We investigate equivalence relations for quadratic forms that can be expressed in terms of algebro-geometric properties of their associated quadrics, more precisely, birational, stably birational and motivic equivalence, and isomorphism of quadrics. We provide some examples and counterexamples and highlight some important open problems.

The talk will describe a new "Brute Force Optimizer" whose objective is to provide a very versatile derivative-free Matlab package for bound-constrained optimization, with the distinctive feature that it can be trained to improve its own performance on classes of problems specified by the user (rather than on a single-but-wide problem class chosen by the algorithm developer).  In addition, BFO can be used to optimize the performance of other algorithms and provides facilities for mixed-integer and multilevel problems, including constrained equilibrium calculations.

I will explain how algebraic spaces can be presented as Zariski geometries and prove some classical facts about algebraic spaces using the theory of Zariski geometries.

Modelling two-phase, incompressible flow with level set or volume-of-fluid formulations results in a variable coefficient Navier-Stokes system that is challenging to solve computationally. In this talk I will present work from a recent InFoMM CDT mini-project which looked to adapt current preconditioners for one-phase Navier-Stokes flows. In particular we consider systems arising from the application of finite element methodology and preconditioners which are based on approximate block factorisations. A crucial ingredient is a good approximation of the Schur complement arising in the factorisation which can be computed efficiently.

Much of the arithmetic behaviour of an elliptic curve can be understood by examining its mod p reduction at some prime p. In this talk, we will aim to explain some of the ways we can define the mod p reduction, and the classifications of which reduction types occur.

Topics to be covered include the classical reduction types (good/multiplicative/additive), the Kodaira-Neron reduction types that refine them, and the Raynaud parametrisation of a semistable abelian variety. Time permitting, we may also discuss joint work with Vladimir Dokchitser classifying the semistable reduction types of 2-dimensional abelian varieties.

I will discuss the many appearances of the class of IP sets in classical theorems of combinatorial number theory and ergodic theory. Our point of departure will be the celebrated theorem of Hindman on partition regularity of IP sets, which is crucial for the introduction of IP-limits. We then discuss how existence of certain IP-limits translates into recurrence statements, which in turn give rise to results in number theory via the Furstenberg correspondence principle. Throughout the talk, the methods of ergodic theory will play an important role - however, no prior familiarity with them is required.