(Oxford University)

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.

Formation of singularities for the relativistic Euler equations (N. Athanasiou): An archetypal phenomenon in the study of hyperbolic systems of conservation laws is the development of singularities (in particular shocks) in finite time, no matter how smooth or small the initial data are. A series of works by Lax, John et al confirmed that for some important systems, when the initial data is a smooth small perturbation of a constant state, singularity formation in finite time is equivalent to the existence of compression in the initial data. Our talk will address the question of whether this dichotomy persists for large data problems, at least for the system of the Relativistic Euler equations in (1+1) dimensions. We shall also give some interesting studies in (3+1) dimensions. This is joint work with Dr. Shengguo Zhu.

Global Well-Posedness for a Class of Stochastic McKean-Vlasov Equations in One Dimension (A. Mayorcas): We show global well-posedness for a family of parabolic McKean--Vlasov SPDEs with additive space-time white noise. The family of interactions we consider are those given by convolution with kernels that are at least integrable. We show that global well-posedness holds in both the repulsive/defocussing and attractive/focussing cases. Our strategy relies on both pathwise and probabilistic techniques which leverage the Gaussian structure of the noise and well known properties of the deterministic PDEs.

We consider the problem of global minimization with bound constraints. The problem is known to be intractable for large dimensions due to the exponential increase in the computational time for a linear increase in the dimension (also known as the “curse of dimensionality”). In this talk, we demonstrate that such challenges can be overcome for functions with low effective dimensionality — functions which are constant along certain linear subspaces. Such functions can often be found in applications, for example, in hyper-parameter optimization for neural networks, heuristic algorithms for combinatorial optimization problems and complex engineering simulations.

Extending the idea of random subspace embeddings in Wang et al. (2013), we introduce a new framework (called REGO) compatible with any global min- imization algorithm. Within REGO, a new low-dimensional problem is for- mulated with bound constraints in the reduced space. We provide probabilistic bounds for the success of REGO; these results indicate that the success is depen- dent upon the dimension of the embedded subspace and the intrinsic dimension of the function, but independent of the ambient dimension. Numerical results show that high success rates can be achieved with only one embedding and that rates are independent of the ambient dimension of the problem.

In 2010 Coecke, Sadrzadeh, and Clark formulated a new model of natural language which operates by combining the syntactics of grammar and the semantics of individual words to produce a unified ''meaning'' of sentences. This they did by using category theory to understand the component parts of language and to amalgamate the components together to form what they called a ''distributional compositional categorical model of meaning''. In this talk I shall introduce the model of Coecke et. al., and use it to compare the meaning of sentences in Irish and in English (and thus ascertain when a sentence is the translation of another sentence) using a cosine similarity score.

The Irish language is a member of the Gaelic family of languages, originating in Ireland and is the official language of the Republic of Ireland.