LSGNT Lunch Seminar 2022

The Lunch Seminar is a weekly seminar organised to encourage discussion between first year LSGNT students and potential supervisors from UCL, KCL, and Imperial. Each week there are going to be two talks (one from a geometer and one from a number theorist), in which each speaker will discuss their research and potential projects.

Students are encouraged to ask questions and discuss ideas between and after the talks.

The seminar will run in a hybrid format on Tuesdays 12pm-2pm for 10 weeks starting the 4th of October. The seminar will be held in King's room S3.31, Strand building KCL. Food will be provided.

For the online access to the seminar, please reach out to Enric Solé Farré.

This seminar is organized by Inés García Redondo and Enric Solé Farré, and funded by the LSGNT.

Schedule

Click on a row to show an abstract of the talk.

Date Speaker Title Material

Abstract

I will give a brief introduction to the theory of birational geometry of algebraic varieties, discussing some recent progress and several open questions in this direction.

Abstract

What do quadratic forms have to do with closed geodesics? What do closed geodesics have to do with prime numbers? What do automorphic forms have to do with analysis? This will be a talk at the interface of Geometry and Number Theory.

Abstract

Hyperkahler manifolds arise in the different areas of differential and algebraic geometry. They are also essentially holomorphically symplectic and they have a non-Kahler analog. I will give construction of main classes of holomorphically symplectic manifolds and discuss Kahler and non-Kahler cases.

Abstract

I will give an overview of the development of the arithmetic of elliptic curves over the past century. At the end, I'll say a bit about my current research and possible LSGNT mini projects.

Abstract

Let X be an algebraic variety over the rationals. If X has a rational point then it has a point over the reals and over the p-adics for all primes p, i.e. a point over every completion of the rationals. The Hasse principle holds if the existence of points over the reals and over the p-adics for all primes p guarantees the existence of a rational point. This talk will discuss the Brauer-Manin obstruction, which explains some failures of the Hasse principle, and also how often the Hasse principle fails in some families of varieties. This will then lead on to some related questions in arithmetic statistics.

Abstract

I'm going to ramble about some stuff I work on.

Abstract

We will sketch what the idea of a moduli space for some algebraic object (curves, varieties, sheaves, etc.) is, and then recall one of the most striking examples of a moduli space, namely, the moduli space of genus $g>2$ curves. We will then turn our attention to constructing moduli spaces of higher dimensional objects, especially moduli spaces of higher dimensional varieties and foliations on those varieties. The goal here is to show how the minimal model program plays a central role in these higher dimensional moduli problems.

Abstract

Realisations of motives is just a fancy name for the category of objects where various cohomological theories take values. For general bases, not just fields, these include l-adic sheaves, locally constant sheaves, vector bundles with integrable connection and F-isocrystals. Characterisation of the image of these functors are subjects of major conjectures of arithmetic and algebraic geometry. Simpson’s conjecture claims that every rigid locally constant sheaf is motivic, while the Fontaine-Mazur conjecture characterises Galois representations of motivic origin over number fields via p-adic Hodge theory. Not surprisingly the Langlands correspondence is used as a powerful tool to prove special cases of such claims, see for example the recent work of Snowden-Tsimerman and Krishnamoorthy-Pal. Our focus however will be the Bombieri-Dwork conjecture, which answers the question for vector bundles with integrable connection. We will look at the current state of affairs and the relations to other main conjectures, such as the p-curvature conjecture of Grothendieck-Katz.

Abstract

CANCELLED

Abstract

Over the last few years, more and more research mathematicians have started playing with interactive theorem provers such as Lean. I'll give an overview of what is going on, explain how we taught a computer a hard theorem of Scholze last summer, and show you how you can join in (you don't have to do a PhD in Lean to take part in the fun!) No background in computers will be necessary to follow the talk.

Abstract

CANCELLED

Abstract

I will give a brief introduction through examples to L-functions (over Q). Two important problems on these are the subconvexity problem and moment problem. These problems, even for the most classical of L-functions, lead us to consider modular forms and automorphic forms, which lead us to more examples of L-functions, now attached to automorphic forms. The subconvexity problem for automorphic L-functions has applications to counting integer points on certain varieties. Time permitting, I will also discuss some recent progress on these problems.

Abstract

The Lindeman-Weierstrass theorem states that the exponentials of nonzero algebraic numbers are agebraically independent as long as the algebraic numbers are Q-linearly independent. The Siegel-Shidlovskii theorem is a vast generalisation of this result. I will give a complete statement of the theorem and some neat examples, and talk about some open questions in the area.

Abstract

Though nominally an analytic number theorist, I have worked on questions with a variety of different flavours. To give LSGNT students a general sense of my research, rather than discussing a single project I will try to give a brief introduction to three different topics. These will be: analysing the sizes of sets with a constraint on their greatest common divisors (combinatorial); identifying obstructions to having so-called poissonian pair correlations for dilated sequences modulo 1 (probabilistic); and using correlations of sieve weights to understand the distribution of the primes in short intervals (number-theoretic, but with some probability and analysis thrown in).

Abstract

Given a Riemannian surface, the study of sharp upper bounds for Laplacian eigenvalues under the volume constraint is a classical problem of spectral geometry. The particular interest in this problem stems from the surprising fact that the optimal metrics for such bounds arise as metrics on minimal surfaces in spheres. For surfaces with boundary a similar story connects Steklov eigenvalues with free boundary minimal surfaces in balls. In this talk I will describe the general idea behind this correspondence and provide an example that leads to a family of minimal surfaces with surprising properties.

Abstract

Many important problems in number theory and other areas of mathematics turn out to lie within the analytic theory of L-functions. One approach to understanding the analytic behaviour of L-functions is through their value distribution. In this talk I will describe some results on the distribution of values of L-functions as well as discuss some of my research in this area.

Abstract

Hyperkähler metrics in dimension 4 are the lowest dimensional examples of manifolds with special holonomy and therefore they provide a non-trivial test case for many constructions and phenomena in higher dimensional hyperkähler, Calabi-Yau, G2 and Spin(7) geometries. In 1978 the physicists Gibbons and Hawking discovered that one can produce a 4-dimensional hyperkähler metric from a single positive harmonic function on an open set of Euclidean 3-dimensional space. A lot of my current research interests revolve around higher dimensional generalizations of the Gibbons-Hawking Ansatz. As a snapshot of this story, I will explain how one can use the Gibbons-Hawking Ansatz to describe some beautiful 4-dimensional geometries very concretely in terms of points in Euclidean space and a non-negative constant.

Abstract

The local Langlands correspondence (LLC) is a kaleidoscope of conjectures relating local Galois theory, complex Lie theory, and representations of p-adic groups. I'll give an introduction to aspects of the LLC related to my work. In particular, I'll explain how we can organize the LLC by depth and give an idea of what's known at the various depths.

Abstract

In the last few decades, there has been a huge interplay between geometry (algebraic, symplectic, homogeneous spaces) and algebra (representation theory=study of linear symmetry). Two spectacular examples are the Borel—Weil—Bott and Beilinson—Bernstein theorems, giving equivalences between representations of semisimple Lie algebras and line bundles or D-modules on flag varieties (homogeneous spaces). I will briefly explain some examples of this and describe new vistas this opens up in algebra, geometry, and physics (eg “3D mirror symmetry”).

Abstract

I will give a motivated introduction to some aspects of the Langlands program, focusing on the modularity conjecture for abelian surfaces.