Deflation is a technique to remove a solution to a problem so that other solutions to this problem can subsequently be found. The most prominent instance is deflation we see in eigenvalue solvers, but recent interest has been in deflation of rootfinding problems from nonlinear PDEs with many isolated solutions (spearheaded by Farrell and collaborators). 

In this talk I’ll show you recent results on deflation techniques for optimisation algorithms with many local minima, focusing on the Gauss—Newton algorithm for nonlinear least squares problems.  I will demonstrate advantages of these techniques instead of the more obvious approach of applying deflated Newton’s method to the first order optimality conditions and present some proofs that these algorithms will avoid the deflated solutions. Along the way we will see an interesting generalisation of Woodbury’s formula to least squares problems, something that should be more well known in Numerical Linear Algebra (joint work with Güttel, Nakatsukasa and Bloor Riley).

Main preprint: https://arxiv.org/abs/2409.14438. 

WoodburyLS preprint: https://arxiv.org/abs/2406.15120

A topological quantum field theory (TQFT) is a functor from a category of bordisms to a category of vector spaces. Classifying low-dimensional TQFTs often involves considering presentations of bordism categories in terms of generators and relations. In this talk, we will introduce these concepts and outline a program for obtaining such presentations using Morse–Cerf theory.

In a recent breakthrough Campos, Griffiths, Morris and Sahasrabudhe obtained the first exponential improvement of the upper bound on the classical Ramsey numbers since 1935. I will outline a reinterpretation of their proof, replacing the underlying book algorithm with a simple inductive statement. In particular, I will present a complete proof of an improved upper bound on the off-diagonal Ramsey numbers and describe the main steps involved in improving their upper bound for the diagonal Ramsey numbers to $R(k,k)\le(3.8)^k$ for sufficiently large $k$.

Based on joint work with Parth Gupta, Ndiame Ndiaye, and Louis Wei.

Let $(V(u))$ be a branching random walk and $(\eta(u))$ be i.i.d marks on the vertices. To each path $\xi$ of the tree, we associate the discounted sum $D(\xi)$ which is the sum of the $\exp(V(u))\eta_u$ along the path. We study conditions under which $\sup_\xi D(\xi)$ is finite, answering an open question of Aldous and Bandyopadhyay. We will see that this problem is related to the study of the local time process of the branching random walk along a path. It is a joint work with Yueyun Hu and Zhan Shi.

The celebrated Szemeredi-Trotter theorem states that the maximum number of incidences between $n$ points and $n$ lines in the plane is $\mathcal{O}(n^{4/3})$, which is asymptotically tight.

Solymosi conjectured that this bound drops to $o(n^{4/3})$ if we exclude subconfigurations isomorphic to any fixed point-line configuration. We describe a construction disproving this conjecture. On the other hand, we prove new upper bounds on the number of incidences for configurations that avoid certain subconfigurations. Joint work with Martin Balko.