Past

Neurodegenerative diseases such as Alzheimer’s or Parkinson’s are devastating conditions with poorly understood mechanisms and no known cure. Yet a striking feature of these conditions is the characteristic pattern of invasion throughout the brain, leading to well-codified disease stages visible to neuropathology and associated with various cognitive deficits and pathologies. In this talk, I will show that by linking new mathematical theories to recent progress in imaging, we can unravel some of the universal features associated with dementia and, more generally, brain functions. In particular, I will outline interesting mathematical problems and ideas that naturally appear in the process.

In this talk, we shall propose a relaxed control regularization with general exploration rewards to design robust feedback controls for multi-dimensional continuous-time stochastic exit time problems. We establish that the regularized control problem admits a H\”{o}lder continuous feedback control, and demonstrate that both the value function and the feedback control of the regularized control problem are Lipschitz stable with respect to parameter perturbations. Moreover, we show that a pre-computed feedback relaxed control has a robust performance in a perturbed system, and derive a first-order sensitivity equation for both the value function and optimal feedback relaxed control. These stability results provide a theoretical justification for recent reinforcement learning heuristics that including an exploration reward in the optimization objective leads to more robust decision making. We finally prove first-order monotone convergence of the value functions for relaxed control problems with vanishing exploration parameters, which subsequently enables us to construct the pure exploitation strategy of the original control problem based on the feedback relaxed controls. This is joint work with Christoph Reisinger (available at https://arxiv.org/abs/2001.03148).
 

The finite element method (FEM) is one of the great triumphs of applied mathematics, numerical analysis and software development. Recent developments in sensor and signalling technologies enable the phenomenological study of systems. The connection between sensor data and FEM is restricted to solving inverse problems placing unwarranted faith in the fidelity of the mathematical description of the system. If one concedes mis-specification between generative reality and the FEM then a framework to systematically characterise this uncertainty is required. This talk will present a statistical construction of the FEM which systematically blends mathematical description with observations.

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A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs.  Given a hereditary graph property H, the speed of H is the function which sends an integer n to the number of distinct elements in H with underlying set {1,...,n}.  Not just any function can occur as the speed of hereditary graph property.  Specifically, there are discrete ``jumps" in the possible speeds.  Study of these jumps began with work of Scheinerman and Zito in the 90's, and culminated in a series of papers from the 2000's by Balogh, Bollob\'{a}s, and Weinreich, in which essentially all possible speeds of a hereditary graph property were characterized.  In contrast to this, many aspects of this problem in the hypergraph setting remained unknown.  In this talk we present new hypergraph analogues of many of the jumps from the graph setting, specifically those involving the polynomial, exponential, and factorial speeds.  The jumps in the factorial range turned out to have surprising connections to the model theoretic notion of mutual algebricity, which we also discuss.  This is joint work with Chris Laskowski.

Gromov hyperbolicity is a property to metric spaces that generalises the notion of negative curvature for manifolds.
After an introduction about these spaces, we will explain the construction of horocyclic products related to lamplighter groups, Baumslag solitar groups and the Sol geometry.
We will describe the shape of geodesics in them, and present rigidity results on their quasi-isometries due to Farb, Mosher, Eskin, Fisher and Whyte.

Pick's theorem is a century-old theorem in complex analysis about interpolation with bounded analytic functions. The Kadison-Singer problem was a question about states on $C^*$-algebras originating in the work of Dirac on the mathematical description of quantum mechanics. It was solved by Marcus, Spielman and Srivastava a few years ago.

I will talk about Pick's theorem, the Kadison-Singer problem and how the two can be brought together to solve interpolation problems with infinitely many nodes. This talk is based on joint work with Alexandru Aleman, John McCarthy and Stefan Richter.

The rows of a Young diagram chosen at random with respect to the Plancherel measure are known to share some features with the eigenvalues of the Gaussian Unitary Ensemble. We shall discuss several ideas, going back to the work of Kerov and developed by Biane and by Okounkov, which to some extent clarify this similarity. Partially based on joint work with Jeong and on joint works in progress with Feldheim and Jeong and with Täufer.

Since Weierstrass it has been known that there are functions that are continuous but nowhere differentiable.  A beautiful example (with probability 1) is any Brownian path.  Brownian paths can be constructed either in space, via Brownian bridge, or in Fourier space, via random Fourier series.

What about functions, which we call "smoothies", that are $C^\infty$ but nowhere analytic?  This case is less familiar but analogous, and again one can do the construction either in space or Fourier space.  We present the ideas and illustrate them with the new Chebfun $\tt{smoothie}$ command.  In the complex plane, the same idea gives functions analytic in the open unit disk and $C^\infty$ on the unit circle, which is a natural boundary.

Mittag-Leffler condition ensures the exactness of the inverse limit of short exact sequences indexed on a partially ordered set admitting a countable cofinal subset. We extend Mittag-Leffler condition by relatively relaxing the countability assumption. As an application we prove an exactness result about the completion functor in the category of ultrametric locally convex vector spaces, and in particular we prove that a strict morphism between these spaces has closed image if its kernel is Fréchet.

Given a tournament with $d{n \choose 3}$ cycles of length three, how many cycles of length four must there be? Linial and Morgenstern (2016) conjectured that the minimum is asymptotically attained by "blowing up" a transitive tournament and orienting the edges randomly within the parts. This is reminiscent of the tight examples for the famous Triangle and Clique Density Theorems of Razborov, Nikiforov and Reiher. We prove the conjecture for $d \geq \frac{1}{36}$ using spectral methods. We also show that the family of tight examples is more complex than expected and fully characterise it for $d \geq \frac{1}{16}$. Joint work with Timothy Chan, Andrzej Grzesik and Daniel Král'.

Reconstruction of 3D images from a set of 2D X-ray projections is a standard inverse problem, particularly in medical imaging. Improvements in imaging technologies have enabled the development of a flat-panel X-ray source, comprised of an array of low-power emitters that are fired in quick succession. During a complete firing sequence, there may be shifts in the patient’s resting position which ultimately create artifacts in the final reconstruction. We present a method for correcting images with respect to unknown body motion, focusing on the case of simple rigid body motion. Image reconstructions are obtained by solving a sparse linear inverse problem, with respect to not only the underlying body but also the unknown velocity. Results find that reconstructions of a moving body can be much better than those obtained by measuring a stationary body, as long as the underlying motion is well approximated.

In this talk we will discuss a problem that was worked on during MISGSA 2020, a Study Group held in January at The University of Zululand, South Africa.

We look at a communication network with two types of users - Primary users (PU) and Secondary users (SU) - such that we reduce the network to a set of overlapping sub-graphs consisting of SUs indexed by a specific PU. Within any given sub-graph, the PU may be communicating at a certain fixed frequency F. The respective SUs also wish to communicate at the same frequency F, but not at the expense of interfering with the PU signal. Therefore if the PU is active then the SUs will not communicate.

In an attempt to increase information throughput in the network, we instead allow the SUs to communicate at a different frequency G, which may or may not interfere with a different sub-graph PU in the network, leading to a multi-objective optimisation problem.

We will discuss not only the problem formulation and possible approaches for solving it, but also the pitfalls that can be easily fallen into during study groups.

The asymptotically locally Euclidean Ricci-flat self-dual 4-manifolds were classified and constructed by Kronheimer as hyperkahler quotients. Each belongs to a finite-dimensional family and a particularly interesting subfamily consists of manifolds with a circle action which can be identified with the minimal resolution of a quotient singularity C^2/G where G is a finite subgroup of SU(2). The resolved singularity is a configuration of rational curves and there is a distinguished one which is pointwise fixed by the circle action. The talk will give an explicit description of the induced metric on this central sphere, and involves twistor theory and the geometry of the lines on a cubic surface.
 

The notion of emergence is at the core of many of the most challenging open scientific questions, being so much a cause of wonder as a perennial source of philosophical headaches. Two classes of emergent phenomena are usually distinguished: strong emergence, which corresponds to supervenient properties with irreducible causal power; and weak emergence, which are properties generated by the lower levels in such "complicated" ways that they can only be derived by exhaustive simulation. While weak emergence is generally accepted, a large portion of the scientific community considers causal emergence to be either impossible, logically inconsistent, or scientifically irrelevant.

In this talk we present a novel, quantitative framework that assesses emergence by studying the high-order interactions of the system's dynamics. By leveraging the Integrated Information Decomposition (ΦID) framework [1], our approach distinguishes two types of emergent phenomena: downward causation, where macroscopic variables determine the future of microscopic degrees of freedom; and causal decoupling, where macroscopic variables influence other macroscopic variables without affecting their corresponding microscopic constituents. Our framework also provides practical tools that are applicable on a range of scenarios of practical interest, enabling to test -- and possibly reject -- hypotheses about emergence in a data-driven fashion. We illustrate our findings by discussing minimal examples of emergent behaviour, and present a few case studies of systems with emergent dynamics, including Conway’s Game of Life, neural population coding, and flocking models.
[1] Mediano, Pedro AM, Fernando Rosas, Robin L. Carhart-Harris, Anil K. Seth, and Adam B. Barrett. "Beyond integrated information: A taxonomy of information dynamics phenomena." arXiv preprint arXiv:1909.02297 (2019).
 

The classical Minkowski problem consists in finding a convex polyhedron from data consisting of normals to their faces and their surface areas. In the smooth case, the corresponding problem for convex bodies is to find the convex body given the Gauss curvature of its boundary, as a function of the unit normal. The proof consists of three parts: existence, uniqueness and regularity. 

In this talk, we study a Minkowski problem for certain measure, called p-capacitary surface area measure, associated to a compact convex set $E$ with nonempty interior and its $p-$harmonic capacitary function (solution to the p-Laplace equation in the complement of $E$).  If $\mu_p$ denotes this measure, then the Minkowski problem we consider in this setting is that; for a given finite Borel positive measure $\mu$ on $\mathbb{S}^{n-1}$, find necessary and sufficient conditions for which there exists a convex body $E$ with $\mu_p =\mu$. We will discuss the existence, uniqueness, and regularity of this problem which have deep connections with the Brunn-Minkowski inequality for p-capacity and Monge-Amp{\`e}re equation.

The mapping class group of a surface is associated to its Teichmüller space. In turn, its boundary consists of projective measured laminations. Similarly, the group of outer automorphisms of a free group is associated to its Outer space. Now the boundary contains equivalence classes of arationaltrees as a subset. There exist distinct projective measured laminations that have the same underlying geodesic lamination, which is also minimal and filling. Such geodesic laminations are called `non-uniquely ergodic'. I will talk briefly about laminations on surfaces and then present a construction of non-uniquely ergodic phenomenon for arational trees. This is joint work with Mladen Bestvina and Jing Tao.

In practice, it is standard to initialize Artificial Neural Networks (ANN) with random parameters. We will see that this allows to describe, in the functional space, the limit of the evolution of (fully connected) ANN when their width tends towards infinity. Within this limit, an ANN is initially a Gaussian process and follows, during learning, a gradient descent convoluted by a kernel called the Neural Tangent Kernel. 

This description allows a better understanding of the convergence properties of neural networks, of how they generalize to examples during learning and has 

practical implications on the training of wide ANNs. 

During this talk we will be interested in a one-dimensional exclusion process subject to strong kinetic constraints, which belongs to the class of cooperative kinetically constrained lattice gases. More precisely, its stochastic short range interaction exhibits a continuous phase transition to an absorbing state at a critical value of the particle density. We will see that the macroscopic behavior of this microscopic dynamics, under periodic boundary conditions and diffusive time scaling, is ruled by a non-linear PDE belonging to free boundary problems (or Stefan problems). One of the ingredients is to show that the system typically reaches an ergodic component in subdiffusive time.

Based on joint works with O. Blondel, C. Erignoux and M. Sasada

Exceptional holonomy manifolds come with certain geometric data that include a Ricci flat metric. Finding examples is therefore very difficult. The task can be made more tractable by imposing symmetry.  The focus of this talk is the case of torus symmetry. For a particular rank of the torus, one gets a natural parameterisation of the orbit space in terms of so-called multi-moment maps. I will discuss aspects of the local and global geometry of these 'toric' exceptional holonomy manifolds. The talk is based on joint work with Andrew Swann.

In this talk, I will argue how the observation that four-dimensional N=2 superconformal field theories are interconnected via the operation of Higgsing can be turned into an effective method to construct such SCFTs. A fundamental role is played by the (generalized) free field realization of the associated VOAs.

The reach is an important geometric invariant of submanifolds of Euclidean space. It is a real-valued global invariant incorporating information about the second fundamental form of the embedding and the location of the first critical point of the distance from the submanifold. In the subject of geometric inference, the reach plays a crucial role. I will give a new method of estimating the reach of a submanifold, developed jointly with Clément Berenfeld, Marc Hoffmann and Krishnan Shankar.

Have you ever had to explain mathematics to someone who isn’t a mathematician? Maybe you’ve been cornered at a family gathering by an interested relative. Maybe you’d like to explain to a potential employer what you’ve been doing for the last three years. Maybe you’ve agreed to explain vector calculus to a room of 13-year-olds. We’ve all been there. This session will cover some top tips for talking about maths in a way that makes sense to non-mathematicians, with specific examples from the outreach team.