Abstracts GQT Colloquium

Fracesca Fedele (Leeds) – Towards “super’’ cluster algebras of type A

Abstract: In the study of cluster algebras, computing cluster variables explicitly is an important problem. For surface cluster algebras, one can do so combinatorially using dimer covers of snake graphs. Recent work by Musiker, Ovenhouse and Zhang extend the theory in an attempt to define “super’’ cluster algebras of type A. The authors give a combinatorial formula, using double dimer covers of snake graphs to compute super lambda lengths in Penner-Zeitlin’s super Teichmuller spaces. In the classic surface cluster algebras setting, one can alternatively use a representation theoretic approach to compute cluster variables using the CC-map. Motivated by this, we introduce a representation theoretic interpretation of super lambda-lengths and a super CC-map which agrees with the combinatorial formula by Musiker, Ovenhouse and Zhang. This is a joint work in progress project with Canakci, Garcia Elsener and Serhiyenko.

Abstract: We discuss cluster algebras of infinite rank, and how we can view them as ind-objects of a category of finite rank cluster algebras. Examples arise from the Sato Grassmannian, and from triangulations of an infinity-gon. Taking a dual approach leads us to the notion of a pro-cluster algebra; a cluster-like structure on a dense subring of an appropriate limit of cluster algebras. A running example will lead us to a pro-cluster algebra from a completed infinity-gon. This talk is based on work in progress with Christian Korff, and on Damian Wierzbicki’s PhD thesis.

Abstract: I will talk about work with Hacking, Keel and Siebert on using mirror constructions to provide partial compactifications of the moduli of K3 surfaces. Starting with a one-parameter maximally unipotent degeneration of Picard rank 19 K3 surfaces, we construct, using methods of myself and Siebert, a mirror family which is defined in a formal neighbourhood of a union of strata of a toric variety whose fan is defined, to first approximation, as the Mori fan of the original degeneration. This formal family may then be glued in to the moduli space of polarized K3 surfaces to obtain a partial compactification. Perhaps the most significant by-product of this construction is the existence of theta functions in this formal neighbourhood, certain canonical bases for sections of powers of the polarizing line bundle.

Abstract: The quadratic Euler characteristic of an algebraic variety is a (virtual) symmetric bilinear form which refines the topological Euler characteristic and contains interesting arithmetic information when the base field is not algebraically closed. For smooth projective varieties, it has a quite concrete expression in terms of Serre duality for Kähler differentials. However, for singular varieties, it is defined abstractly using motivic homotopy theory and is still rather mysterious. I will explain some progress on concrete computations, first for Hilbert schemes and symmetric powers of algebraic surfaces (joint with Lenny Taelman) and second for conductor formulas for hypersurface singularities (older results with Marc Levine and Vasudevan Srinivas on the one hand, and joint work in progress with Ran Azouri, Niels Feld, Yonathan Harpaz and Tasos Moulinos on the other). This last project involves non-commutative techniques, based on recent developments on dg-categories of matrix factorisations and hermitian K-theory.

Ariyan Javan Peykar (Nijmegen) – The conjectures of Campana, Lang and Vojta

Abstract: One of the deepest conjectures on arithmetic geometry and complex analysis is that finiteness (resp. density) of rational points on an algebraic variety is conjecturally equivalent to the absence of an entire curve (resp. presence of a dense entire curve). In this talk I will explain this conjecture of Campana, Lang, and Vojta, as well as its natural extensions and predictions.

Ailsa Keating (Cambridge) – Structural properties of symplectic mapping class groups

Abstract: Given a symplectic manifold, an important invariant is its symplectic mapping class group, a generalisation of the classical mapping class group for real surfaces. In higher dimensions, relatively little is known; a natural question is to understand to what extent structural properties of classical mapping class groups do and don’t generalise. We will present a biased selection of results in this direction, focusing on the case of Weinstein manifold (loosely, complex submanifolds of Cn). Partially based on joint work with Paul Hacking and with Ivan Smith.

Abstract: The Laplace-Beltrami operator is a fundamental tool in the study of compact
Riemannian manifolds. In this talk I will discuss how, despite the absence of
differential- or algebro-geometric structure, any Ahlfors regular metric
measure space carries an intrinsically defined Laplace type operator. The
spectral properties of this operator are analogous to those of elliptic
differential operators on a manifold. Moreover, it is compatible, in the sense
of non-commutative geometry, with the action of a large class of non-isometric
homeomorphisms. Examples include fractals, boundaries of discrete groups,
compact symmetric spaces and manifolds with boundary. This is joint work with
Dimitris Gerontogiannis (Leiden).

Abstract: In this talk I will explain how, over a field of characteristic 0, the homotopy category of commutative algebras embeds into the homotopy category of associative algebras. This is joint work with Ricardo Campos, Dan Petersen and Daniel Robert-Nicoud.