Heegner points are a powerful tool for understanding the structure of the group of rational points on elliptic curves. In this talk, I will describe these points and the ideas surrounding their generalisation to other situations.
June Huh proved in 2012 that the Betti numbers of the complement of a complex hyperplane arrangement form a log concave sequence. But what if the arrangement has symmetries, and we regard the cohomology as a representation of the symmetry group? The motivating example is the braid arrangement, where the complement is the configuration space of n points in the plane, and the symmetric group acts by permuting the points. I will present an equivariant log concavity conjecture, and show that one can use representation stability to prove infinitely many cases of this conjecture for configuration spaces.
The talk will focus on recent developments regarding the (near-)critical behaviour of certain statistical physics models with long-range dependence in dimensions larger than 2, but smaller than 6, above which mean-field behaviour is known to set in. This “intermediate” regime remains a great challenge for mathematicians. The models revolve around a certain percolation phase transition that brings into play very natural probabilistic objects, such as random walk traces and the Gaussian free field.
I will discuss recent and ongoing work with Davesh Maulik that explains how Gromov-Witten invariants behave under simple normal crossings degenerations. The main outcome of the study is that if a projective manifold $X$ undergoes a simple normal crossings degeneration, the Gromov-Witten theory of $X$ is determined, via universal formulas, by the Gromov-Witten theory of the strata of the degeneration. Although the proof proceeds via logarithmic geometry, the statement involves only traditional Gromov-Witten cycles. Indeed, one consequence is a folklore conjecture of Abramovich-Wise, that logarithmic Gromov-Witten theory “does not contain new invariants”. I will also discuss applications of this to a conjecture of Levine and Pandharipande, concerning the relationship between Gromov-Witten theory and the cohomology of the moduli space of curves.
In recent years, a new class of deep neural networks has emerged, which finds its roots at model-based iterative algorithms solving inverse problems. We call these model-based neural networks deep unfolding networks (DUNs). The term is coined due to their formulation: the iterations of optimization algorithms are “unfolded” as layers of neural networks, which solve the inverse problem at hand. Ever since their advent, DUNs have been employed for tackling assorted problems, e.g., compressed sensing (CS), denoising, super-resolution, pansharpening.
In this talk, we will revisit the application of DUNs on the CS problem, which pertains to reconstructing data from incomplete observations. We will present recent trends regarding the broader family of DUNs for CS and dive into their theory, which mainly revolves around their generalization performance; the latter is important, because it informs us about the behaviour of a neural network on examples it has never been trained on before.
Particularly, we will focus our interest on overparameterized DUNs, which exhibit remarkable performance in terms of reconstruction and generalization error. As supported by our theoretical and empirical findings, the generalization performance of overparameterized DUNs depends on their structural properties. Our analysis sets a solid mathematical ground for developing more stable, robust, and efficient DUNs, boosting their real-world performance.
I will discuss the preprint arXiv:2409.18130 describing an interesting connection between 4d N=2 SCFTs and 2d chiral CFTs (alias: Vertex Operator Algebras) by way of 3d EFTs.
This session is particularly aimed at fourth-year and OMMS students who are completing a dissertation this year. For many of you this will be the first time you have written such an extended piece on mathematics. The talk will include advice on planning a timetable, managing the workload, presenting mathematics, structuring the dissertation and creating a narrative, and avoiding plagiarism.
After recalling how Hecke algebras occur in the representation theory of reductive groups, we will introduce affine Hecke algebras through a combinatorial object called a root datum. Through a worked example we will construct a filtration on the affine Hecke algebra from which we obtain the graded Hecke algebra. This has a role analogous to the Lie algebra of an algebraic group.
We will discuss star operations on these rings, with a view towards the classical problem of studying unitary representations of reductive groups.
We consider a system of interacting particles as a model for a foraging ant colony, where each ant is represented as an active Brownian particle. The interactions among ants are mediated through chemotaxis, aligning their orientations with the upward gradient of a pheromone field. Unlike conventional models, our study introduces a parameter that enables the reproduction of two distinctive behaviours: the conventional Keller-Segel aggregation and the formation of travelling clusters without relying on external constraints such as food sources or nests. We consider the associated mean-field limit of this system and establish the analytical and numerical foundations for understanding these particle behaviours.
The monadic second-order theory S1S of (ℕ,<) is decidable (it essentially describes ω-automata). Undecidability of the monadic theory of (ℝ,<) was proven by Shelah. Previously, Rabin proved decidability if the monadic quantifier is restricted to Fσ-sets.
We discuss decidability for Borel sets, or even σ-combinations of analytic sets. Moreover, the Boolean combinations of Fσ-sets form an elementary substructure. Under determinacy hypotheses, the proof extends to larger classes of sets.
A non-local game involves two non-communicating players who cooperatively play to give winning pairs of answers to questions posed by an external referee. Non-local games provide a convenient framework for exhibiting quantum supremacy in accomplishing certain tasks and have become increasingly useful in quantum information theory, mathematics, computer science, and physics in recent years. Within mathematics, non-local games have deep connections with the field of operator algebras, group theory, graph theory, and combinatorics. In this talk, I will provide an introduction to the theory of non-local games and quantum correlation classes and show their connections to different branches of mathematics. We will discuss how entanglement-assisted strategies for non-local games may be interpreted and studied using tools from operator algebras, group theory, and combinatorics. I will then present a general framework of non-local games involving quantum questions and answers.
Given an elliptic curve over the rationals, a natural problem is to find an explicit point of infinite order over a given number field when there is expected to be one. Geometric constructions are known in only two different settings. That of Heegner points, developed since the 1950s, which yields points over abelian extensions of imaginary quadratic fields. And that of Stark-Heegner points, from the late 1990s: here the points constructed are conjectured to be defined over abelian extensions of real quadratic fields. I will describe a new analytic formula which encompasses both of these, and conjecturally yields points in many other settings. This is joint work with Henri Darmon and Victor Rotger.
Please join us for refreshments outside the lecture room from 15:30.
In this talk we study the numerical approximation of the jump-diffusion McKean--Vlasov SDEs with super-linearly growing drift, diffusion and jump-coefficient. In the first step, we derive the corresponding interacting particle system and define a Milstein-type approximation for this. Making use of the propagation of chaos result and investigating the error of the Milstein-type scheme we provide convergence results for the scheme. In a second step, we discuss potential simplifications of the numerical approximation scheme for the direct approximation of the jump-diffusion McKean--Vlasov SDE. Lastly, we present the results of our numerical simulations.
AI tools like ChatGPT, Microsoft Copilot, GitHub Copilot, Claude and even older AI-enabled tools like Grammarly and MS Word, are becoming an everyday part of our research environment. This last-minute opening up of a seminar slot due to the unfortunate illness of our intended speaker (who will hopefully re-schedule for next term) gives us an opportunity to discuss what this means for us as researchers; what are good helpful uses of AI, and are there uses of AI which we might view as inappropriate? Please come ready to participate with examples of things which you have done yourselves with AI tools.
The Parker conjecture, which explores whether magnetic fields in perfectly conducting plasmas can develop tangential discontinuities during magnetic relaxation, remains an open question in astrophysics. Helicity conservation provides a topological barrier against topologically nontrivial initial data relaxing to a trivial solution. Preserving this mechanism is therefore crucial for numerical simulation.
This paper presents an energy- and helicity-preserving finite element discretization for the magneto-frictional system for investigating the Parker conjecture. The algorithm enjoys a discrete version of the topological mechanism and a discrete Arnold inequality.
We will also discuss extensions to domains with nontrivial topology.
This is joint work with Prof Patrick Farrell, Dr Kaibo Hu, and Boris Andrews
The McCullough-Miller space is a contractible simplicial complex that admits an action of the pure symmetric (outer) automorphisms of the free group, with stabilizers that are free abelian. This space has been used to derive several cohomological properties of these groups, such as computing their cohomology ring and proving that they are duality groups. In this talk, we will generalize the construction to right-angled Artin groups (RAAGs), and use it to obtain some interesting cohomological results about the pure symmetric (outer) automorphisms of RAAGs.
I will present the basic idea of the flow equation approach developed by Paweł Duch to study singular stochastic partial differential equations. In particular, I will show how it can be used to prove the existence of a solution of the stochastic Burgers equation on the one-dimensional torus.
The Aldous-Lyons conjecture from probability theory states that every (unimodular) infinite graph can be (Benjamini-Schramm) approximated by finite graphs. This conjecture is an analogue of other influential conjectures in mathematics concerning how well certain infinite objects can be approximated by finite ones; examples include Connes' embedding problem (CEP) in functional analysis and the soficity problem of Gromov-Weiss in group theory. These became major open problems in their respective fields, as many other long-standing open problems, that seem unrelated to any approximation property, were shown to be true for the class of finitely-approximated objects. For example, Gottschalk's conjecture and Kaplansky's direct finiteness conjecture are known to be true for sofic groups, but are still wide open for general groups.
In 2019, Ji, Natarajan, Vidick, Wright and Yuen resolved CEP in the negative. Quite remarkably, their result is deduced from complexity theory, and specifically from undecidability in certain quantum interactive proof systems. Inspired by their work, we suggest a novel interactive proof system which is related to the Aldous-Lyons conjecture in the following way: If the Aldous-Lyons conjecture was true, then every language in this interactive proof system is decidable. A key concept we introduce for this purpose is that of a Subgroup Test, which is our analogue of a Non-local Game. By providing a reduction from the Halting Problem to this new proof system, we refute the Aldous-Lyons conjecture.
This talk is based on joint work with Lewis Bowen, Alex Lubotzky, and Thomas Vidick.
No special background in probability theory or complexity theory will be assumed.
A C*-correspondence between two C*-algebras is a generalization of a *-homomorphism. Laca and Neshveyev showed that, like a *-homomorphism, there is an induced map between traces of the algebras. Given sufficient regularity conditions, the map defines a bounded operator between the spaces of (bounded) tracial linear functionals.
This operator can be of independent interest - a special case of correspondence gives Ruelle's operator associated to a non-invertible discrete-time dynamical system, and the study of Ruelle's operator is the basis of his thermodynamic formalism. Moreover, by the work of Laca and Neshveyev, the operator's positive eigenvectors determine the KMS states of the gauge action on the Cuntz-Pimsner algebra of the correspondence.
Given a C*-correspondence from a C*-algebra to itself, we will present a sufficient condition on the C*-correspondence that implies the operator on traces has a unique positive eigenvector, and moreover a spectral gap. This result recovers the Perron-Frobenius theorem, aspects of Ruelle's thermodynamic formalism, and unique KMS state results for a variety of constructions of Cuntz-Pimsner algebras, including the C*-algebras associated to self-similar groupoids. The talk is based on work in progress.
The celebrated Szemeredi-Trotter theorem states that the maximum number of incidences between $n$ points and $n$ lines in the plane is $\mathcal{O}(n^{4/3})$, which is asymptotically tight.
Solymosi conjectured that this bound drops to $o(n^{4/3})$ if we exclude subconfigurations isomorphic to any fixed point-line configuration. We describe a construction disproving this conjecture. On the other hand, we prove new upper bounds on the number of incidences for configurations that avoid certain subconfigurations. Joint work with Martin Balko.
Let G be the points of a reductive group over a p-adic field. According to Bernstein, the category of smooth complex representations of G decomposes as a product of indecomposable subcategories (blocks), each determined by inertial supercuspidal support. Moreover, each of these blocks is equivalent to the category of modules over a Hecke algebra, which is understood in many (most) cases. However, when the coefficients of the representations are now allowed to be in a more general ring (in which p is invertible), much of this fails in general. I will survey some of what is known, and not known.