Oxford

In this talk I will explain a category theoretic perspective on geometry.  Starting with a category of local objects (of and algebraic nature), and a (Grothendieck) 
topology on it, one can define global objects such as schemes and stacks. Examples of this  approach are algebraic, analytic, differential geometries and also more exotic geometries  such as analytic and differential geometry over the integers and analytic geometry over  the field with one element. In this approach the notion of a point is not primary but is  derived from the local to global structure. The Zariski and Huber spectra are recovered  in this way, and we also get new spectra which might be of interest in model theory.

Feynman integrals govern the perturbative expansion in quantum field theories. As periods, these integrals generate representations of a motivic Galois group. I will explain this idea and illustrate the 'coaction principle', a mechanism that constrains which periods can appear at any loop order.
 

Let n be a random integer (sampled from {1,..,X} for some large X). It is a classical fact that, typically, n will have around (log n)^{log 2} divisors. Must some of these be close together? Hooley's Delta function Delta(n) is the maximum, over all dyadic intervals I = [t,2t], of the number of divisors of n in I. I will report on joint work with Kevin Ford and Dimitris Koukoulopoulos where we conjecture that typically Delta(n) is about (log log n)^c for some c = 0.353.... given by an equation involving an exotic recurrence relation, and then prove (in some sense) half of this conjecture, establishing that Delta(n) is at least this big almost surely.

I will discuss work in progress with Sary Drappeau and Maksym Radziwill on low-lying zeros of Dirichlet L-functions. By way of motivation I will discuss some results on the spacings of zeros of the Riemann zeta function, and the conjectures of Katz and Sarnak relating the distribution of low-lying zeros of L-functions to eigenvalues of random matrices. I will then describe some ideas behind the proof of our theorem.
 

We present a programme towards a combinatorial language for higher (stratified) Morse-Cerf theory. Our starting point will be the interpretation of a Morse function as a constructible bundle (of manifolds) over R^1. Generalising this, we describe a surprising combinatorial classification of constructible bundles on flag foliated R^n (the latter structure of a "flag foliation” is needed for us to capture the notions of "singularities of higher Morse-Cerf functions" independently of differentiable structure). We remark that flag foliations can also be seen to provide a notion of directed topology and in this sense higher Morse-Cerf singularities are closely related to coherences in higher category theory. The main result we will present is the algorithmic decidability of existence of mutual refinements of constructible bundles. Using this result, we discuss how "combinatorial stratified higher Morse-Cerf theory" opens up novel paths to the computational treatment of interesting questions in manifold topology.

The Gibbons-Hawking ansatz is a powerful method for constructing a large family of hyperkaehler 4-manifolds (which are thus Ricci-flat), which appears in a variety of contexts in mathematics and theoretical physics. I will describe work in progress to understand the theory of minimal surfaces and mean curvature flow in these 4-manifolds. In particular, I will explain a proof of a version of the Thomas-Yau Conjecture in Lagrangian mean curvature flow in this setting. This is joint work with G. Oliveira.

We study nearly singular PDEs that arise in the solution of linear elasticity and incompressible flow. We will demonstrate, that due to the nearly singular nature, standard methods for the solution of the linear systems arising in a finite element discretisation for these problems fail. We motivate two key ingredients required for a robust multigrid scheme for these equations and construct robust relaxation and prolongation operators for a particular choice of discretisation.
 

I will give a brief survey of some problems in curvature free geometry and sketch

a new proof of the following:

Theorem (Guth). There is some $\delta (n)>0$ such that if $(M^n,g)$ is a closed aspherical Riemannian manifold and $V(R)$ is the volume of the largest ball of radius $R$ in the universal cover of $M$, then $V(R)\geq \delta(n)R^n$ for all $R$.

I will also discuss some recent related questions and results.

There are various notions of rank, which measure the complexity of a tensor or polynomial. Cubic surfaces can be viewed as symmetric tensors.  We consider the non-symmetric tensor rank and the symmetric Waring rank of cubic surfaces, and show that the two notions coincide over the complex numbers. The results extend to order three tensors of all sizes, implying the equality of rank and symmetric rank when the symmetric rank is at most seven. We then explore the connection between the rank of a polynomial and the singularities of its vanishing locus, and we find the possible singular loci of a cubic surface of given rank. This talk is based on joint work with Eunice Sukarto.
 

The Alternating Directions Method of Multipliers (ADMM) is a widely popular first-order method for solving convex optimization problems. Its simplicity is arguably one of the main reasons for its popularity. For a broad class of problems, ADMM iterates by repeatedly solving perhaps the two most standard Linear Algebra problems: linear systems and symmetric eigenproblems. In this talk, we discuss how employing standard Krylov-subspace methods for ADMM can lead to tenfold speedups while retaining convergence guarantees.