Abstract

Manifolds with special holonomy are an active area of research bringing together a number of different parts of geometry and theoretical physics. As a taste of the rich geometry of manifolds with special holonomy, I will describe a simple explicit construction of 4-dimensional manifolds with special holonomy (more precisely, hyperkähler 4-manifolds) in terms of points in Euclidean 3-space.

Abstract

The first problem is about the minimal ramification conjecture, a refinement of the famous inverse Galois problem over Q. The aim is to derive an upper bound on the number of primes needed to ramify for a Galois extension with a Galois group in a large class of groups by combining deep work in arithmetic geometry, using the middle convolution, with deep results in analytic number theory, using sieving methods. The second problem is in arithmetic dynamics, the study of arithmetic properties of iterations of polynomials. In my visits to Africa I saw that mathematicians on that continent are fascinated by recursively defined sequences of polynomials whose complex roots are distributed on real algebraic plane curves. The basic example is the roots of unity on the unit circle. We will try to generalise some of the classical results on the arithmetic of roots of unity, such as Ihara’s theorem, to this more general setting.

Abstract

I'll give a very vague explanation of what a derived category is and then explain some stuff involving them that I find interesting.

Abstract

L-functions are one of the central objects of study in number theory. There are many beautiful theorems and many more open conjectures linking their values to arithmetic problems. The most famous example is the conjecture of Birch and Swinnerton-Dyer, which is one of the Clay Millenium Prize Problems. I will discuss this conjecture and some related open problems, and I will describe some recent progress on these conjectures, using a tool called `Euler systems’.

Abstract

I'll speak about joint work with Alex Eremenko and Gabriele Mondello on the moduli space of tori with spherical metric and one conical point of angle 2piϑ. For ϑ∈(2m−1,2m+1), the moduli space is connected and has orbifold Euler characteristic −m^2/12. For ϑ=2m the moduli space has a natural holomorphic structure and is biholomorphic to the quotient of Poincare disk by a certain subgroup of SL(2,Z) of index m^2.

Abstract

I will discuss the motivation behind the idea of a p-adic family of modular forms and how it leads to the notion of overconvergent modular forms, and the construction of the eigencurve.

Abstract

In the calculus of variations one typically has a notion of energy and looks at configurations that are optimal in the sense that they "do not waste energy". The resulting objects play a key role in geometry (and physics). We will mostly focus on the case in which the energy is related to the notion of area: we will thus look at minimal surfaces (soap films), constant-mean-curvature surfaces (soap bubbles), etc.

Abstract

Formalizing mathematics means typing it into a computer and getting a computer to check it. The area has existed for decades but is only now getting to the point where it's usable by people with no computer science background (e.g. me). I will talk about how a group of mathematicians taught a computer what a scheme is, and what the future holds.

Abstract

I will survey about some old and new directions in the Minimal Model Program.

Abstract

I will present a number of results on the distribution of standard lattice points. Towards the end I will mention a relation of lattice points to a problem in mathematical physics, namely, the study of Laplace eigenfunctions of the standard torus.

Abstract

I will discuss how to relate the geometry of a compact manifold to the growth of its Laplace eigenfunctions. In the process, I will describe some recent results pertaining to the remainder in the Weyl law and various measures of eigenfunction growth.

Abstract

I will discuss the current state of the arithmetic of elliptic curves, and the long and winding road that led us there.

Abstract

I shall give examples of discrete quotients of nilpotent Lie groups and explain how they can be used to construct symplectic forms and explicit metrics with special holonomy. The talk will focus on examples rather than theory.

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

Abstract

One of the basic problems in Diophantine geometry is to find methods to determine whether an algebraic variety over the rational numbers has a rational point. In this talk I will discuss some approaches to this, mostly focusing on the case of curves.

Abstract

There are a number of theorems and conjectures in number theory relating the values of zeta and Dirichlet L-functions with interesting arithmetic objects such as class numbers. I will explain that these values are sometimes given by linking numbers and so have topological meaning. I will describe how this interaction between arithmetic and topology is mediated by an object called an Eisenstein series, and will give some applications of the resulting picure.

Abstract

Log symplectic manifolds are manifolds with a symplectic form which is allowed to have poles along a divisor in the mildest possible way (called "logarithmic"). For example, even-dimensional complex projective space, while not holomorphic symplectic, is log symplectic. More generally, every even-dimensional toric variety admits many such structures. Many moduli spaces in geometry and physics (which are not literally symplectic) admit these structures, and they are also closely linked to Calabi-Yau algebras via quantization.

Abstract

I will explain the notion of translating solitons of the mean curvature flow and expose a new way of studying it for manifolds admitting pseudo-Riemannian submersions and cohomegeneity one actions by isometries on suitable open subsets. This general setting also covers the classical Euclidean examples. As an application, we completely classify the rotationally invariant translating solitons in Minkowski space, obtaining six types.

Abstract

Theta operators are certain weight-shifting differential operators on modular forms, defined in various contexts. I'll describe the classical construction over C (due to Ramanujan), its analogue for mod p modular forms (due to Serre and Swinnerton-Dyer) and its connection with Galois representations.

Abstract

Instanton Floer homology is a powerful 3-manifold invariant, but it’s very difficult to compute because it’s built by counting solutions to a nonlinear elliptic PDE. In this talk, I’ll explain how it can nonetheless be used to prove some very down-to-earth, combinatorial theorems about knots.

Abstract

My talk will be a walk through the forest of indefinite quadratic forms, class numbers, their relation to hyperbolic space and counting. Some of the mathematics I will discuss are really ancient, some are classical, but some are at the edge of current research.