Oxford

Stable commutator length (scl) is a well established invariant of elements g in the commutator subgroup (write scl(g)) and has both geometric and algebraic meaning.  A group has a \emph{gap} in stable commutator length if for every non-trivial element g, scl(g) > C for some C > 0.
SCL may be interpreted as an 'algebraic translation length' and such a gap may be thus interpreted an 'algebraic injectivity radius'.
Many classes of groups have such a gap, like hyperbolic groups, mapping class groups, Baumslag-Solitar groups and graph of groups.
In this talk I will show that Right-Angled Artin Groups have the optimal scl-gap of 1/2. This yields a new invariant for the vast class of subgroups of Right-Angled Artin Groups.

We begin by reviewing numerical methods for problems in one variable and find that univariate polynomials are the starting point for most of them.  A similar review in several variables, however, reveals that multivariate polynomials are not so important.  Why?  On the other hand in pure mathematics, the field of algebraic geometry is precisely the study of multivariate polynomials.  Why?

We recall the impact of stabilizing a 4-manifold with $S^2 \times S^2$. The corresponding local situation concerns knots in the 3-sphere which bound (nullhomotopic) discs in a stabilized 4-ball. We explain how these discs arise, and discuss bounds on the minimal number of stabilizations needed. Then we compare this minimal number to the 4-genus.
This is joint work with A. Conway.

In this talk I will present a large class of non-supersymmetric AdS7 solutions of IIA supergravity, and their (in)stabilities. I will start by reviewing supersymmetric AdS7 solutions of 10D supergravity dual to 6D (1,0) SCFTs. I will then focus on their non-supersymmetric counterpart, discussing how they are related. The connection between supersymmetric and non-supersymmetric solutions leads to a hint for the SUSY breaking mechanism, which potentially allows to evade some of the assumptions of the Ooguri-Vafa Conjecture about the AdS landscape. A big subset of these solutions shows a curious pattern of perturbative instabilities whenever many open-string modes are considered. On the other hand an infinite class remains apparently stable.

The idea of adjusting prices in order to sell goods at the highest acceptable price, such as haggling in a market, is as old as money itself. We consider the problem of pricing multiple products on a network of resources, such as that faced by an airline selling tickets on its flight network. In this talk I will consider various optimization relaxations to the deterministic dynamic pricing problem on a network. This is joint work with Raphael Hauser.

In recent years, there has been an increased interest in exploiting rank structure of matrices arising from the discretization of partial differential equations to develop fast direct solvers. In this talk, I will outline the fundamental ideas of this topic in the context of solving the integral equation formulation of the Helmholtz equation, known as the Lippmann-Schwinger equation, and will discuss some plans for future work to develop new, higher-order solvers. This is joint work with Gunnar Martinsson.

This talk will feature a brief introduction to persistent homology, the vanguard technique in topological data analysis. Nothing will be required of the audience beyond a willingness to row-reduce enormous matrices (by hand if we can, by machine if we must).

For decades, researchers have been studying efficient numerical methods to solve differential equations, most of them optimised for one-core processors. However, we are about to reach the limit in the amount of processing power we can squeeze into a single processor. This explains the trend in today's computing industry to design high-performance processors looking at parallel architectures. As a result, there is a need to develop low-complexity parallel algorithms capable of running efficiently in terms of computational time and electric power.

Parallelisation across time appears to be a promising way to provide more parallelism. In this talk, we will introduce the main algorithms, following (Gander, 2015), with a particular focus on the parareal algorithm.

The focus of this talk is how to tackle huge linear least square problems via sketching, a dimensionality reduction technique from randomised numerical linear algebra. The technique allows us to project the huge problem to a smaller dimension that captures essential information of the original problem. We can then solve the projected problem directly to obtain a low accuracy solution or using the projected problem to construct a preconditioner for the original problem to obtain a high accuracy solution. I will survey the existing projection techniques and evaluate the performance of sketching for linear least square problems by comparing it to the state-of-the-art traditional solution methods. More than ten-fold speed-up has been observed in some cases.