Gravitational instantons are Riemannian solutions to Einstein equations in four dimensions which yield complete metrics on non-compact four-manifolds, and which asymptotically `look like' flat space. Their study has been initiated by Stephen Hawking in his quest for Euclidean quantum gravity, and since then lot of effort has been put to make the term ‘look– like’ into a precise mathematical statement. While Euclidean quantum gravity does not any-more aspire to a status of a fundamental theory, the study of gravitational instantons has influenced both theoretical physics and pure mathematics. I will give an elementary introduction to the subject, and focus on the recent retirement of the Riemannian black-hole uniqueness conjecture: It is now known that there exist asymptotically flat gravitational instantons which can not be obtained as analytic continuations of black hole solutions to imaginary time.
Many proofs in additive combinatorics regarding the structure of subsets of positive upper (Banach) density have been successfully solved using techniques and methods from ergodic theory. In this talk we will present ongoing work with Daniel Palacín (Madrid) on how to tackle some variations of these statements using model-theory as an alternative approach.
The curve shortening flow evolves an embedded curve in the plane (say) in order to reduce its length as quickly as possible. It is simple to visualise for non-experts and yet hosts a beautiful and often surprising theory. It is a great model for other geometric flows such as Ricci flow and yet is useful in applications in its own right. I plan to survey some of what is known, including some of my recent work with Arjun Sobnack. The talk should be accessible to a general mathematician.
Let (X, d) be a compact metric space and (X, A, μ, T) a measure preserving dynamical system. Furthermore, given a real, positive function ψ, let W (T, ψ) and R(T, ψ) respectively denote the shrinking target set and the recurrent set associated with the dynamical system. Under certain mixing properties it is known that if the natural measure sum diverges then the recurrent and shrinking target sets are of full μ-measure. The purpose of this talk is to give a brief overview of these results, to discuss possible quantitative refinements of the full-measure conclusions, and to highlight important structural differences between the shrinking-target and recurrence theories. These differences lead to a natural and strikingly simple problem that, perhaps surprisingly, remains unresolved.
The harmonic locus consists of the monodromy-free Schroedinger operators with rational potential quadratically growing at infinity. It is known after Oblomkov that it can be identified with the set of all partitions via Wronskian map for Hermite polynomials.
We show that the harmonic locus can also be identified with the subset of the CalogeroMoser spaces, introduced by Wilson, which is invariant under a natural symplectic action of C*. As a corollary, for the multiplicity-free part of the locus we effectively solve the inverse problem for the Wronskian map by describing partition in terms of the spectrum of the corresponding Moser's matrix. We also compute the characters of the C*-action at the fixed points, proving a conjecture of Conti and Masoero.
Convex geometry has long been influenced by the study of dualities and extremal inequalities, with origins in classical affine geometry and functional analysis. In this talk, we will explore an abstract concept of duality, focusing on the classical idea of the polar set, which captures the duality of finite-dimensional normed spaces. This notion leads to fundamental questions about volume products, inspiring some of the most famous inequalities in the field. While we will mention Mahler’s influential 1939 conjecture regarding the minimiser of the volume product, the emphasis will be on the Blaschke–Santaló inequality, which identifies the maximiser, along with its modern extensions.
Model Theory is having an increasing impact in research in classical areas of mathematics, ranging from Complex Geometry through Number Theory to Combinatorics. In this talk I begin by discussing its potential applications in Foundations of Physics and subsequently present some results in this direction. Among these, I will explain how Hilbert-space axiomatisation of quantum physics is in effect an axiomatisation in the language of Continuous Logic. In particular, I prove that in this interpretation Dirac - von Neumann axioms of quantum mechanics admit approximate models with (very large) finite universes.
The fields of set theory and homological algebra are both centrally concerned with
questions of compactness, regarding the extent to which a structure's global properties are
determined by its local properties. It is thus no surprise that there has been considerable interplay
between these two fields. In this talk we will discuss some recent applications of set-theoretic
techniques to the study of the derived functor of the inverse limit, with further applications to the study
of strong homology and to the developing field of condensed mathematics. We then relate these
applications back to one of the oldest questions in set theory, that of the cardinality of the continuum.
At their core, these applications reduce to simple, purely combinatorial problems that are of interest in
their own right. No prior knowledge of either set theory or homological algebra will be assumed.
Representation theory involves studying mathematical objects by interpreting them in linear algebra: for example, we interpret group elements as matrices. It can be useful to abstract our linear algebra problem using a quiver, which is a directed graph where the vertices correspond to vector spaces and the edges to linear transformations. Gabriel showed that a quiver has finitely many representations precisely when its underlying graph is of ADE Dynkin type and noticed a pattern which Kontsevich later formalised as the fractionally Calabi-Yau property, based on categorical properties occurring in geometry. I will explain work with Mathew Pugh where we show how this is a shadow of a property of the Temperley-Lieb category, formed from non-crossing lines between dots in the plane, when the quantum parameter is a complex root of unity. This involves working with new definitions of Frobenius algebra objects and Nakayama morphisms in monoidal categories.
SubRiemannian geometry represents a vast generalisation of Riemannian geometry and it is meant to model motions that are permitted only along a limited set of directions at any point. The aim of this talk is to give an intuition of how subRiemannian geometry naturally arises from modelling different mathematical and physical problems (e.g. optimal control, image processing, thermodynamics). Surprisingly, for such a setting many fundamental mathematical objects are not yet defined or understood. If time allows, I will present what obstacles appear when trying to extend the most basic tools of geometric analysis and differential geometry to subRiemannian manifolds.
