University of Oxford

Adaptive cubic regularization methods have recently emerged as a credible alternative to line search and trust-region for smooth nonconvex optimization, with optimal complexity amongst second-order methods. Here we consider a general class of adaptive regularization methods, that use first- or higher-order local Taylor models of the objective regularized by a(ny) power of the step size. We investigate the worst-case complexity/global rate of convergence of these algorithms, in the presence of varying (unknown) smoothness of the objective. We find that some methods automatically adapt their complexity to the degree of smoothness of the objective; while others take advantage of the power of the regularization step to satisfy increasingly better bounds with the order of the models. This work is joint with Nick Gould (RAL) and Philippe Toint (Namur).

Computing distinct solutions of differential equations -- Patrick Farrell

Abstract: TBA

Triangles and equations -- Yufei Zhao

Abstract: I will explain how tools in graph theory can be useful for understanding certain problems in additive combinatorics, in particular the existence of arithmetic progressions in sets of integers. 

TBA

We shall describe a new proof of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety. 
Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on  Langer's theorem that the Harder-Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs. Our proof produces a numerical upper-bound for the degree of the finite morphism from an isotrivial variety appearing in the statement of the Mordell-Lang conjecture. This upper-bound is given in terms of the Frobenius-stabilised slopes of the cotangent bundle of the variety.

I will describe some diophantine problems and results motivated by the analogy between powers of the modular curve and powers of the multiplicative group in the context of the Zilber-Pink conjecture.