I'll consider recent results concerning the stability of the classic Brunn-Minkowski inequality. In particular, I will focus on the linear stability for homothetic sets. Resolving a conjecture of Figalli and Jerison, we showed there are constants $C,d>0$ depending only on $n$ such that for every subset $A$ of $\mathbb{R}^n$ of positive measure, if $|(A+A)/2 - A| \leq d |A|$, then $|co(A) - A| \leq C |(A+A)/2 - A|$ where $co(A)$ is the convex hull of $A$. The talk is based on joint work with Hunter Spink and Marius Tiba.

In the event of food-borne disease outbreaks, conventional epidemiological approaches to identify the causative food product are time-intensive and often inconclusive. Data-driven tools could help to reduce the number of products under suspicion by efficiently generating food-source hypotheses. We frame the problem of generating hypotheses about the food-source as one of identifying the source network from a set of food supply networks (e.g. vegetables, eggs) that most likely gave rise to the illness outbreak distribution over consumers at the terminal stage of the supply network. We introduce an information-theoretic measure that quantifies the degree to which an outbreak distribution can be explained by a supply network’s structure and allows comparison across networks. The method leverages a previously-developed food-borne contamination diffusion model and probability distribution for the source location in the supply chain, quantifying the amount of information in the probability distribution produced by a particular network-outbreak combination. We illustrate the method using supply network models from Germany and demonstrate its application potential for outbreak investigations through simulated outbreak scenarios and a retrospective analysis of a real-world outbreak.

Many real world graphs have edges correlated to the distance between them, but, in an inhomogeneous manner. While the Chung-Lu model and geometric random graph models both are elegant in their simplicity, they are insufficient to capture the complexity of these networks. For instance, the Chung-Lu model captures the inhomogeneity of the nodes but does not address the geometric nature of the nodes and simple geometric models treat names homogeneously.

In this talk, we develop a generalized geometric random graph model that preserves many graph-theoretic aspects of these models. Notably, each node is assigned a weight based on its desired expected degree; nodes are then adjacent based on a function of their weight and geometric distance. We will discuss the mathematical properties of this model. We also test the validity of this model on a graphical representation of the Drosophila Medulla connectome, a natural real-world inhomogeneous graph where spatial information is known.

This is joint work with Susama Agarwala, Johns Hopkins, Applied Physics Lab.

arXiv link: https://arxiv.org/abs/2109.00061

We propose a decentralised “local2global" approach to graph representation learning, that one can a-priori use to scale any embedding technique. Our local2global approach proceeds by first dividing the input graph into overlapping subgraphs (or “patches") and training local representations for each patch independently. In a second step, we combine the local representations into a globally consistent representation by estimating the set of rigid motions that best align the local representations using information from the patch overlaps, via group synchronization.  A key distinguishing feature of local2global relative to existing work is that patches are trained independently without the need for the often costly parameter synchronisation during distributed training. This allows local2global to scale to large-scale industrial applications, where the input graph may not even fit into memory and may be stored in a distributed manner.

arXiv link: https://arxiv.org/abs/2107.12224v1

I discuss a novel expansion of superconformal indices of U(N) gauge theories at finite N. When a holographic description is available, the formula expresses the index as a sum over stacks of "giant graviton" branes in the dual string theory. Surprisingly, the expansion turns out to be exact: the sum over strings and branes seems to capture the degeneracy of states expected from saddle geometries such as BPS black holes, while also reproducing the correct degeneracies at lower orders of charges. Based on 2109.02545 with D. Gaiotto.

This session is particularly aimed at fourth-year and OMMS students who are completing a dissertation this year. The talk will be given by Dr Richard Earl who chairs Projects Committee. For many of you this will be the first time you have written such an extended piece on mathematics. The talk will include advice on planning a timetable, managing the  workload, presenting mathematics, structuring the dissertation and creating a narrative, providing references and avoiding plagiarism.

What should you expect in intercollegiate classes?  What can you do to get the most out of them?  In this session, experienced class tutors will share their thoughts, including advice about online classes. 

All undergraduate and masters students welcome, especially Part B and MSc students attending intercollegiate classes. 

We'll discuss what mathematicians are looking for in written solutions.  How can you set out your ideas clearly, and what are the standard mathematical conventions?

This session is likely to be most relevant for first-year undergraduates, but all are welcome.