Mathematics

The semi-random graph process can be thought of as a one player game. Starting with an empty graph on n vertices, in each round a random vertex u is presented to the player, who chooses a vertex v and adds the edge uv to the graph (hence 'semi-random'). The goal of the player is to construct a small fixed graph G as a subgraph of the semi-random graph in as few steps as possible. I will discuss this process, and in particular the asympotically tight bounds we have found on how many steps the player needs to win. This is joint work with Trent Marbach, Pawel Pralat and Andrzej Rucinski.

This panel discussion took part as part of the 2022 Celebration of Women in Mathematics event.

Wasserstein distances, or Optimal Transport methods more generally, offer a powerful non-parametric toolbox to conceptualise and quantify model uncertainty in diverse applications. Importantly, they work across the spectrum: from small uncertainty around a selected model (e.g., the empirical measure) to large uncertainty of considering all models consistent with the data. I will showcase this using examples from mathematical finance (pricing and hedging of options, optimal investment) and statistics (non-parametric estimators, regularised regression methods). I will illustrate the large uncertainty regime using Martingale OT problems. For the small uncertainty regime I will consider a generic stochastic optimization problem and its distributionally robust version using Wasserstein balls. I will derive explicit formulae for the first order correction to both the value function and the optimizer. Throughout, I will present both theoretical result, as well as comments on the available numerical methods.

The talk will be borrow from many joint works, including with Daniel Bartl, Samuel Drapeau, Stephan Eckstein, Gaoyue Guo, Tongseok Lim and Johannes Wiesel.

Large sets in Euclidean space should have large projections in most directions. Projection theorems in geometric measure theory make this intuition precise, by quantifying the words “large” and “most”.

How large can a planar set be if it contains a circle of every radius? This is the quintessential example of a curvilinear Kakeya problem, central to many areas of harmonic analysis and incidence geometry.

What do projections have to do with circles?

The talk will survey a few landmark results in these areas and point to a newly discovered connection between the two.

Monitoring marked individuals is a common strategy in studies of wild animals (referred to as mark-recapture or capture-recapture experiments) and hard to track human populations (referred to as multi-list methods or multiple-systems estimation). A standard assumption of these techniques is that individuals can be identified uniquely and without error, but this can be violated in many ways. In some cases, it may not be possible to identify individuals uniquely because of the study design or the choice of marks. Other times, errors may occur so that individuals are incorrectly identified. I will discuss work with my collaborators over the past 10 years developing methods to account for problems that arise when are only individuals are only partially identified. I will present theoretical aspects of this research, including an introduction to the latent multinomial model and algebraic statistics, and also describe applications to studies of species ranging from the golden mantella (an endangered frog endemic to Madagascar measuring only 20 mm) to the whale shark (the largest know species of fish, measuring up to 19m).