-uniqueness. In this talk, I will discuss the existence of global (dissipative weak martingale) solutions to the CH equation with multiplicative, gradient type noise, derived as an inviscid limit. The goal of the talk is twofold. The stochastic CH equation will be used to illustrate aspects of a stochastic compactness and renormalisation method which is popularly used to derive well-posedness and continuous dependence results in SPDEs. I shall also discuss how a lack of temporal compactness introduces fundamental difficulties in the case of the stochastic CH equation.

This talk is based on joint works with L. Galimbert and H. Holden, both at NTNU, and with K.H. Karlsen at the University of Oslo. 

The handlebody group is defined as the mapping class group of a three-dimensional handlebody. We will survey some geometric and algebraic properties of the handlebody groups and compare them to those of two of the most studied (classes of) groups in geometric group theory, namely mapping class groups of surfaces, and ${\rm Out}(F_n)$. We will also introduce the disk graph, the handlebody-analogon of the curve graph of a surface, and discuss some of its properties.

Since Segal's formulation of axioms for 2d CFTs in the 80s, it has remained a major problem to construct examples of CFTs that satisfy the axioms.

I will report on ongoing joint work with James Tener in that direction.

I discuss the set of rates of growth of a finitely generated 
group with respect to all its finite generating sets. In a joint work 
with Sela, for a hyperbolic group, we showed that the set is 
well-ordered, and that each number can be the rate of growth of at most 
finitely many generating sets up to automorphism of the group. I may 
discuss its generalization to acylindrically hyperbolic groups.

We prove that every finite colouring of the plane contains a monochromatic pair of points at an odd (integral) distance from each other. We will also discuss some further results with Rose McCarty and Michal Pilipczuk concerning prime and polynomial distances.

Fibres coming from the Springer resolution on the nilpotent cone are incredibly rich algebraic varieties that have many applications in representation theory and combinatorics. Though their geometry can be very difficult to describe in general, in type A at least, their irreducible components can be described using standard Young tableaux, and this can help describe their geometry in small dimensions. In this talk, I will report on recent and ongoing work with Lewis Topley and separately Daniele Rosso on geometrical and combinatorial applications of the classical ‘type A’ Springer fibres and the ‘exotic’ type C Springer fibres coming from Kato’s exotic Springer correspondence.

Baseline T cell infiltration and the spatial distribution of T cells within a tumour has been found to be a significant indicator of patient outcomes. This observation, coupled with the increasing availability of spatially-resolved imaging data of individual cells within the tumour tissue, motivates the development of mathematical models which capture the spatial dynamics of T cells. Agent-based models allow the simulation of complex biological systems with detailed spatial resolution, and generate rich spatio-temporal datasets. In order to fully leverage the information contained within these simulated datasets, spatial statistics provide methods of analysis and insight into the biological system modelled, by quantifying inherent spatial heterogeneity within the system. We present a cellular automaton model of interactions between tumour cells and cytotoxic T cells, and an analysis of the model dynamics, considering both the temporal and spatial evolution of the system. We use the model to investigate some of the standard assumptions made in these models, to assess the suitability of the models to accurately describe tumour-immune dynamics.

We present a new approach to the gluing problem in General Relativity, that is, the problem of matching two solutions of the Einstein equations along a spacelike or characteristic (null) hypersurface. In contrast to previous constructions, the new perspective actively utilizes the nonlinearity of the constraint equations. As a result, we are able to remove the 10-dimensional spaces of obstructions to gluing present in the literature. As application, we show that any asymptotically flat spacelike initial data set can be glued to Schwarzschild initial data of sufficiently large mass. This is joint work with I. Rodnianski.

A modular functor is defined as a system of mapping class group representations on vector spaces (the so-called conformal blocks) that is compatible with the gluing of surfaces. The notion plays an important role in the representation theory of quantum groups and conformal field theory. In my talk, I will give an introduction to the theory of modular functors and recall some classical constructions. Afterwards, I will explain the approach to modular functors via cyclic and modular operads and their bicategorical algebras. This will allow us to extend the known constructions of modular functors and to classify modular functors by certain cyclic algebras over the little disk operad for which an obstruction formulated in terms of factorization homology vanishes. (The talk is based to a different extent on different joint works with Adrien Brochier, Lukas Müller and Christoph Schweigert.)

We introduce so-called functional input neural networks defined on infinite dimensional weighted spaces, where we use an additive family as hidden layer maps and a non-linear activation function applied to each hidden layer. Relying on approximation theory based on Stone-Weierstrass and Nachbin type theorems on weighted spaces, we can prove global universal approximation results for (differentiable and) continuous functions going beyond approximation on compact sets. This applies in particular to approximation of (non-anticipative) path space functionals via functional input neural networks but also via linear maps of the signature of the respective paths. We apply these results in the context of stochastic portfolio theory to generate path dependent portfolios that are trained to outperform the market portfolio. The talk is based on joint works with Philipp Schmocker and Josef Teichmann.

I will present two new results. The first concerns minimal surfaces of the hyperbolic space and is a relation between their renormalised area (in the sense of Graham and Witten) and the length of their ideal boundary measured in different metrics of the conformal infinity. The second result concerns minimal submanifolds of the sphere and is a relation between their volume and antipodal-ness. Both results were obtained from the same framework, which involves new monotonicity theorems and a comparison principle for them. If time permits, I will discuss how to use these to answer questions about uniqueness and non-existence of minimal surfaces.

Conformal Field Theories (CFT) are believed to be exactly solvable once their primary scaling fields and their 3-point functions are known. This input is called the spectrum and structure constants of the CFT respectively. I will review recent work where this conformal bootstrap program can be rigorously carried out for the case of Liouville CFT, a theory that plays a fundamental role in 2d random surface theory and many other fields in physics and mathematics. Liouville CFT has a probabilistic formulation on an arbitrary Riemann surface and the bootstrap formula can be seen as a "quantization" of the plumbing construction of surfaces with marked points axiomatically discussed earlier by Graeme Segal. Joint work with Colin Guillarmou, Remi Rhodes and Vincent Vargas

Conformal Field Theories (CFT) are believed to be exactly solvable once their primary scaling fields and their 3-point functions are known. This input is called the spectrum and structure constants of the CFT respectively. I will review recent work where this conformal bootstrap program can be rigorously carried out for the case of Liouville CFT, a theory that plays a fundamental role in 2d random surface theory and many other fields in physics and mathematics. Liouville CFT has a probabilistic formulation on an arbitrary Riemann surface and the bootstrap formula can be seen as a "quantization" of the plumbing construction of surfaces with marked points axiomatically discussed earlier by Graeme Segal. Joint work with Colin Guillarmou, Remi Rhodes and Vincent Vargas.

Matthew Buckland 
Branching Interval Partition Diffusions

We construct an interval-partition-valued diffusion from a collection of excursions sampled from the excursion measure of a real-valued diffusion, and we use a spectrally positive Lévy process to order both these excursions and their start times. At any point in time, the interval partition generated is the concatenation of intervals where each excursion alive at that point contributes an interval of size given by its value. Previous work by Forman, Pal, Rizzolo and Winkel considers self-similar interval partition diffusions – and the key aim of this work is to generalise these results by dropping the self-similarity condition. The interval partition can be interpreted as an ordered collection of individuals (intervals) alive that have varying characteristics and generate new intervals during their finite lifetimes, and hence can be viewed as a class of Crump-Mode-Jagers-type processes.

Ofir Gorodetsky
Smooth and rough numbers

We all know and love prime numbers, but what about smooth and rough numbers?
We'll define y-smooth numbers -- numbers whose prime factors are all less than y. We'll explain their application in cryptography, specifically to factorization of integers.
We'll shed light on their density, which is modelled using a peculiar differential equation. This equation appears naturally in probability theory.
We'll also explain the dual notion to smooth numbers, that of y-rough numbers: numbers whose prime factors are all bigger than y, and in some sense generalize primes.
We'll explain their importance in sieve theory. Like smooth numbers, their density has interesting properties and will be surveyed.

At the intersection of Topological Data Analysis and machine learning, the field of cellular signal processing has advanced rapidly in recent years. In this context, each signal on the cells of a complex is processed using the combinatorial Laplacian and the resulting Hodge decomposition. Meanwhile, discrete Morse theory has been widely used to speed up computations by reducing the size of complexes while preserving their global topological properties. In this talk, we introduce an approach to signal compression and reconstruction on complexes that leverages the tools of discrete Morse theory. The main goal is to reduce and reconstruct a cell complex together with a set of signals on its cells while preserving their global topological structure as much as possible. This is joint work with Stefania Ebli and Kelly Maggs.

Donald Knutson proposed the conjecture, later disproven and refined by Savitskii, that for every irreducible character of a finite group, there existed a virtual character such their tensor product was the regular character. In this talk, we disprove both this conjecture and its refinement. We then introduce the Knutson Index as a measure of the failure of Knutson's Conjecture and discuss its algebraic properties.

Abstract: We give a generalized Kuznetsov formula that allows one to impose additional conditions at finitely many primes.  The formula arises from the relative trace formula. I will discuss applications to spectral large sieve inequalities and subconvexity. This is work in progress with M.P. Young and Y. Hu.

We have recently proposed a new kind of neural network, called a Legendre Memory Unit (LMU) that is provably optimal for compressing streaming time series data. In this talk, I describe this network, and a variety of state-of-the-art results that have been set using the LMU. I will include recent results on speech and language applications that demonstrate significant improvements over transformers. I will discuss variants of the original LMU that permit effective scaling on current GPUs and hold promise to provide extremely efficient edge time series processing.

We forecast the realized covariance matrix of asset returns in the U.S. equity market by exploiting the predictive information of graphs in volatility and correlation. Specifically, we augment the Heterogeneous Autoregressive (HAR) model via neighborhood aggregation on these graphs. Our proposed method allows for the modeling of interdependence in volatility (also known as spillover effect) and correlation, while maintaining parsimony and interpretability. We explore various graph construction methods, including sector membership and graphical LASSO (for modeling volatility), and line graph (for modeling correlation). The results generally suggest that the augmented model incorporating graph information yields both statistically and economically significant improvements for out-of-sample performance over the traditional models. Such improvements remain significant over horizons up to one month ahead, but decay in time. The robustness tests demonstrate that the forecast improvements are obtained consistently over the different out-of-sample sub-periods, and are insensitive to measurement errors of volatilities.

Cayley submanifolds in Spin(7) geometry are an analogue and generalisation of complex submanifolds in Kähler geometry. In this talk we provide a glimpse into calibrated geometry, which encompasses both of these, and how it ties into the study of manifolds of special holonomy. We then focus on the deformation theory of compact and conically singular Cayleys. Finally we explain how to remove conical singularities via a gluing construction.

We study an infinite family of Massive Type IIA backgrounds that holographically describe the twisted compactification of N=(1,0) six-dimensional SCFTs to four dimensions. The analysis of the branes involved suggests a four dimensional linear quiver QFT, that deconstructs the theory in six dimensions. For the case in which the system reaches a strongly coupled fixed point, we calculate some observables that we compare with holographic results. Two quantities measuring the number of degrees of freedom for the flow across dimensions are studied.

Surface water waves of significant interest, such as tsunamis and solitary waves, are nonlinear and dispersive waves. Unluckily, the equations derived from first principles that describe the propagation of surface water waves, known as Euler's equations, are immensely hard to study. For this reason, several approximate systems have been proposed as mathematical alternatives. We show that among the numerous simplified systems of PDEs of water wave theory there is only one that is provably well-posed (in Hadamard’s sense) in bounded domains with slip-wall boundary conditions. We also show that the particular well-posed system obeys most of the physical laws that acceptable water wave equations must obey, and it is consistent with the Euler equations. For the numerical solution of our system we rely on a Galerkin/finite element method based on Nitsche's method for which we have proved its convergence. Validation with laboratory data is also presented.

We will finish off the term with a discussion of allies and community. You are encouraged to bring a friend who has never been to Mathematrix before!

We will show asymptotic analysis for hydrodynamic system, as Navier-Stokes-Fourier system, as a useful tool in in the situation when certain parameters in the system – called characteristic numbers – vanish or become infinite. The choice of proper scaling, namely proper system of reference units, the parameters determining the behaviour of the system under consideration allow to eliminate unwanted or unimportant for particular phenomena modes of motion. The main goal of many studies devoted to asymptotic analysis of various physical systems is to derive a simplified set of equations - simpler for mathematical or numerical analysis. Such systems may be derived in a very formal way, however we will concentrate on rigorous mathematical analysis. I will concentrate on low Mach number limits with so called ill-prepared data and I will present some results which concerns passage from compressible to incompressible models of fluid flow emphasising difficulties characteristic for particular problems. In particular we will discuss Navier-Stokes-Fourier system on varying domains, a multi-scale problem for viscous heat-conducting fluids in fast rotation and the incompressible limit of compressible finitely extensible nonlinear bead-spring chain models for dilute polymeric fluids.

Hypergraphs for multiscale cycles in structured data

Iris Yoon

Understanding the spatial structure of data from complex systems is a challenge of rapidly increasing importance. Even when data is restricted to curves in three-dimensional space, the spatial structure of data provides valuable insight into many scientific disciplines, including finance, neuroscience, ecology, biophysics, and biology. Motivated by concrete examples arising in nature, I will introduce hyperTDA, a topological pipeline for analyzing the structure of spatial curves that combines persistent homology, hypergraph theory, and network science. I will show that the method highlights important segments and structural units of the data. I will demonstrate hyperTDA on both simulated and experimental data. This is joint work with Agnese Barbensi, Christian Degnbol Madsen, Deborah O. Ajayi, Michael Stumpf, and Heather Harrington.

Minmax Connectivity and Persistent Homology 

Ambrose Yim

We give a pipeline for extracting features measuring the connectivity between two points in a porous material. For a material represented by a density field f, we derive persistent homology related features by exploiting the relationship between dimension zero persistent homology of the density field and the min-max connectivity between two points. We measure how the min-max connectivity varies when spurious topological features of the porous material are removed under persistent homology guided topological simplification. Furthermore, we show how dimension one persistent homology encodes a relaxed notion of min-max connectivity, and demonstrate how we can summarise the multiplicity of connections between a pair of points by associating to the pair a sub-diagram of the dimension one persistence diagram.

Given a group of type ${\rm FP}_n$, one may ask if this property also holds for its subgroups. The BNS invariant is a subset of the character sphere that fully captures this information for subgroups that are kernels of characters. It also provides an interesting connection of finiteness properties of subgroups and group homology. In this talk I am going to give an introduction to this problem and present an attempt to generalize the BNS invariant to more subgroups than just the kernels of characters.

Tom Hutchcroft and I have been working to develop a general theory of percolation on arbitrary finite transitive graphs. This extends from percolation on local approximations to infinite graphs, such as a sequence of tori, to percolation on the complete graphs - the Erdős-Rényi model. I will summarise our progress on the basic questions: When is there a phase transition for the emergence of a giant cluster? When is the giant cluster unique? How does this relate to percolation on infinite graphs? I will then sketch our proof that for finite transitive graphs with uniformly bounded vertex degrees, the supercritical giant cluster is unique, verifying a conjecture of Benjamini from 2001.

A coarse structure is a way of talking about "large-scale" properties.  It is encoded in a family of relations that often, but not always, come from a metric.  A coarse structure naturally gives rise to Hilbert space operators that in turn generate a so-called uniform Roe algebra.

In work with Bruno Braga and Joe Eisner, we use ideas of Weaver to construct "quantum" coarse structures and uniform Roe algebras in which the underlying set is replaced with an arbitrary represented von Neumann algebra.  The general theory immediately applies to quantum metrics (suitably defined), but it is much richer.  We explain another source based on measure instead of metric, leading to the new, large, and easy-to-understand class of support expansion C*-algebras.

I will present the big picture: where uniform Roe algebras come from, how Weaver's framework facilitates our definitions.  I will focus on a few illustrative examples and will not presume familiarity with coarse structures or von Neumann algebras.

In 1973, Erdős conjectured the existence of high girth $(n,3,2)$-Steiner systems. Recently, Glock, Kühn, Lo, and Osthus and independently Bohman and Warnke proved the approximate version of Erdős' conjecture. Just this year, Kwan, Sah, Sawhney, and Simkin proved Erdős' conjecture. As for Steiner systems with more general parameters, Glock, Kühn, Lo, and Osthus conjectured the existence of high girth $(n,q,r)$-Steiner systems. We prove the approximate version of their conjecture. This result follows from our general main results which concern finding perfect or almost perfect matchings in a hypergraph $G$ avoiding a given set of submatchings (which we view as a hypergraph $H$ where $V(H)=E(G)$). Our first main result is a common generalization of the classical theorems of Pippenger (for finding an almost perfect matching) and Ajtai, Komlós, Pintz, Spencer, and Szemerédi (for finding an independent set in girth five hypergraphs). More generally, we prove this for coloring and even list coloring, and also generalize this further to when $H$ is a hypergraph with small codegrees (for which high girth designs is a specific instance). A number of applications in various areas follow from our main results including: Latin squares, high dimensional permutations, and rainbow matchings. This is joint work with Luke Postle.

This talk is motivated by computing correlations for domino tilings of the Aztec diamond.  It is inspired by two of the three distinct methods that have recently been used in the simplest case of a doubly periodic weighting, that is the two-periodic Aztec diamond. This model is of particular probabilistic interest due to being one of the few models having a boundary between polynomially and exponentially decaying macroscopic regions in the limit. One of the methods to compute correlations, powered by the domino shuffle, involves inverting the Kasteleyn matrix giving correlations through the local statistics formula. Another of the methods, driven by a Wiener-Hopf factorization for two- by-two matrix valued functions, involves the Eynard-Mehta theorem. For arbitrary weights the Wiener-Hopf factorization can be replaced by an LU- and UL-decomposition, based on a matrix refactorization, for the product of the transition matrices. In this talk, we present results to say that the evolution of the face weights under the domino shuffle and the matrix refactorization is the same. This is based on joint work with Maurice Duits (Royal Institute of Technology KTH).  

We will describe a new equivariant version of discrete Morse theory designed specially for quotient objects X/G which arise naturally in geometric group theory from actions of finite groups G on finite simplicial complexes X. Our main tools are (A) a reconstruction theorem due to Bridson and Haefliger which recovers X from X/G decorated with stabiliser data, and (B) a 2-categorical upgrade of discrete Morse theory which faithfully captures the underlying homotopy type. Both tools will be introduced during the course of the talk. This is joint work with Naya Yerolemou.

Khovanov-Rozansky homology is a link invariant that categorifies the HOMFLY-PT polynomial. I will describe a geometric model for this invariant, living in the monodromic Hecke category. I will also explain how it allows to identify objects representing graded pieces of Khovanov-Rozansky homology, using a remarkable family of character sheaves. Based on joint works with Roman Bezrukavnikov.

In this talk I will present a new class of algorithms for separable nonlinear inverse problems based on inexact Krylov methods. In particular, I will focus in semi-blind deblurring applications. In this setting, inexactness stems from the uncertainty in the parameters defining the blur, which are computed throughout the iterations. After giving a brief overview of the theoretical properties of these methods, as well as strategies to monitor the amount of inexactness that can be tolerated, the performance of the algorithms will be shown through numerical examples. This is joint work with Silvia Gazzola (University of Bath).