Tag - Moduli spaces

Oscar Randal-Williams: Cohomology of moduli spaces: a case study

25th March, 2024

I will explain recent work of Bergström–Diaconu–Petersen–Westerland, and of Miller-Patzt-Petersen-R-W, which uses methods which have been developed over the last 25 years for studying the topology of certain moduli spaces in order to answer a question in arithmetic statistics (the function field analogue of a conjecture of Conrey-Farmer-Keating-Rubinstein-Snaith on moments of quadratic L-functions). My focus will be on the translation of this question to a problem in topology, and some of the modern methods which go into solving this problem.

Laura Pertusi: Moduli spaces of stable objects in Enriques categories

29th November, 2022

Enriques categories are characterized by the property that their Serre functor is the composition of an involutive autoequivalence and the shift by 2. The bounded derived category of an Enriques surface is an example of Enriques category. Other interesting examples are provided by the Kuznetsov components of Gushel-Mukai threefolds and quartic double solids.

In this talk, we study moduli spaces of semistable objects in the Kuznetsov components of Gushel-Mukai threefolds and quartic double solids with respect to Serre-invariant stability conditions. We provide a result of non-emptiness for these moduli spaces, by using the relation with certain moduli spaces on the associated K3 category.

Ananth Shankar: Special points on moduli spaces

13th January, 2022

The study of moduli spaces, and special points in moduli spaces, has been of arithmetic interest. In this talk, I will speak about results pertaining to the algebraic and analytic distribution of special points, and touch upon topics such as the Andre-Oort conjecture, the p-adic distribution of special points, and the p-adic and mod p distribution of isogeny-orbits in these moduli spaces.

Francesca Carocci: The geometry of moduli spaces of stable maps to projective space and modular desingularisation via log blowups

7th September, 2020

An LMS online lecture course in moduli spaces.

Moduli spaces of stable maps have been of central interest in algebraic geometry for the last 30 years. In spite of that, the geometry of these spaces in genus bigger than zero is poorly understood, as the Kontsevich compactifications include many components of different dimensions meeting each other in complicated ways, and the closure of the smooth locus is difficult to describe.

In recent years a new perspective on the problem of finding better behaved compactifications, ideally smooth ones, has come from log geometry. This approach has proved successful in a series of examples and log geometry is now becoming a natural setting to study modular resolutions of moduli spaces.

The aim of this series of talks will be to see how log geometry techniques apply to give modular smooth compactifications of moduli spaces of stable maps to projective spaces in genus one and two; we will also explain why the latter are interesting from an enumerative point of view.

In more detail: we begin by studying the deformation theory and the global geometry of moduli spaces of genus one and two stable maps; we then give a brief introduction to log schemes, line bundles on log schemes and log blowups and conclude by exhibiting the log modification resolving the moduli spaces of maps in genus one and two and explaining their modular meaning.

Jared Weinstein: Moduli spaces of shtukas over function fields

13th February, 2020

We present some work in progress, on moduli spaces of Drinfeld shtukas. These spaces are the function field analogous to Shimura varieties. In fact they are more versatile; there are r-legged versions for any r. Tate's conjecture predicts some interesting relations between shtuka spaces and function field arithmetic. For instance, there should be a notion of modularity for the r-fold product of an elliptic curve. We verify these predictions in a few cases.

This is partly joint work with Noam Elkies.

Junho Peter Whang: Non-linear descent on moduli of local systems

31st October, 2017

In 1880, Markoff studied a cubic Diophantine equation in three variables now known as the Markoff equation, and observed that its integral solutions satisfy a form of non-linear descent. Generalizing this, we consider families of log Calabi-Yau varieties arising as moduli spaces for local systems on topological surfaces, and prove a structure theorem for their integral points using mapping class group dynamics. The result is reminiscent of the finiteness of class numbers for linear arithmetic group actions on homogeneous varieties, and this Diophantine perspective guides us to obtain new extensions of classical results on hyperbolic surfaces along the way.