Past

Multilayer networks are a way to represent dependent connectivity patterns — e.g., time-dependence, multiple types of interactions, or both — that arise in many applications and which are difficult to incorporate into standard network representations. In the study of multilayer networks, it is important to investigate mesoscale (i.e., intermediate-scale) structures, such as communities, to discover features that lie between the microscale and the macroscale. We introduce a framework for the construction of generative models for mesoscale structure in multilayer networks.  We model dependency at the level of partitions rather than with respect to edges, and treat the process of generating a multilayer partition separately from the process of generating edges for a given multilayer partition. Our framework can admit many features of empirical multilayer networks and explicitly incorporates a user-specified interlayer dependency structure. We discuss the parameters and some properties of our framework, and illustrate an example of its use with benchmark models for multilayer community-detection tools. 

Spatial navigation in preclinical and clinical Alzheimer’s disease - Relevance for topological data analysis?

Spatial navigation changes are one of the first symptoms of Alzheimer’s disease and also lead to significant safeguarding issues in patients after diagnosis. Despite their significant implications, spatial navigation changes in preclinical and clinical Alzheimer’s disease are still poorly understood. In the current talk, I will explain the spatial navigation processes in the brain and their relevance to Alzheimer’s disease. I will then introduce our Sea Hero Quest project, which created the first global benchmark data for spatial navigation in ~4.5 million people worldwide via a VR-based game. I will present data from the game, which has allowed to create personalised benchmark data for at-risk-of-Alzheimer’s people. The final part of my talk will explore how real-world environment & entropy impacts on dementia patients getting lost and how this has relevance for GPS technology based safeguarding and car driving in Alzheimer’s disease.

Understanding the size of the rational points on a curve of higher genus is one of the major open problems in the theory of Diophantine equations. In this talk I will discuss the related problem of understanding how close together rational points can get. I will also discuss the relation to the subject of (generalised) Wieferich primes.

I will present a $\Gamma$-convergence for degenerate integral functionals related to homogenisation problems  in the Heisenberg group. In our  case, both the rescaling and the notion of invariance or periodicity are chosen in a way motivated by the geometry of the Heisenberg group. Without using special geometric features, these functionals would be neither coercive nor periodic, so classic results do not apply.  All the results apply to the more general case of Carnot groups. Joint with Nicolas Dirr, Paola Mannucci and Claudio Marchi.

The square peg problem asks whether every Jordan curve in the
plane contains the vertices of a square. Inspired by Hugelmeyer's approach
for smooth curves, we give a topological proof for "locally 1-Lipschitz"
curves using 4-dimensional topology.

We will talk about some stochastic parabolic and hyperbolic partial differential equations (SPDEs), which arise naturally in the context of Liouville quantum gravity. These dynamics are proposed to preserve the Liouville measure, which has been constructed recently in the series of works by David-Kupiainen-Rhodes-Vargas. We construct global solutions to these equations under some conditions and then show the invariance of the Liouville measure under the resulting dynamics. As a by-product, we also answer an open problem proposed by Sun-Tzvetkov recently.
 

Stochastic processes subject to weak noise often show a metastable
behaviour, meaning that they converge to equilibrium extremely slowly;
typically, the convergence time is exponentially large in the inverse
of the variance of the noise (Arrhenius law).
  
In the case of finite-dimensional Ito stochastic differential
equations, the large-deviation theory developed in the 1970s by
Freidlin and Wentzell allows to prove such Arrhenius laws and compute
their exponent. Sharper asymptotics for relaxation times, including the
prefactor of the exponential term (Eyring–Kramers laws) are known, for
instance, if the stochastic differential equation involves a gradient
drift term and homogeneous noise. One approach that has been very
successful in proving Eyring–Kramers laws, developed by Bovier,
Eckhoff, Gayrard and Klein around 2005, relies on potential theory.
  
I will describe Eyring–Kramers laws for some parabolic stochastic PDEs
such as the Allen–Cahn equation on the torus. In dimension 1, an
Arrhenius law was obtained in the 1980s by Faris and Jona-Lasinio,
using a large-deviation principle. The potential-theoretic approach
allows us to compute the prefactor, which turns out to involve a
Fredholm determinant. In dimensions 2 and 3, the equation needs to be
renormalized, which turns the Fredholm determinant into a
Carleman–Fredholm determinant.
  
Based on joint work with Barbara Gentz (Bielefeld), and with Ajay
Chandra (Imperial College), Giacomo Di Gesù (Vienna) and Hendrik Weber
(Warwick). 

References: 
https://dx.doi.org/10.1214/EJP.v18-1802
https://dx.doi.org/10.1214/17-EJP60

It is well-known that the Teichmüller space of a compact surface can be identified with a connected component of the moduli space of representations of the fundamental group of the surface in PSL(2,R). Higher Teichmüller components are generalizations of this that exist for the moduli space of representations of the fundamental group into certain real simple Lie groups of higher rank. As for the usual Teichmüller space, these components consist entirely  of discrete and faithful representations. Several cases have been identified over the years. First, the Hitchin components for split groups, then the maximal Toledo invariant components for Hermitian groups, and more recently certain components for SO(p,q). In this talk, I will describe a general construction of (still somewhat conjecturally) all possible Teichmüller components, and a parametrization of them using Higgs bundles.

String sigma-models relevant in the AdS/CFT correspondence are highly non-trivial two-dimensional field theories for which predictions at finite coupling exist, assuming integrability and/or the duality itself.  I will discuss general features of the perturbative approach to these models, and present progress on how to go extract finite coupling information in the most possibly general way, namely via the use of lattice field theory techniques. I will also present new results on certain ``defect-CFT’' correlators  at strong coupling. 

The talk will introduce two general models of random simplicial complexes which extend the highly studied Erdös-Rényi model for random graphs. These models include the well known probabilistic models of random simplicial complexes from Costa-Farber, Kahle, and Linial-Meshulam as special cases. These models turn out to have a satisfying Alexander duality relation between them prompting the hope that information can be transferred for free between them. This turns out to not quite be the case with vanishing probability parameters, but when all parameters are uniformly bounded the duality relation works a treat. Time permitting I may talk about the Rado simplicial complex, the unique (with probability one) infinite random simplicial complex.
This talk is based on various bits of joint work with Michael Farber, Tahl Nowik, and Lewin Strauss.

Mathematicians need to talk and write about their mathematics.  This includes undergraduates and MSc students, who may be writing a dissertation or project report, preparing a presentation on a summer research project, or preparing for a job interview.  We think that it can be helpful to think of this as a form of storytelling, as this can lead to more effective communication.  For a story to be engaging you also need to know your audience.  In this session, we'll discuss what we mean by telling a mathematical story, give you some top tips from our experience, and give you a chance to think about how you might put this into practice.

Tensors are higher dimensional analogues of matrices, used to record data with multiple changing variables. Interpreting tensor data requires finding multi-linear stucture that depends on the application or context. I will describe a tensor-based clustering method for multi-dimensional data. The multi-linear structure is encoded as algebraic constraints in a linear program. I apply the method to a collection of experiments measuring the response of genetically diverse breast cancer cell lines to an array of ligands. In the second part of the talk, I will discuss low-rank decompositions of tensors that arise in statistics, focusing on two graphical models with hidden variables. I describe how the implicit semi-algebraic description of the statistical models can be used to obtain a closed form expression for the maximum likelihood estimate.

The famous Birch & Swinnerton-Dyer conjecture predicts that the (algebraic) rank of an elliptic curve is equal to the so-called analytic rank, which is the order of vanishing of the associated L-functions at the central point. In this talk, we shall discuss the analytic rank of automorphic L-functions in an "alternate universe". This is joint work with Kyle Pratt and Alexandru Zaharescu.

Your brain has its own waterscape: whether you are reading or sleeping, fluid flows around or through the brain tissue and clears waste in the process. These physiological processes are crucial for the well-being of the brain. In spite of their importance we understand them but little. Mathematics and numerics could play a crucial role in gaining new insight. Indeed, medical doctors express an urgent need for modeling of water transport through the brain, to overcome limitations in traditional techniques. Surprisingly little attention has been paid to the numerics of the brain’s waterscape however, and fundamental knowledge is missing. In this talk, I will discuss mathematical models and numerical methods for the brain's waterscape across scales - from viewing the brain as a poroelastic medium at the macroscale and zooming in to studying electrical, chemical and mechanical interactions between brain cells at the microscale.
 

This talk aims to provide a simple introduction on how to probe the
explicit geometry of certain moduli schemes arising in enumerative
geometry. Stable pairs, introduced by Pandharipande and Thomas in 2009, offer a curve-counting theory which is tamer than the Hilbert scheme of
curves used in Donaldson-Thomas theory. In particular, they exclude
curves with zero-dimensional or embedded components.

Ribbons are non-reduced schemes of dimension one, whose non-reduced
structure has multiplicity two in a precise sense. Following Ferrand, Banica, and Forster, there are several results on how to construct
ribbons (and higher non-reduced structures) from the data of line
bundles on a reduced scheme. With this approach, we can consider stable
pairs whose underlying curve is a ribbon: the remaining data is
determined by allowing devenerations of the line bundle defining the
double structure.

In this talk I will present a Perron-Frobenius type result for nonlinear eigenvector problems which allows us to compute the global maximum of a class of constrained nonconvex optimization problems involving multihomogeneous functions.

I will structure the talk into three main parts:

First, I will motivate the optimization of homogeneous functions from a graph partitioning point of view, showing an intriguing generalization of the famous Cheeger inequality.

Second, I will define the concept of multihomogeneous function and I will state our main Perron-Frobenious theorem. This theorem exploits the connection between optimization of multihomogeneous functions and nonlinear eigenvectors to provide an optimization scheme that has global convergence guarantees.

Third, I will discuss a few example applications in network science and machine learning that require the optimization of multihomogeneous functions and that can be solved using nonlinear Perron eigenvectors.

Computers can now beat humans at chess and at go. Surely one day they will beat us at proving theorems. But when will it happen, how will it happen, and what should humans be doing in order to make it happen? Furthermore -- do we actually want it to happen? Will they generate incomprehensible proofs, which teach us nothing? Will they find mistakes in the human literature?

I will talk about how I am training undergraduates at Imperial College London to do their problem sheets in a formal proof verification system, and how this gamifies mathematics. I will talk about mistakes in the modern pure mathematics literature, and ask what the point of modern pure mathematics is.

In this talk I will discuss some aspects of the potential theory, fine properties and boundary behaviour of the solutions to the Total Variation Flow. Instead of the classical Euclidean setting, we intend to work mostly in the general setting of metric measure spaces. During the past two decades, a theory of Sobolev functions and BV functions has been developed in this abstract setting.  A central motivation for developing such a theory has been the desire to unify the assumptions and methods employed in various specific spaces, such as weighted Euclidean spaces, Riemannian manifolds, Heisenberg groups, graphs, etc.

The total variation flow can be understood as a process diminishing the total variation using the gradient descent method.  This idea can be reformulated using parabolic minimizers, and it gives rise to a definition of variational solutions.  The advantages of the approach using a minimization formulation include much better convergence and stability properties.  This is a very essential advantage as the solutions naturally lie only in the space of BV functions. Our main goal is to give a necessary and sufficient condition for continuity at a given point for proper solutions to the total variation flow in metric spaces. This is joint work with Vito Buffa and Juha Kinnunen.

Limit groups are a powerful tool in the study of free and hyperbolic groups (and even broader classes of groups). I will define limit groups in various ways: algebraic, logical and topological, and draw connections between the different definitions. We will also see how one can equip a limit group with an action on a real tree, and analyze this action using the Rips machine, a generalization of Bass-Serre theory to real trees. As a conclusion, we will obtain that hyperbolic groups whose outer automorphism group is infinite, split non-trivially as graphs of groups.

Let $X$ be a countable discrete metric space, and think of operators on $\ell^2(X)$ in terms of their $X$-by-$X$ matrix. Band operators are ones whose matrix is supported on a "band" along the main diagonal; all norm-limits of these form a C*-algebra, called uniform Roe algebra of $X$. This algebra "encodes" the large-scale (a.k.a. coarse) structure of $X$. Quasi-locality, coined by John Roe in '88, is a property of an operator on $\ell^2(X)$, designed as a condition to check whether the operator belongs to the uniform Roe algebra (without producing band operators nearby). The talk is about our attempt to make this work, and an expander-ish condition on graphs that came out of trying to find a counterexample. (Joint with: A. Tikuisis, J. Zhang, K. Li and P. Nowak.)