Past

Formation of singularities for the relativistic Euler equations (N. Athanasiou): An archetypal phenomenon in the study of hyperbolic systems of conservation laws is the development of singularities (in particular shocks) in finite time, no matter how smooth or small the initial data are. A series of works by Lax, John et al confirmed that for some important systems, when the initial data is a smooth small perturbation of a constant state, singularity formation in finite time is equivalent to the existence of compression in the initial data. Our talk will address the question of whether this dichotomy persists for large data problems, at least for the system of the Relativistic Euler equations in (1+1) dimensions. We shall also give some interesting studies in (3+1) dimensions. This is joint work with Dr. Shengguo Zhu.

Global Well-Posedness for a Class of Stochastic McKean-Vlasov Equations in One Dimension (A. Mayorcas): We show global well-posedness for a family of parabolic McKean--Vlasov SPDEs with additive space-time white noise. The family of interactions we consider are those given by convolution with kernels that are at least integrable. We show that global well-posedness holds in both the repulsive/defocussing and attractive/focussing cases. Our strategy relies on both pathwise and probabilistic techniques which leverage the Gaussian structure of the noise and well known properties of the deterministic PDEs.

I will be talking of recent results by Ayse Berkman and myself, as well as about a more general program of research in this area.

A group splits as an HNN-extension if and only if the rank of its abelianisation is strictly positive. If we fix a class of groups one may ask a few questions about these splittings: How distorted are the vertex and edge groups? What form can the vertex and edge groups take? If they remain in our fixed class, do they also split? If so, under iteration will we terminate at something nice? In this talk we will answer all these questions for the class of one-relator groups and go through an example or two. Time permitting, we will also discuss possible generalisations to groups with staggered presentations.

I will give an overview of semigroup C*-algebras, which are C*-algebras generated by left regular representations of semigroups. The main focus will be on examples from number theory and group theory.

An important property of Gromov hyperbolic spaces is the fact that every path for which all sufficiently long subpaths are quasi-geodesics is itself a quasi-geodesic. Gromov showed that this property is actually a characterization of hyperbolic spaces. In this talk, we will consider a weakened version of this local-to-global behaviour, called the Morse local-to-global property. The class of spaces that satisfy the Morse local-to-global property include several examples of interest, such as CAT(0) spaces, Mapping Class Groups, fundamental groups of closed 3-manifolds and more. The leverage offered by knowing that a space satisfies this property allows us to import several results and techniques from the theory of hyperbolic groups. In particular, we obtain results relating to stable subgroups, normal subgroups and algorithmic properties.

I will consider the problem of reconstructing a signal from its encrypted and corrupted image
by a Least Square Scheme. For a certain class of random encryption the problem is equivalent to finding the
configuration of minimal energy in a (unusual) version of spherical spin
glass model.  The Parisi replica symmetry breaking (RSB) scheme is then employed for evaluating
the quality of the reconstruction. It  reveals a phase transition controlled
by RSB and reflecting impossibility of the signal retrieval beyond certain level of noise.

This talk is an update on joint work with Geoff Penington on extending Morse theory to smooth functions on compact manifolds with very mild nondegeneracy assumptions. The only requirement is that the critical locus should have just finitely many connected components. To such a function we associate a quiver with vertices labelled by the connected components of the critical locus. The analogue of the Morse–Witten complex in this situation is a spectral sequence of multicomplexes supported on this quiver which abuts to the homology of the manifold.

The Alternating Directions Method of Multipliers (ADMM) is a widely popular first-order method for solving convex optimization problems. Its simplicity is arguably one of the main reasons for its popularity. For a broad class of problems, ADMM iterates by repeatedly solving perhaps the two most standard Linear Algebra problems: linear systems and symmetric eigenproblems. In this talk, we discuss how employing standard Krylov-subspace methods for ADMM can lead to tenfold speedups while retaining convergence guarantees.

I will talk about a way of building graded Lie algebras from certain Heisenberg groups. The input for this construction arises naturally when studying families of algebraic curves, and we'll look at some examples in which Lie theory interacts with number theory in an illuminating way. 

Subspace methods have the potential to outperform conventional methods, as the derivatives only need to be computed in a smaller dimensional subspace. The sub-problem that needs to be solved at each iteration is also smaller in size, and thus the Linear Algebra cost is also lower. However, if the subspace is not selected "properly", the progress per iteration can be significantly much lower than the progress of the equivalent full-space method. Thus, "improper" selection of the subspace results in subspace methods which are actually more computationally expensive per unit of progress than their full-space alternatives. The popular subspace selection methods (such as randomized) fall into this category when the objective function does not have a known (exploitable) structure. We provide a simple and effective rule to choose the subspace in the "right way" when the objective function does not have a structure. We focus on Gauss-Newton and Least-Squares, but the idea can be generalised to any other solvers and/or objective functions. We show theoretically that the cost of this strategy per unit progress lies in between (approximately) 50% and 100% of the cost of Gauss-Newton, and give an intuition why in practice, it should be closer to the favorable end of the spectrum. We confirm these expectations by running numerical experiments on the CUTEst32 test set. We also compare the proposed selection method with randomized subspace selection. We briefly show that the method is globally convergent and has a 2-step quadratic asymptotic rate of convergence for zero-residual problems.
 

Measurement outcomes in quantum theory are randomly distributed, and local measurements of the energy density of a QFT exhibit nontrivial fluctuations even in a vacuum state. This talk will present recent progress in determining the probability distribution for such measurements. In the specific case of 1+1 dimensional CFT, there are two methods (one based on Ward identities, the other on "conformal welding") which can lead to explicit closed-form results in some cases. The analogous problem for the free field in 1+3 dimensions will also be discussed.

Persistent homology has been applied to graph classification problems as a way of generating vectorizable features of graphs that can be fed into machine learning algorithms, such as neural networks. A key ingredient of this approach is a filter constructor that assigns vector features to nodes to generate a filtration. In the case where the filter constructor is smoothly tuned by a set of real parameters, we can train a neural network graph classifier on data to learn an optimal set of parameters via the backpropagation of gradients that factor through persistence diagrams [Leygonie et al., arXiv:1910.00960]. We propose a flexible, spectral-based filter constructor that parses standalone graphs, generalizing methods proposed in [Carrière et al., arXiv: 1904.09378]. Our method has an advantage over optimizable filter constructors based on iterative message passing schemes (`graph neural networks’) [Hofer et al., arXiv: 1905.10996] which rely on heuristic user inputs of vertex features to initialise the scheme for datasets where vertex features are absent. We apply our methods to several benchmark datasets and demonstrate results comparable to current state-of-the-art graph classification methods.

American mathematics was experiencing growing pains in the 1920s. It had looked to Europe at least since the 1890s when many Americans had gone abroad to pursue their advanced mathematical studies.  It was anxious to assert itself on the international—that is, at least at this moment in time, European—mathematical scene. How, though, could the Americans change the European perception from one of apprentice/master to one of mathematical equals? How could Europe, especially Germany but to a lesser extent France, Italy, England, and elsewhere, come fully to sense the development of the mathematical United States?  If such changes could be effected at all, they would likely involve American and European mathematicians in active dialogue, working shoulder to shoulder in Europe and in the United States, and publishing side by side in journals on both sides of the Atlantic. This talk will explore one side of this “equation”: European mathematicians and their experiences in the United States in the 1920s.

Let $[N] = \{1,..., N\}$ and let $A$ be a subset of $[N]$. A result of Sárközy in 1978 showed that if the difference set $A-A = \{ a - a’: a, a’ \in A\}$ does not contain any number which is one less than a prime, then $A = o(N)$. The quantitative upper bound on $A$ obtained from Sárközy’s proof has be improved subsequently by Lucier, and by Ruzsa and Sanders. In this talk, I will discuss my work on this problem. I will give a brief introduction of the iteration scheme and the Hardy-Littlewood method used in the known proofs, and our major arc estimate which leads to an improved bound.

We investigate properties of minimisers of a variational model describing the shape of charged liquid droplets. Roughly speaking, the shape of a charged liquid droplet is determined by the competition between an ”aggerating” term, due to surface tension forces, and to a ”disaggergating” term due to the repulsive effect between charged particles.

In my talk I want to present our ”first” analysis of the so called Deby-Hückel-type free energy. In particular we show that minimisers satisfy a partial regularity result, a first step of understanding the further properties of a minimiser. The presented results are joint work with Guido De Philippis and Giulia Vescovo.

The homological monodromy of the universal family of cubic threefolds defines a representation of a certain Artin-type group into the symplectic group Sp(10;\Z). We use Thurston’s classification of surface automorphisms to prove this does not factor through the genus five mapping class group.  This gives a geometric group theory perspective on the well-known irrationality of cubic threefolds, as established by Clemens and Griffiths.
 

Stochastic impulse control problems are continuous-time optimization problems in which a stochastic system is controlled through finitely many impulses causing a discontinuous displacement of the state process. The objective is to construct impulses which optimize a given performance functional of the state process. This type of optimization problem arises in many branches of applied probability and economics such as optimal portfolio management under transaction costs, optimal forest harvesting, inventory control, and valuation of real options.

In this talk, I will give an introduction to stochastic impulse control and discuss classical solution techniques. I will then introduce a new method to solve impulse control problems based on superharmonic functions and a stochastic analogue of Perron's method, which allows to construct optimal impulse controls under a very general set of assumptions. Finally, I will show how the general results can be applied to optimal investment problems in the presence of transaction costs.

This talk is based on joint work with Sören Christensen (Christian-Albrechts-University Kiel), Lukas Mich (Trier University), and Frank T. Seifried (Trier University).

References:
C. Belak, S. Christensen, F. T. Seifried: A General Verification Result for Stochastic Impulse Control Problems. SIAM Journal on Control and Optimization, Vol. 55, No. 2, pp. 627--649, 2017.
C. Belak, S. Christensen: Utility Maximisation in a Factor Model with Constant and Proportional Transaction Costs. Finance and Stochastics, Vol. 23, No. 1, pp. 29--96, 2019.
C. Belak, L. Mich, F. T. Seifried: Optimal Investment for Retail Investors with Floored and Capped Costs. Preprint, available at http://ssrn.com/abstract=3447346, 2019.

Mean-field games with absorption is a class of games, that have been introduced in Campi and Fischer (2018) and that can be viewed as natural limits of symmetric stochastic differential games with a large number of players who, interacting through a mean-field, leave the game as soon as their private states hit some given boundary. In this talk, we push the study of such games further, extending their scope along two main directions. First, a direct dependence on past absorptions has been introduced in the drift of players' state dynamics. Second, the boundedness of coefficients and costs has been considerably relaxed including drift and costs with linear growth. Therefore, the mean-field interaction among the players takes place in two ways: via the empirical sub-probability measure of the surviving players and through a process representing the fraction of past absorptions over time. Moreover, relaxing the boundedness of the coefficients allows for more realistic dynamics for players' private states. We prove existence of solutions of the mean-field game in strict as well as relaxed feedback form. Finally, we show that such solutions induce approximate Nash equilibria for the N-player game with vanishing error in the mean-field limit as N goes to infinity. This is based on a joint work with Maddalena Ghio and Giulia Livieri (SNS Pisa). 

In this talk we will discuss a new approach to non rationality of projective varieties based on HMS. Examples will be discussed.

I will elaborate on some recent developments on the theory of special functions which are relevant to the calculation of Feynman integrals in perturbative quantum field theory, highlighting the connections with some recent ideas in pure mathematics.

Speaker: Daniel Woodhouse (North)
Title: Generalizing Leighton's Graph Covering Theorem
Abstract: Before he ran off and became a multimillionaire, exploiting his knowledge of network optimisation, the computer scientist F. Thomas Leighton proved an innocuous looking result about finite graphs. The result states that any pair of finite graphs with isomorphic universal covers have isomorphic finite covers. I will explain what all this means, and why this should be of tremendous interest to group theorists and topologists.

Speaker: Benjamin Fehrman (South)
Title: Large deviations for particle processes and stochastic PDE
Abstract: In this talk, we will introduce the theory of large deviations through a simple example based on flipping a coin.  We will then define the zero range particle process, and show that its diffusive scaling limit solves a nonlinear diffusion equation.  The large deviations of the particle process about its scaling limit formally coincide with the large deviations of a certain ill-posed, singular stochastic PDE.  We will explain in what sense this relationship has been made mathematically precise.

Configuration spaces of points in Euclidean space or on a manifold are well studied in algebraic topology. But what if the points have some positive thickness? This is a natural setting from the point of view of physics, since this the energy landscape of a hard-spheres system. Such systems are observed experimentally to go through phase transitions, but little is known mathematically.

In this talk, I will focus on two special cases where we have started to learn some things about the homology: (1) hard disks in an infinite strip, and (2) hard squares in a square or rectangle. We will discuss some theorems and conjectures, and also some computational results. We suggest definitions for "homological solid, liquid, and gas" regimes based on what we have learned so far.

This is joint work with Hannah Alpert, Ulrich Bauer, Robert MacPherson, and Kelly Spendlove.

Outbreaks and epidemics from Ebola to influenza and measles are often in the news. Statistical analysis and modelling are frequently used to understand the transmission dynamics of epidemics as well as to inform and evaluate control measures, with real-time analysis being the most challenging but potentially most impactful. Examples will be drawn from diseases affecting both humans and animals.

Understanding the mechanisms of mutagenesis is important for prevention and treatment of numerous diseases, most prominently cancer. Large sequencing datasets revealed a substantial number of mutational processes in recent years, many of which are poorly understood or of completely unknown aetiology. These mutational processes leave characteristic sequence patterns in the DNA, often called "mutational signatures". We use bioinformatics methods to characterise the mutational signatures with respect to different genomic features and processes in order to unravel the aetiology and mechanisms of mutagenesis. 

In this talk, I will present our results on how mutational processes might be modulated by DNA replication. We developed a linear-algebra-based method to quantify the magnitude of replication strand asymmetry of mutational signatures in individual patients, followed by detection of these signatures in early and late replicating regions. Our analysis shows that a surprisingly high proportion (more than 75 %) of mutational signatures exhibits a significant replication strand asymmetry or correlation with replication timing. However, distinct groups of signatures have distinct replication-associated properties, capturing differences in DNA repair related to replication, and how different types of DNA damage are translated into mutations during replication. These findings shed new light on the aetiology of several common but poorly explained mutational signatures, such as suggesting a novel role of replication in the mutagenesis due to 5-methylcytosine (signature 1), or supporting involvement of oxidative damage in the aetiology of a signature characteristic for oesophageal cancers (signature 17). I will conclude with our ongoing work of wet-lab validations of some of these hypotheses and usage of computational methods (such as genetic algorithms) in guiding the development of experimental protocols.