Below are the future seminars organised by the School of Mathematics.
Please note that only some seminar series are advertised here.
Results 1 to 10 of 23
We will talk about our recent work, a resolution of a longstanding problem on the sharp curl-Sobolev inequality for differential forms on the round sphere. We then present its geometric and variational applications that settle several open problems and conjectures in geometry and mathematical physics. This is a joint work with Mingwei Zhang.
While powerful methods have been developed for high-dimensional hypothesis testing assuming orthogonal predictors, current approaches struggle to generalise to non-orthogonal designs. Procedures such as Generalised Higher Criticism and the Cauchy Combination Test pay a substantial price in reduced power for providing an approximate orthogonalisation.
I will be describing a novel paradigm for global null testing, Stable Distillation (SD), that introduces auxiliary randomisation -- in a way different from, but broadly analogous to established model-selection and post-selection procedures such as Knockoffs and Stable Algorithms -- with a focus on minimising the noise added. SD iteratively extracts independent p-values corresponding to hypotheses according to evolving expectations about the nature of the underlying signals. This allows rapid calculation of p-values deep into the tail, enabling us to identify rare signals on a background of large numbers of null predictors in large regression models.
While it is in principle a very general method, SD was created with the particular exigencies of genomic testing in mind, and we present two applications of this method: First, testing a whole gene for association with a phenotype by combining signals across its many genomic markers. Second, as one component of an omnibus test for inferred haplotype genealogies, it can identify rare haplotypes of strong effect, while more broadly distributed associations are measured by a more conventional quadratic-form-based test. This application has been implemented in the LOCATER procedure, achieving substantial power gains over traditional single marker testing and window-based testing in cases of allelic heterogeneity.
Preprint on Stable Distillation: https://arxiv.org/pdf/2212.12539
Paper on LOCATER: https://academic.oup.com/genetics/article/232/1/iyaf158/8226070
Mathematicians have long been interested in the question of which problems can be automated, that is, which mathematical tasks can be solved using a computer. Famously, in 1900 Hilbert put forward the challenge of finding an algorithm which for any given Diophantine equation (a polynomial equation with integer coefficients and a finite number of unknowns) can decide whether or not the equation has a solution where the unknowns take integer values. In 1970 Matiyasevich showed that no such algorithm exists, that is, the problem is undecidable. One can consider equations for other algebraic structures, and in group theory equations capture many natural and well-studied notions e.g. the question of whether two elements g and h of a group G are conjugate is equivalent to asking whether the equation Xg = hX has a solution. The Diophantine problem for a group G asks whether there is an algorithm that takes any finite system of equations over G as input and decides whether or not it has a solution. The first major breakthrough in this area was achieved by Makanin in the 1970s who proved that free groups have decidable Diophantine problem. Since then the Diophantine problem has been studied for various other classes of groups and a range of decidability and undecidability results have been proved. In particular, the Diophantine problem has been investigated for one-relator groups, that is, groups defined by presentations <A|w=1> with a finite generating set A and just a single defining relation w=1. The study of one-relator groups has long been a central topic in combinatorial and geometric group theory. Magnus proved in the 1930s that all one-relator groups have decidable word problem. However, many algorithmic problems remain open for this class. For example, it is still not know whether all one-relator groups have decidable conjugacy problem. In this talk I will speak about recent joint work with Alex Levine in which we studied the Diophantine problem for one-relator groups. In particular, I will explain some of the ideas behind our recent result which shows that there exist one-relator groups with undecidable Diophantine problem.
In this talk, I will introduce spherical horoconvexity, a new notion of convexity inspired by its hyperbolic counterpart. The focus will be on its definition and geometric meaning, with geodesic spheres tangent to the equator playing the role of horospheres. I will also briefly discuss applications to curvature flows and geometric inequalities, together with a capillary analogue in the Euclidean unit ball. This is based on joint works with Julian Scheuer (Frankfurt).
In last passage percolation, we independently put weights on the vertices of the square lattice according to some probability distribution. We then study the up-right paths between points on the lattice that collect as much weight as possible. A phenomenon known as KPZ universality postulates that the essential behavior of the maximum weight and the corresponding optimal paths does not depend on the choice of weight distribution (subject to a few regularity conditions). So far however, this can only be verified for geometric and exponential distributions. In this talk, we look at general continuous distributions with finite exponential moments and prove upper probability bounds on various events where the weight maximizing paths coalesce.
More precisely, we show the following under a uniform curvature assumption: Locally for a deterministic full-Lebesgue-measure set of directions, we give a polynomial upper probability bound for the event that a Busemann function exhibits a jump in an interval of directions. The proof of the result has three parts: First, we give a bound on the coalescence time of a point-to-point geodesic and an infinite geodesic going in the same direction and second, for two infinite geodesics of distinct directions. Third, we bound the coalescence time for two finite geodesics with nearby endpoints. The latter result is an adaptation of a theorem for first passage percolation by Dembin, Elboim and Peled. It does not require any assumptions on the limit shape and gives stronger bounds the closer the endpoints are to each other.
De Fenetti’s Theorem states that all N-indexed exchangeable sequences of real-valued random variables are mixings of i.i.d sequences. For real-valued random processes with exchangeable increments on [0, 1], Kallenberg’s 1973 result provides a complete characterisation of these processes via another mixing relationship. This talk is about an analogue of Kallenberg’s Theorem for random trees. We will explore the connection between random trees and stochastic processes, leading to a conjecture by Aldous, Miermont and Pitman on a mixing relationship of continuum random trees, and a recent proof of the conjecture.
Automorphic Lie algebras are infinite-dimensional Lie algebras of meromorphic maps (usually with prescribed poles) from a Riemann surface $X$ into a finite-dimensional Lie algebra $\mathfrak{g}$, equivariant with respect to a group acting by automorphisms on both $X$ and $\mathfrak{g}$. They arise naturally in integrable systems through symmetry reductions of Lax pairs and, from an algebraic viewpoint, provide examples of equivariant map algebras.
In this talk, we discuss some algebraic aspects of elliptic automorphic Lie algebras and their connections to integrable systems. We show that well-known algebras in this context, such as Holod's hidden symmetry algebra of the Landau-Lifshitz equation and the Wahlquist-Estabrook prolongation algebra of the same equation, admit, from the automorphic perspective, a particularly simple description in terms of current algebras. The key tools are normal forms for automorphic Lie algebras, analogous to Chevalley normal forms for finite-dimensional simple Lie algebras. Moreover, these normal forms classify up to Lie algebra isomorphism a class of automorphic Lie algebras associated with complex reductive $\mathfrak{g}$, the Klein four-group acting by translations on a complex torus, and arbitrary finite invariant pole sets. This talk is based on joint work with Sara Lombardo and Vincent Knibbeler.