University of Oxford

The second fundamental form of an embedded manifold must satisfy a set of constraint equations known as the Gauß-Codazzi equations. Since work of Chen-Slemrod-Wang, these equations are known to satisfy a particular div-curl structure: under suitable L^p bound on the second fundamental form, the curvatures are weakly continuous. In this talk we explore generalisations of this original result under weaker assumptions. We show how techniques from fluid dynamics can yield interesting insight into the weak continuity properties of isometric embeddings.

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Teichmüller harmonic map flow is a geometric flow designed to evolve combinations of maps and metrics on a surface into minimal surfaces in a Riemannian manifold. I will introduce the flow and describe known existence results, and discuss recent joint work with M. Rupflin that demonstrates how singularities can develop in the metric component in finite time.

Limit groups are a powerful tool in the study of free and hyperbolic groups (and even broader classes of groups). I will define limit groups in various ways: algebraic, logical and topological, and draw connections between the different definitions. We will also see how one can equip a limit group with an action on a real tree, and analyze this action using the Rips machine, a generalization of Bass-Serre theory to real trees. As a conclusion, we will obtain that hyperbolic groups whose outer automorphism group is infinite, split non-trivially as graphs of groups.

How would we get a powerful AI to align itself with human preferences? What are human preferences anyway? And how can you code all this?
It turns out that maths give you the grounding to answer these fascinating and vital questions.
 

Let's take a step back to understand what it means to use maths in society: Which maths, and whose society? I'll talk about some of the options I've come across, including time I spent at the US Census Bureau, and we will hear your ideas too. We might even crowdsource a document of maths in society opportunities together...

In this talk, we consider the random version of some classical extremal problems in the context of long cycles. This type of problems can also be seen as random analogues of the Turán number of long cycles, established by Woodall in 1972.

For a graph $G$ on $n$ vertices and a graph $H$, denote by $\text{ex}(G,H)$ the maximal number of edges in an $H$-free subgraph of $G$. We consider a random graph $G\sim G(n,p)$ where $p>C/n$, and determine the asymptotic value of $\text{ex}(G,C_t)$, for every $A\log(n)< t< (1- \varepsilon)n$. The behaviour of $\text{ex}(G,C_t)$ can depend substantially on the parity of $t$. In particular, our results match the classical result of Woodall, and demonstrate the transference principle in the context of long cycles.

Using similar techniques, we also prove a robustness-type result, showing the likely existence of cycles of prescribed lengths in a random subgraph of a graph with a nearly optimal density (a nearly ''Woodall graph"). If time permits, we will present some connections to size-Ramsey numbers of long cycles.

Based on joint works with Michael Krivelevich and Adva Mond.

(Joint work with: Jüri Lember, Heinrich Matzinger, Raul Kangro)

Principal component analysis is an important pattern recognition and dimensionality reduction tool in many applications and are computed as eigenvectors

of a maximum likelihood covariance that approximates a population covariance. The eigenvectors are often used to extract structural information about the variables (or attributes) of the studied population. Since PCA is based on the eigen-decomposition of the proxy covariance rather than the ground-truth, it is important to understand the approximation error in each individual eigenvector as a function of the number of available samples. The combination of recent results of Koltchinskii & Lounici [8] and Yu, Wang & Samworth [11] yields such bounds. In the presented work we sharpen these bounds and show that eigenvectors can often be reconstructed to a required accuracy from a sample of strictly smaller size order.