(Oxford University)

Do you feel unable to explain why maths are cool? Are you looking for fun and affordable theorems for your non-mathematician friends? This is your topic.

This talk aims to be a rigorous introduction to Social Choice Theory, a sub-branch of Game Theory with natural applications to economics, sociology and politics that tries to understand how to determine, based on the personal opinions of all individuals, the collective opinion of society. The goal is to prove the three famous and pessimistic impossibility theorems: Arrow's theorem, Gibbard's theorem and Balinski-Young's theorem. Our blunt conclusion will be that, unfortunately, there are no ideally fair social choice systems. Is there any hope yet?

 

Given a finite set X, is an easy exercise to show that a binary operation * from XxX to X which is injective in each variable separately, and which is also associative, makes (X,*) into a group. Hrushovski and others have asked what happens if * is only partially associative - do we still get something resembling a group? The answer is known to be yes (in a strong sense) if almost all triples satisfy the associative law. In joint work with Tim Gowers, we consider the so-called `1%' regime, in which we only have an epsilon fraction of triples satisfying the associative law. In this regime, the answer turns out to be rather more subtle, involving certain group-like structures which we call rough approximate groups. I will discuss these objects, and try to give a sense of how they arise, by describing a somewhat combinatorial interpretation of partial associativity.
 

Scrolls play a central role in the construction of varieties; they are ambient spaces for K3 surfaces and Fano 3-folds. In this talk, I will say in two ways what scrolls are and give examples of some embedded varieties in them.

The $k$-th complete homogeneous symmetric polynomial in $m$ variables $h_{k,m}$ is the sum of all the monomials of degree $k$ in $m$ variables. They are related to the Symmetric powers of vector spaces. In this talk we will present some of their standard properties, some classic combinatorial results using the "stars and bars" argument, as well as an interesting result: the complete homogeneous symmetric polynomial applied to $(1+X_i)$ can be written as a linear combination of complete homogeneous symmetric poynomials in the $X_i$. To compute the coefficients of this linear combination, we extend the classic "stars and bars" argument.

The Aldous diffusion is a conjectured Markov process on the
space of real trees that is the continuum analogue of discrete Markov
chains on binary trees. We construct this conjectured process via a
consistent system of stationary evolutions of binary trees with k
labelled leaves and edges decorated with diffusions on a space of
interval partitions constructed in previous work by the same authors.
This pathwise construction allows us to study and compute path
properties of the Aldous diffusion including evolutions of projected
masses and distances between branch points. A key part of proving the
consistency of the projective system is Rogers and Pitman’s notion of
intertwining. This is joint work with Noah Forman, Soumik Pal and
Douglas Rizzolo.                            

Topology optimisation finds the optimal material distribution of a fluid or solid in a domain, subject to PDE and volume constraints. There are many formulations and we opt for the density approach which results in a PDE, volume and inequality constrained, non-convex, infinite-dimensional optimisation problem without a priori knowledge of a good initial guess. Such problems can exhibit many local minima or even no minima. In practice, heuristics are used to obtain the global minimum, but these can fail even in the simplest of cases. In this talk, we will present an algorithm that solves such problems and systematically discovers as many of these local minima as possible along the way.

Recent advances in experimental imaging techniques have allowed us to observe the fine details of how droplets behave upon impact onto a substrate. However, these are highly non-linear, multiscale phenomena and are thus a formidable challenge to model. In addition, when the substrate is deformable, such as an elastic sheet, the fluid-structure interaction introduces an extra layer of complexity.

We present two modeling approaches for droplet impact onto deformable substrates: matched asymptotics and direct numerical simulations. In the former, we use Wagner's theory of impact to derive analytical expressions which approximate the behavior during the early time of impact. In the latter, we use the open source volume-of-fluid code Basilisk to conduct simulations designed to give insight into the later times of impact.

We conclude by showing how these methods are complementary, and a combination of both can give a thorough understanding of the droplet impact across timescales. 

In the seminar, I will talk about a parabolic toy-model for the incompressible Navier-Stokes equations, that satisfies the same energy inequality, same scaling symmetry and which is also super-critical in dimension 3. I will present some partial regularity results that this model shares with the incompressible model and other results that occur only for our model.