Oxford

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.

The aim of this talk is to describe joint work with Gergely Berczi using a recent extension to non-reductive actions of geometric invariant theory, and its links with moment maps in symplectic geometry, to study hyperbolicity of generic hypersurfaces in a projective space. Using intersection theory for non-reductive GIT quotients applied to  compactifications of bundles of invariant jet differentials over complex manifolds leads to a proof of the Green-Griffiths-Lang conjecture for a generic projective hypersurface of dimension n whose degree is greater than n^6. A recent result of Riedl and Yang then implies the Kobayashi conjecture for generic hypersurfaces of degree greater than (2n-1)^6.

Given a smooth variety X with a normal crossings divisor D (or more generally a smooth log variety) we consider the ring of logarithmic differential operators: the subring of differential operators on X generated by vector fields tangent to D. Modules over this ring are called logarithmic D-modules and generalize the classical theory of regular meromorphic connections. They arise naturally when considering compactifications.

We will discuss which parts of the theory of D-modules generalize to the logarithmic setting and how to overcome new challenges arising from the logarithmic structure. In particular, we will define holonomicity for log D-modules and state a conjectural extension of the famous Riemann-Hilbert correspondence. This talk will be very example-focused and will not require any previous knowledge of D-modules or logarithmic geometry. This is joint work with Mattia Talpo.
 

Stability conditions on triangulated categories were introduced by Bridgeland, based on ideas from string theory. Conjecturally, they control existence of solutions to the deformed Hermitian Yang-Mills equation and the special Lagrangian equation (on the A-side and B-side of mirror symmetry, respectively). I will focus on the symplectic side and sketch a program which replaces special Lagrangians by "spectral networks", certain graphs enhanced with algebraic data. Based on joint work in progress with Katzarkov, Konstevich, Pandit, and Simpson.