Past

The Zilber-Pink Conjecture, which should rule the behaviour of intersections between an algebraic variety and a countable family of "special varieties", does not take into account double intersections; some results related to tangencies with special subvarieties have been obtained by Marché-Maurin in 2014 in the case of powers of the multiplicative group and by Corvaja-Demeio-Masser-Zannier in 2019 in the case of elliptic schemes. We prove that any algebraic curve contained in Y(1)^2 is tangent to finitely many modular curves, which are the one-codimensional special subvarieties. The proof uses the Pila-Zannier strategy: the Pila-Wilkie counting theorem is combined with a degree bound coming from a Weakly Bounded Height estimate. The seminar will be divided into two talks: in the first one, we will explain the general Zilber-Pink Conjecture philosophy, we will describe the main tools used in this context and we will see what the differences in the double intersection case are; in the second one, we will focus on the proofs and we will see how o-minimality plays a main role here. In the case of a curve in Y(1)^2, o-minimality is also used for height estimates (which are then ineffective, which is usually not the case).

The space of subgroups of a countable group is a compact topological space which encodes many of the properties of its non-free actions. We will discuss some approaches to studying the Cantor-Bendixson decomposition of this space in the context of hyperbolic groups and groups which act (nicely) on trees. We will also give some conditions under which the conjugation action on the perfect kernel is highly topologically transitive and see how this can be applied to find new examples of groups (including all virtually compact special groups) which admit faithful transitive amenable actions. This is joint work with Damien Gaboriau.

The study of topologically ordered quantum phases has led to interesting connections with, for example, the study of subfactors. In this talk, I will introduce a new axiomatisation of such quantum models defined on d-dimensional square lattices in terms of nets of projections. These local topological order axioms are satisfied by known 2D models such as the toric code and Levin-Wen models built on a unitary fusion category. We show that these axioms lead to a definition of boundary algebras naturally living on a hyperplane. This boundary algebra encodes information about the excitations in the bulk theory, leading to a bulk-boundary correspondence. I will outline the main points, with an emphasis on interesting connections to operator algebras and fusion categories. Based on joint work with C. Jones, Penneys, and Wallick (arXiv:2307.12552).

It was proved by Selberg's in the 1940's that the typical values of the logarithm of the Riemann zeta function on the critical line is distributed like a complex Gaussian random variable. In this talk, I will present recent work with Emma Bailey that extends the Gaussian behavior for the real part to the large deviation regime. This gives a new proof of unconditional upper bounds of the $2k$-moments of zeta for $0\leq k\leq 2$, and lower bounds for $k>0$. I will also discuss the connections with random matrix theory and with the Moments Conjecture of Keating & Snaith. 

Given a group $G$, finding a geometric action of $G$ on a CAT(0) cube complex can be used to say some rather strong things about $G$. Such actions are not always easy to find, however, which makes it useful to have sufficient conditions, both for existence and for non-existence. This talk concerns the latter: we shall see a coarse geometric obstruction to a group admitting a cocompact cubulation. Based on joint work with Zach Munro.

In representation theory, the characters of induced representations are explicitly known in terms of the character of the inducing representation. This leads to the question of understanding the elliptic representation space, i.e., the space of representations modulo the properly (parabolically) induced characters. I will give an overview of the description of the elliptic space for finite Weyl groups, affine Weyl groups, affine Hecke algebras, and their connection with the geometry of the nilpotent cone of a semisimple complex Lie algebra. These results fit together in the representation theory of semisimple p-adic groups, where they lead to a new description of the elliptic space within the framework of the local Langlands parameterisation.

Consider the following "controlled" random graph process: edges of the complete graph are revealed one by one in random order to an online algorithm, which immediately decides whether to retain each observed edge. The algorithm's objective is to construct a graph property within specified constraints on the total number of observed edges ("time") and the total number of retained edges ("budget").

During this talk, I will present results in this model for natural graph properties, such as connectivity, Hamiltonicity, and containment of fixed-size subgraphs. Specifically, I will describe a strategy to construct a Hamilton cycle at the hitting time for minimum degree 2 by retaining a linear number of edges. This extends the classical hitting time result for Hamiltonicity originally established by Ajtai–Komlós–Szemerédi and Bollobás.

The talk is based on joint work with Alan Frieze and Michael Krivelevich.

I will talk about supersymmetric quantum field theories with 8 supercharges in dimensions 3-6. After a brief introduction I will mostly speak about the moduli space of vacua of such theories, and in particular their Higgs branches, which are so called symplectic singularities (or mild generalisations thereof). Powerful theorems from mathematics say that a singular Higgs branch is stratified into a disjoint union of smooth open subsets, so called symplectic leaves. This stratification matches exactly the pattern of partial Higgsings of the theory in question. After introducing the stratification and explaining its physical interpretation, I will show how brane systems and so called magnetic quivers can be used to compute it.

Random walks on random graphs are associated with diffusion in disordered media. In this talk, the graphs of interest are uniform spanning tree (UST) and loop-erased random walk (LERW). First I will demonstrate some asymptotic behavior of the simple random walk on the three-dimensional UST. Next I will discuss annealed transition probability of the simple random walk on high-dimensional LERWs.

 

This talk presents some recent progress for models coupling large-strain, geometrically nonlinear elasto-plasticity with the movement of dislocations. In particular, a new geometric language is introduced that yields a natural mathematical framework for dislocation evolutions. In this approach, the fundamental notion is that of 2-dimensional "slip trajectories" in space-time (realized as integral 2-currents), and the dislocations at a given time are recovered via slicing. This modelling approach allows one to prove the existence of solutions to an evolutionary system describing a crystal undergoing large-strain elasto-plastic deformations, where the plastic part of the deformation is driven directly by the movement of dislocations. This is joint work with T. Hudson (Warwick).

I will introduce the Hodge-de-Rham spectral sequence and formulate an algebraic Hodge decomposition theorem. Time permitting, I will sketch Deligne and Illusie’s proof of the Hodge decomposition using positive characteristic methods.

We prove convergence results of the simulation of the density solution to the McKean-Vlasov equation, when the measure variable is in the drift. Our method builds upon adaptive nonparametric results in statistics that enable us to obtain a data-driven selection of the smoothing parameter in a kernel-type estimator. In particular, we give a generalised Bernstein inequality for Euler schemes with interacting particles and obtain sharp deviation inequalities for the estimated classical solution. We complete our theoretical results with a systematic numerical study and gather empirical evidence of the benefit of using high-order kernels and data-driven smoothing parameters. This is a joint work with M. Hoffmann.

Until the 2010’s the only « comparison geometry » result for compact Riemannian manifolds (M^n,g) with scal≥n(n-1) was Greene’s upper bound on the injectivity radius. Moreover, it is known that classical metric invariants (volume, diameter) cannot be controlled by a lower bound on the scalar curvature alone. It has only recently been discovered that some more subtle invariants, such as 2-systoles, can be controlled under a lower bounds on scal provided M has enough topology. We will present some results of Bray-Brendle-Neves (in dim 3), Zhu (in dim≤7) for S^2xT^(n-2), some version for S^2xS^2 and some conjecture with more general topology which we show to hold true under the additional assumption of Kaehlerness.

Harmonic maps from surfaces to other manifolds is a fundamental object of geometric analysis with many applications, for example to minimal surfaces. In particular, there are many available methods of constructing them such, such as using complex geometry, min-max methods or flow techniques. By contrast, much less is known for harmonic maps from higher dimensional manifolds. In the present talk I will explain the role of dimension in this problem and outline the recent joint work with D. Stern, where we provide a min-max construction for higher-dimensional harmonic maps. If time permits, an application to eigenvalue optimisation problems will be discussed. Based on joint work with D. Stern.

Job applications involve a lot of work and can be overwhelming. Join us for a workshop and Q+A session focused on breaking down academic applications: we’ll talk about approaching reference letter writers, writing research statements, and discussing what makes a great CV and covering letter.

In this talk, I will present joint work with Omer Bobrowski:  a series of statements regarding the behaviour of persistence diagrams arising from random point-clouds. I will present evidence that, viewed in the right way, persistence values obey a universal probability law, that depends on neither the underlying space nor the original distribution of the point-cloud.  I will present two versions of this universality: “weak” and “strong” along with progress which has been made in proving the statements.  Finally, I will also discuss some applications of this phenomena based on detecting structure in data.

Mathematical models are useful tools for guiding infectious disease outbreak control measures. Before a pathogen has even entered a host population, models can be used to determine the locations that are most at risk of outbreaks, allowing limited surveillance resources to be deployed effectively. Early in an outbreak, key questions for policy advisors include whether initial cases will lead on to a major epidemic or fade out as a minor outbreak. When a major epidemic is ongoing, models can be applied to track pathogen transmissibility and inform interventions. And towards the end of (or after) an outbreak, models can be used to estimate the probability that the outbreak is over and that no cases will be detected in future, with implications for when interventions can be lifted safely. In this talk, I will summarise the work done by my research group on modelling different stages of infectious disease outbreaks. This includes: i) Before an outbreak: Projections of the locations at-risk from vector-borne pathogens towards the end of the 21^st century under a changing climate; ii) Early in an outbreak: Methods for estimating the risk that introduced cases will lead to a major epidemic; and iii) During a major epidemic: A novel approach for inferring the time-dependent reproduction number during outbreaks when disease incidence time series are aggregated temporally (e.g. weekly case numbers are reported rather than daily case numbers). In addition to discussing this work, I will suggest areas for further research that will allow effective interventions to be planned during future infectious disease outbreaks.

Kaplansky's Zerodivisor Conjecture predicts that the group algebra kG is a domain, where k is a field and G is a torsion-free group. Though the general sentiment is that the conjecture is false, it still remains wide open after more than 70 years. In this talk we will survey known positive results surrounding the Zerodivisor Conjecture, with a focus on the technique of embedding group algebras into division rings. We will also present some new results in this direction, which are joint with Pablo Sánchez Peralta.

A key ingredient in the proof of the model completeness of the real exponential field was a valuation inequality for polynomially bounded o-minimal structures. I shall briefly describe the argument, and then move on to the complex exponential field and Zilber's quasiminimality conjecture for this structure. Here, one can reduce the problem to that of establishing an analytic continuation property for (complex) germs definable in a certain o-minimal expansion of the real field and in order to study this question I propose notions of "complex Hardy fields" and "complex valuations".   Here, the value group is not necessarily ordered but, nevertheless, one can still prove a valuation inequality.

I will survey Cartan respectively diagonal subalgebras of nuclear C*-algebras. This setup corresponds to a presentation of the ambient C*-algebra as an amenable groupoid C*-algebra, which in turn means that there is an underlying structure akin to an amenable topological dynamical system.

The existence of such subalgebras is tightly connected to the UCT problem; the classification of Cartan pairs is largely uncharted territory. I will present new constructions of diagonals of the Jiang-Su algebra Z and of the Cuntz algebra O_2, and will then focus on distinguishing Cantor Cartan subalgebras of O_2.

It was asked by E. Szemerédi if, for a finite set $A\subset \mathbf{Z}$, one can improve estimates for $\max\{|A+A|,|A\cdot A|\}$, under the constraint that all integers involved have a bounded number of prime factors -- that is, each $a\in A$ satisfies $\omega(a)\leq k$. In this paper we show that this maximum is at least of order $|A|^{\frac{5}{3}-o(1)}$ provided $k\leq (\log|A|)^{1-\varepsilon}$ for some $\varepsilon>0$. In fact, this will follow from an estimate for additive energy which is best possible up to factors of size $|A|^{o(1)}$. Our proof consists of three parts: combinatorial, analytical and number theoretical.

Central clearing counterparty houses (CCPs) play a fundamental role in mitigating the counterparty risk for exchange traded options. CCPs cover for possible losses during the liquidation of a defaulting member's portfolio by collecting initial margins from their members. In this article we analyze the current state of the art in the industry for computing initial margins for options, whose core component is generally based on a VaR or Expected Shortfall risk measure. We derive an approximation formula for the VaR at short horizons in a model-free setting. This innovating formula has promising features and behaves in a much more satisfactory way than the classical Filtered Historical Simulation-based VaR in our numerical experiments. In addition, we consider the neural-SDE model for normalized call prices proposed by [Cohen et al., arXiv:2202.07148, 2022] and obtain a quasi-explicit formula for the VaR and a closed formula for the short term VaR in this model, due to its conditional affine structure.