L4

My talk will attempt to capture the imperfect state of the art in high-resolution ice sheet modelling, aiming to expose the core performance-limiting issues.  The essential equations for modeling ice flow in a changing climate will be presented, assuming no prior knowledge of the problem.  These geophysical/climate problems are of both free-boundary and algebraic-equation-constrained character.  Current-technology models usually solve non-linear Stokes equations, or approximations thereof, at every explicit time-step.  Scale analysis shows why this current design paradigm is expensive, but building significantly faster high-resolution ice sheet models requires new techniques.  I'll survey some recently-arrived tools, some near-term improvements, and sketch some new ideas.

We consider the varifolds associated to phase transitions whose first variation of Allen-Cahn energy is $L^p$ integrable with respect to the energy measure. We can see that the Dirichlet and potential part of the energy are almost equidistributed. After passing to the phase field limit, one can obtain an integer rectifiable varifold with bounded $L^p$ mean curvature. This is joint work with Huy Nguyen.

It is well understood by works of Fukaya and Teleman that the Fukaya category of a symplectic reduction at a regular value of the moment map can be computed before taking the quotient as an equivariant Fukaya category. Informed by mirror calculations,  we will give a new geometric interpretation of the equivariant Fukaya category corresponding to a singular value of the moment map where the equivariance is traded with wrapping.

Joint work in progress with Ed Segal.

In this talk I'll discuss some applications of the mean curvature flow to self shrinkers in R^3 and R^4.

Glueing construction has been used extensively to construct solutions to nonlinear geometric PDEs. In this talk, I will focus on the glueing construction of solutions to Lagrangian mean curvature flow. Specifically, I will explain the construction of Lagrangian translating solitons by glueing a small special Lagrangian 'Lawlor neck' into the intersection point of suitably rotated Lagrangian Grim Reaper cylinders. I will also discuss an ongoing joint project with Chung-Jun Tsai and Albert Wood, where we investigate the construction of solutions to Lagrangian mean curvature flow with infinite time singularities.

First we recall the mirror symmetry identification of the coordinate ring of certain very stable upward flows in the Hitchin system and the Kirillov algebra for the minuscule representation of the Langlands dual group via the equivariant cohomology of the cominuscule flag variety (e.g. complex Grassmannian). In turn we discuss a conjectural extension of this picture to non-very stable upward flows in terms of a big commutative subalgebra of the Kirillov algebra, which also ringifies the equivariant intersection cohomology of the corresponding affine Schubert variety.

The moduli space M of stable bundles on a Riemann surface possesses a natural family of holomorphic trivector fields. The talk will introduce these objects with examples and then use them to gain information about the Hochschild cohomology of M.

Mike will briefly describe the scope and shape of science within the Met Office and of his career there. He will also outline the coordination of the development of the NEMO ocean model, which he leads, and work to ensure the marine systems at the Met Office work efficiently on modern High Performance Computers (HPCs).  In the second half of the talk, Mike will focus on two of his current scientific interests: accurate calculation of horizontal pressure forces in models with steeply sloping coordinates; and dynamical interpretations of meridional overturning circulations and ocean heat uptake.

A pair of elliptic curves is said to be $N$-congruent if their mod $N$ Galois representations are isomorphic. We will discuss a construction of the moduli spaces of $N$-congruent elliptic curves, due to Kani--Schanz, and describe how this can be exploited to compute explicit equations. Finally we will outline a proof that there exist infinitely many pairs of elliptic curves with isomorphic mod $12$ Galois representations, building on previous work of Chen and Fisher (in the case where the underlying isomorphism of torsion subgroups respects the Weil pairing).

The Riemann hypothesis (RH) is one of the great open problems in
mathematics. It arose from the study of prime numbers in an analytic
context, and—as often occurs in mathematics—developed analogies in an
algebraic setting, leading to the influential Weil conjectures. RH for
curves over finite fields was proven in the 1940’s by Weil using
algebraic-geometric methods, and later reproven by Stepanov and
Bombieri by elementary means. In this talk, we use RH for curves to
prove the Weil bound for certain (Kloosterman) exponential sums, which
in turn is a fundamental tool in the study of prime numbers.