The fundamental group of a hyperbolic surface has an infinite number of rank k subgroups. What does it mean, therefore, to pick a 'random' subgroup of this type? In this talk, I will introduce a method for counting subgroups and discuss how counting allows us to study the properties of a random subgroup and its associated cover.
We will explore the connection between Celestial and Euclidean Anti-de Sitter (EAdS) holography in the massive scalar case. Specifically, exploiting the so-called hyperbolic foliation of Minkowski space-time, we will show that each contribution to massive Celestial correlators can be reformulated as a linear combination of contributions to corresponding massive Witten correlators in EAdS. This result will be demonstrated explicitly both for contact diagrams and for the four-point particle exchange diagram, and it extends to all orders in perturbation theory by leveraging the bootstrapping properties of the Celestial CFT (CCFT). Within this framework, the Kantorovic-Lebedev transform plays a central role. This transform will allow us to make broader considerations regarding non-perturbative properties of a CCFT.
Given a singularity with a crepant resolution, a symmetry of the derived
category of coherent sheaves on the resolution may often be constructed
using the formalism of spherical functors. I will introduce this, and
new work (arXiv:2409.19555) on general constructions of such symmetries
for hypersurface singularities. This builds on previous results with
Segal, and is inspired by work of Bodzenta-Bondal.
Joint work with Paul Hacking (U Mass Amherst). We first explain how to
prove homological mirror symmetry for a maximal normal crossing
Calabi-Yau surface Y with split mixed Hodge structure. This includes the
case when Y is a type III K3 surface, in which case this is used to
prove a conjecture of Lekili-Ueda. We then explain how to build on this
to prove an HMS statement for K3 surfaces. On the symplectic side, we
have any K3 surface (X, ω) with ω integral Kaehler; on the algebraic
side, we get a K3 surface Y with Picard rank 19. The talk will aim to be
accessible to audience members with a wide range of mirror symmetric
backgrounds.
A number of moduli problems are, via Hodge theory, closely related to
ball quotients. In this situation there is often a choice of possible
compactifications such as the GIT compactification´and its Kirwan
blow-up or the Baily-Borel compactification and the toroidal
compactificatikon. The relationship between these compactifications is
subtle and often geometrically interesting. In this talk I will discuss
several cases, including cubic surfaces and threefolds and
Deligne-Mostow varieties. This discussion links several areas such as
birational geometry, moduli spaces of pointed curves, modular forms and
derived geometry. This talk is based on joint work with S.
Casalaina-Martin, S. Grushevsky, S. Kondo, R. Laza and Y. Maeda.
A group is coherent if all its finitely generated subgroups are finitely presented. Aside from some easy cases, it appears that coherence is a phenomenon that occurs only among groups of cohomological dimension 2. In this talk, we will give many examples of coherent and incoherent groups, discuss techniques to prove a group is coherent, and mention some open problems in the area.
How much can the order type - the list of element orders (with multiplicities)—reveal about the structure of a finite group G? Can it tell us whether G is abelian, nilpotent? Can it always determine whether G is solvable?
This last question was posed in 1987 by John G. Thompson and I answered it negatively this year. The search for a counterexample was quite a puzzle hunt! It involved turning the problem into linear algebra and solving an integer matrix equation Ax=b. This would be easy if not for the fact that the size of A was 100,000 by 10,000…
Consider a connected graph and choose a subset of its vertices. From this simple setup, Iyama and Wemyss define a collection of real hyperplanes known as an intersection arrangement, going on to classify all tilings of the affine plane that arise in this way. These "local" generalisations of Coxeter combinatorics also admit a nice wall-crossing structure via Dynkin involutions and longest Weyl elements. In this talk I give an analogous classification in the hyperbolic setting using the data of an "overextended" ADE diagram with three distinguished vertices. I then discuss ongoing work applying intersection arrangements to parametrise notions of stability conditions for preprojective algebras.
Suppose a finite group satisfies the following property: If you take two random elements, then with probability bigger than 5/8 they commute. Then this group is commutative.
Starting from this well-known result, it is natural to ask: Do similar results hold for other laws (p-groups, nilpotent groups...)? Are there analogous results for infinite groups? Are there phenomena specific to the infinite setup?
We will survey known and new results in this area. New results are joint with Gideon Amir, Maria Gerasimova and Gady Kozma.