Past

Let K be a number field and let L/K be an abelian extension. The genus field of L/K is the largest extension of L which is unramified at all places of L and abelian as an extension of K. The genus group is its Galois group over L, which is a quotient of the class group of L, and the genus number is the size of the genus group. We study the quantitative behaviour of genus numbers as one varies over abelian extensions L/K with fixed Galois group. We give an asymptotic formula for the average value of the genus number and show that any given genus number appears only 0% of the time. This is joint work with Christopher Frei and Daniel Loughran.

Learning dynamical models from data plays a vital role in engineering design, optimization, and predictions. Building models describing the dynamics of complex processes (e.g., weather dynamics, reactive flows, brain/neural activity, etc.) using empirical knowledge or first principles is frequently onerous or infeasible. Therefore, system identification has evolved as a scientific discipline for this task since the 1960ies. Due to the obvious similarity of approximating unknown functions by artificial neural networks, system identification was an early adopter of machine learning methods. In the first part of the talk, we will review the development in this area until now.

For complex systems, identifying the full dynamics using system identification may still lead to high-dimensional models. For engineering tasks like optimization and control synthesis as well as in the context of digital twins, such learned models might still be computationally too challenging in the aforementioned multi-query scenarios. Therefore, it is desirable to identify compact approximate models from the available data. In the second part of this talk, we will therefore exploit that the dynamics of high-fidelity models often evolve in lowdimensional manifolds. We will discuss approaches learning representations of these lowdimensional manifolds using several ideas, including the lifting principle and autoencoders. In particular, we will focus on learning state-space representations that can be used in classical tools for computational engineering. Several numerical examples will illustrate the performance and limitations of the suggested approaches.

This talk will explore a series of topics and example applications at the interface of graph theory and dynamics, from synchronization and diffusion dynamics on networks, to graph-based data clustering, to graph convolutional neural networks. The underlying links are provided by concepts in spectral graph theory.

Group cohomology is a powerful tool which has found many applications in modern group theory. It can be calculated and interpreted through geometric, algebraic and topological means, and as such it encodes the relationships between these different aspects of infinite groups. The aim of this talk is to introduce a circle of ideas which link group cohomology with the theory of BNS invariants, and the property of being subgroup separable. No prior knowledge of any of these topics will be assumed.

In the operator algebraic approach to quantum field theory, the DHR category is a braided tensor category describing topological point defects of a theory with at least 1 (+1) dimensions. A single von Neumann algebra with no extra structure can be thought of as a 0 (+1) dimensional quantum field theory. In this case, we would not expect a braided tensor category of point defects since there are not enough dimensions to implement a braiding. We show, however, that one can think of central sequence algebras as operators localized ``at infinity", and apply the DHR recipe to obtain a braided tensor category of bimodules of a von Neumann algebra M, which is a Morita invariant. When M is a II_1 factor, the braided subcategory of automorphic objects recovers Connes' chi(M) and Jones' kappa(M). We compute this for II_1 factors arising naturally from subfactor theory and show that any Drinfeld center of a fusion category can be realized. Based on joint work with Quan Chen and Dave Penneys.

In recent years, duality approaches have yielded new results about the high-dimensional cohomology of several groups and moduli spaces, such as $\operatorname{SL}_n(\mathbb{Z})$ and $\mathcal{M}_g$. I will explain the general strategy of these approaches and survey results that have been obtained so far. To give an example, I will first explain how Borel-Serre duality can be used to show that the rational cohomology of $\operatorname{SL}_n(\mathbb{Z})$ vanishes near its virtual cohomological dimension. This is based on joint work with Miller-Patzt-Sroka-Wilson and builds on results by Church-Farb-Putman. I will then put this into a more general context by giving an overview of analogous results for mapping class groups of surfaces, automorphism groups of free groups and further arithmetic groups such as $\operatorname{SL}_n(\mathcal{O}_K)$ and $\operatorname{Sp}_{2n}(\mathbb{Z})$.

We investigate the theoretical properties of subsampling and hashing as tools for approximate Euclidean norm-preserving embeddings for vectors with (unknown) additive Gaussian noises. Such embeddings are called Johnson-Lindenstrauss embeddings due to their celebrated lemma. Previous work shows that as sparse embeddings, if a comparable embedding dimension to the Gaussian matrices is required, the success of subsampling and hashing closely depends on the $l_\infty$ to $l_2$ ratios of the vectors to be mapped. This paper shows that the presence of noise removes such constrain in high-dimensions; in other words, sparse embeddings such as subsampling and hashing with comparable embedding dimensions to dense embeddings have similar norm-preserving dimensionality-reduction properties, regardless of the $l_\infty$ to $l_2$ ratios of the vectors to be mapped. The key idea in our result is that the noise should be treated as information to be exploited, not simply a nuisance to be removed. Numerical illustrations show better performances of sparse embeddings in the presence of noise.

In the last few years, major advances have been made in our understanding of the representation theory of reductive algebraic groups over algebraically closed fields of positive characteristic. Four key tools which are central to this progress are highest weight theory, reduction to the principal block, wall-crossing functors, and tilting modules. When considering instead the representation theory of the Lie algebras of these algebraic groups, more subtleties arise. If we look at those modules whose p-character is in so-called standard Levi form we are able to recover the four tools mentioned above, but they have been less well-studied in this setting. In this talk, we will explore the similarities and differences which arise when employing these tools for the Lie algebras rather than the algebraic groups. This research is funded by a research fellowship from the Royal Commission for the Exhibition of 1851.

In this talk we are presenting a new approach for compatible finite element discretisations for atmospheric flows on a terrain following mesh. In classical compatible finite element discretisations, the H(div)-velocity space involves the application of Piola transforms when mapping from a reference element to the physical element in order to guarantee normal continuity. In the case of a terrain following mesh, this causes an undesired coupling of the horizontal and vertical velocity components. We are proposing a new finite element space, that drops the Piola transform. For solving the equations we introduce a hybridisable formulation with trace variables supported on horizontal cell faces in order to enforce the normal continuity of the velocity in the solution. Alongside the discrete formulation for various fluid equations we discuss solver approaches that are compatible with them and present our latest numerical results.

The triangle removal lemma of Ruzsa and Szemerédi is a fundamental result in extremal graph theory; very roughly speaking, it says that if a graph is "far" from triangle-free, then it contains "many" triangles. Despite decades of research, there is still a lot that we don't understand about this simple statement; for example, our understanding of the quantitative dependencies is very poor.

In this talk, I will discuss asymmetric versions of the triangle removal lemma, where in some cases we can get almost optimal quantitative bounds. The proofs use a mix of ideas coming from graph theory, number theory, probabilistic combinatorics, and Ramsey theory.

Based on joint work with Lior Gishboliner and Asaf Shapira.

In this talk I will discuss Onsager's conjecture for energy conservation. Moreover, in 1949 Onsager conjectured that weak solutions to the incompressible Euler equations, that were Hölder continuous with Hölder exponent greater than 1/3, conserved kinetic energy. Onsager also conjectured that there were weak solutions that were Hölder continuous with Hölder exponent less than 1/3 that didn't conserve kinetic energy. I will discuss the results regarding the former, focusing mainly on the case where the spacial domain is bounded with C^2 boundary, as proved by Bardos and Titi.

This talk will be devoted to our recent developments in the analysis of emerging models for complex flows. I will start from presenting a general PDE system describing two-fluid flows, for which we prove existence of global in time weak solutions for arbitrary large initial data. I will explain where the famous approach of Lions developed for the compressible Navier-Stokes equations fails and how to use a more direct, weighted Kolmogorov criterion to prove compactness of approximating sequences of solutions. Through a formal limit, I will link the two-fluid model to the constrained two-phase models. Applications of such models include modelling of granular flows, crowd motion, or shallow water flow through a channel. The last part of my talk will focus on the rigorous derivation of these models from the compressible Navier-Stokes equations via the vanishing singular pressure or viscosity limit.

In 2018, Keating, Rodgers, Roditty-Gershon and Rudnick conjectured that the variance of sums of the divisor function in short intervals is described by a certain piecewise polynomial coming from a unitary matrix integral. That is to say, this conjecture ties a straightforward arithmetic problem to random matrix theory. They supported their conjecture by analogous results in the setting of polynomials over a finite field rather than in the integer setting. In this talk, we'll discuss arithmetic problems over F_q[T] and their connections to matrix integrals, focusing on variations on the divisor function problem with symplectic and orthogonal distributions. Joint work with Matilde Lalín.

In 2018, Keating, Rodgers, Roditty-Gershon and Rudnick conjectured that the variance of sums of the divisor
function in short intervals is described by a certain piecewise polynomial coming from a unitary matrix integral. That is
to say, this conjecture ties a straightforward arithmetic problem to random matrix theory. They supported their
conjecture by analogous results in the setting of polynomials over a finite field rather than in the integer setting. In this
talk, we'll discuss arithmetic problems over F_q[T] and their connections to matrix integrals, focusing on variations on
the divisor function problem with symplectic and orthogonal distributions. Joint work with Matilde Lalín.

In this talk I will present an exclusion process with different types of particles: A, B and C. This last type can be understood as holes. Two scaling limits will be discussed: hydrodynamic limits in the boundary driven setting; and equilibrium fluctuations for an evolution on the torus. In the later case, we distinguish several cases, that depend on the choice of the jump rates, for which we get in the limit either the stochastic Burgers equation or the Ornstein-Uhlenbeck equation. These results match with predictions from non-linear fluctuating hydrodynamics. 
(Joint work with G. Cannizzaro, A. Occelli, R. Misturini).

I shall discuss my recent work showing that the Bogomolov-Tschinkel universality conjecture holds if and only if the mapping class groups of a punctured surface is large. One consequence of this result is that all genus 2 surface-by-surface (and all genus 2 surface-by-free) groups are virtually algebraically fibered. Moreover, I will explain why simple curve homology does not always generate homology of finite covers of closed surface. I will also mention my work with O. Tosic regarding the Putman-Wieland conjecture, and explain the partial solution to the Prill's problem about algebraic curves.

 

Associated to any smooth projective curve C is its degree d Jacobian variety, parametrising isomorphism classes of degree d line bundles on C. Letting the curve vary as well, one is led to the universal Jacobian stack. This stack admits several compactifications over the stack of marked stable curves $\overline{\mathcal{M}}_{g,n}$, depending on the choice of a stability condition. In this talk I will introduce these compactified universal Jacobians, and explain how their moduli spaces can be constructed using Geometric Invariant Theory (GIT). This talk is based on arXiv:2210.11457.

Characterising intractable high-dimensional random variables is one of the fundamental challenges in stochastic computation. We develop a deep transport map that is suitable for sampling concentrated distributions defined by an unnormalised density function. We approximate the target distribution as the push-forward of a reference distribution under a composition of order-preserving transformations, in which each transformation is formed by a tensor train decomposition. The use of composition of maps moving along a sequence of bridging densities alleviates the difficulty of directly approximating concentrated density functions. We propose two bridging strategies suitable for wide use: tempering the target density with a sequence of increasing powers, and smoothing of an indicator function with a sequence of sigmoids of increasing scales. The latter strategy opens the door to efficient computation of rare event probabilities in Bayesian inference problems.

Numerical experiments on problems constrained by differential equations show little to no increase in the computational complexity with the event probability going to zero, and allow to compute hitherto unattainable estimates of rare event probabilities for complex, high-dimensional posterior densities.
 

We study supersymmetric gauge theories with 8 supercharges in d=3,4,5,6. For these theories, one can perform Higgsings by turning on VEVs of scalar fields. However, this process can often be difficult when dealing with superconformal field theories (SCFTs) where the Lagrangian is often not known. Using techniques of magnetic quivers and a new algorithm we call "Inverted Quiver Subtraction", we show how one can easily obtain the SCFT(s) after Higgsing. This technique can be equally well applied to SCFTs in d=3,4,5,6. 

The basic question in prime number theory is to try to understand the number of primes in some interesting set of integers. Unfortunately many of the most basic and natural examples are famous open problems which are over 100 years old!

We aim to give an accessible survey of (a selection of) the main results and techniques in prime number theory. In particular we highlight progress on some of these famous problems, as well as a selection of our favourite problems for future progress.

The basic question in prime number theory is to try to understand the number of primes in some interesting set of integers. Unfortunately many of the most basic and natural examples are famous open problems which are over 100 years old!

We aim to give an accessible survey of (a selection of) the main results and techniques in prime number theory. In particular we highlight progress on some of these famous problems, as well as a selection of our favourite problems for future progress.

Reaction-diffusion waves have long been used to describe the growth and spread of populations undergoing a spatial range expansion. Such waves are generally classed as either pulled, where the dynamics are driven by the very tip of the front and stochastic fluctuations are high, or pushed, where cooperation in growth or dispersal results in a bulk-driven wave in which fluctuations are suppressed. These concepts have been well studied experimentally in populations where the cooperation leads to a density-dependent growth rate. By contrast, relatively little is known about experimental populations that exhibit a density-dependent dispersal rate.

Using bacteriophage T7 as a test organism, we present novel experimental measurements that demonstrate that the diffusion of phage T7, in a lawn of host E. coli, is hindered by steric interactions with host bacteria cells. The coupling between host density, phage dispersal and cell lysis caused by viral infection results in an effective density-dependent diffusion rate akin to cooperative behavior. Using a system of reaction-diffusion equations, we show that this effect can result in a transition from a pulled to pushed expansion. Moreover, we find that a second, independent density-dependent effect on phage dispersal spontaneously emerges as a result of the viral incubation period, during which phage is trapped inside the host unable to disperse. Our results indicate both that bacteriophage can be used as a controllable laboratory population to investigate the impact of density-dependent dispersal on evolution, and that the genetic diversity and adaptability of expanding viral populations could be much greater than is currently assumed.

Néron models are mathematical objects which play a very important role in contemporary arithmetic geometry. However, they usually behave badly, particularly in respect of exact sequences and base change, which makes most problems regarding their behaviour very delicate. Chai introduced the base change conductor, a rational number associated with a semiabelian variety $G$ which measures the failure of the Néron model of $G$ to commute with (ramified) base change. Moreover, Chai conjectured that this invariant is additive in certain exact sequences. We shall introduce a new method to study the Néron models of Jacobians of proper (possibly singular) curves, and sketch a proof of Chai's conjecture for semiabelian varieties which are also Jacobians.