Week 05.11.2023 – 11.11.2023

Random intersection graph is a simple random graph model that incorporates community structures. To build such a graph, imagine there are n individuals and m potential communities, where each individual joins a community independently with probability p. Two individuals are adjacent to each other in the random intersection graph G(n, m, p) if and only if they belong to a common community. It is known that if m>>n^3, then the model is asymptotically equivalent to an Erdos-Renyi graph G(n, q) with a suitable q. I’ll discuss other ways to compare the two models, particularly regarding the scaling limit of the graphs inside the critical window and number of triangles found in the large components.

In recent years, significant progress has been made in the field of birational geometry for foliations. Notably, the Minimal Model Program (MMP) has been shown to work for foliations on threefolds. In this talk, I will demonstrate that the MMP is applicable to toric foliations as well. Specifically, I will discuss how non-dicritical singularities (and foliated dlt singularities if time permits) are preserved under the MMP. This is a joint work with Chih-Wei Chang.

The purpose of this talk is to introduce a practical method for assessing mathematical proof online. We examine the effect of faded worked examples and reading comprehension questions to student's understanding of proof. By breaking down a given proof, we will demonstrate a checklist that can be used to generate comprehension questions which can be assessed automatically online. We then provide some preliminary results of deploying such questions.
In addition, we shall examine several examples of proof comprehension questions that have been created by following this checklist. In particular, I will discuss some ongoing work related to the creation of proof comprehension tasks used to assessment of proof in a higher-level undergraduate pure mathematics courses.

We consider the Landau equation with Coulomb potentials. We discuss a global existence result close to equilibrium and a short time existence result for general data. In both cases we take initial data in unweighted Lp spaces for p > 3/2. We thus work with the weakest class of initial data that has been considered until now. This is joint work with William Golding (UT Austin) and Maria Gualdani (UT Austin).

Cyber-systems are under near-constant threat from intrusion attempts. Attacks types vary, but each attempt typically has a specific underlying intent, and the perpetrators are typically groups of individuals with similar objectives. Clustering attacks appearing to share a common intent is very valuable to threat-hunting experts. This talk explores topic models for clustering terminal session commands collected from honeypots, which are special network hosts designed to entice malicious attackers. The main practical implications of clustering the sessions are two-fold: finding similar groups of attacks, and identifying outliers. A range of statistical topic models are considered, adapted to the structures of command-line syntax. In particular, concepts of primary and secondary topics, and then session-level and command-level topics, are introduced into the models to improve interpretability. The proposed methods are further extended in a Bayesian nonparametric fashion to allow unboundedness in the vocabulary size and the number of latent intents. The methods are shown to discover an unusual MIRAI variant which attempts to take over existing cryptocurrency coin-mining infrastructure, not detected by traditional topic-modelling approaches. This is joint work with Daniyar Ghani (Imperial College London), Anastasia Mantziou (Alan Turing Institute), Philip Thiede (formerly at Imperial College London, now at Abios), Ross Bevington (Microsoft), Nicholas A. Heard (Imperial College London).

I will discuss the problem of counting states in superconformal field theories, focusing on 4D N=4 SYM. The main focus will be on reviewing background material before briefly commenting on some recent developments in 2306.04693 and 2209.06728. I also hope to make some comments on the history of the problem.