16:00
Parametrising abelian surfaces with RM by Z[√2] using Richelot isogenies
Abstract
Correlations of almost primes
Abstract
The Hardy-Littlewood generalised twin prime conjecture states an asymptotic formula for the number of primes $p\le X$ such that $p+h$ is prime for any non-zero even integer $h$. While this conjecture remains wide open, Matom\"{a}ki, Radziwi{\l}{\l} and Tao proved that it holds on average over $h$, improving on a previous result of Mikawa. In this talk we will discuss an almost prime analogue of the Hardy-Littlewood conjecture for which we can go beyond what is known for primes. We will describe some recent work in which we prove an asymptotic formula for the number of almost primes $n=p_1p_2 \le X$ such that $n+h$ has exactly two prime factors which holds for a very short average over $h$.
16:00
Symmetric power functoriality for modular forms
Abstract
Some of the simplest expected cases of Langlands functoriality are the symmetric power liftings Sym^r from automorphic representations of GL(2) to automorphic representations of GL(r+1). I will discuss some joint work with Jack Thorne on the symmetric power lifting for holomorphic modular forms.
12:45
Twisted BRST quantization and localization in supergravity
Abstract
Supersymmetric localization is a powerful technique to evaluate a class of functional integrals in supersymmetric field theories. It reduces the functional integral over field space to ordinary integrals over the space of solutions of the off-shell BPS equations. The application of this technique to supergravity suffers from some problems, both conceptual and practical. I will discuss one of the main conceptual problems, namely how to construct the fermionic symmetry with which to localize. I will show how a deformation of the BRST technique allows us to do this. As an application I will then sketch a computation of the one-loop determinant of the super-graviton that enters the localization formula for BPS black hole entropy.
14:15
The geometry of constant mean curvature disks embedded in R^3.
Abstract
In this talk I will discuss results on the geometry of constant mean curvature (H\neq 0) disks embedded in R^3. Among other
things I will prove radius and curvature estimates for such disks. It then follows from the radius estimate that the only complete, simply connected surface embedded in R^3 with constant mean curvature is the round sphere. This is joint work with Bill Meeks.
14:00
Floer cohomology and Platonic solids
Abstract
We consider Fano threefolds on which SL(2,C) acts with a dense
open orbit. This is a finite list of threefolds whose classification
follows from the classical work of Mukai-Umemura and Nakano. Inside
these threefolds, there sits a Lagrangian space form given as an orbit
of SU(2). We prove this Lagrangian is non-displaceable by Hamiltonian
isotopies via computing its Floer cohomology over a field of non-zero
characteristic. The computation depends on certain counts of holomorphic
disks with boundary on the Lagrangian, which we explicitly identify.
This is joint work in progress with Jonny Evans.
14:15
Sequential entry and exit decisions with an ergodic criterion
Abstract
We consider an investment model that can operate in two different
modes. The transition from one mode to the other one is immediate and forms a
sequence of costly decisions made by the investment's management. Each of the
two modes is associated with a rate of payoff that is a function of a state
process which can be an economic indicator such as the price of a given
comodity. We model the state process by a general one-dimensional
diffusion. The objective of the problem is to determine the switching
strategy that maximises a long-term average criterion in a pathwise
sense. Our analysis results in analytic solutions that can easily be
computed, and exhibit qualitatively different optimal behaviours.