Week 26.11.2023 – 02.12.2023

In 1953 Yu. Prohorov published a paper on weak convergence of
probability measures on metric spaces, bringing a new, extended context to the Invariance Principle proved by Donsker two years earlier.
Prohorov’s formalism, publicised in books by K.R. Parthasarathy and
P. Billingsley, established the equivalence of the notion of convergence
in law of stochastic processes and the weak convergence of their distributions. This point of view is completely justified in metric spaces,
especially in Polish spaces.

It is, however, much less satisfactory in non-metric spaces, as was
shown by examples due to X. Fernique, given long time ago.
We show that in a large class of submetric spaces there exists a
stronger mode of convergence, coinciding with the weak convergence
on metric spaces, and much more suitable for needs of contemporary
theory of stochastic partial differential equations.

A submetric space is a topological space (X,tau) admitting a continuous metrics d that in turn determines a metric topology $\tau_d \subset \tau$ (where this inclusion is in general strict). As a standard (and the simplest)
example may serve a separable Hilbert space equipped with the weak
topology.

Let $K$ be a knot or link in the 3-sphere, thought of as the ideal boundary of hyperbolic 4-space $\mathbb{H}^4$. I will describe a programme to count minimal surfaces in $\mathbb{H}^4$ which have $K$ as their asymptotic boundary. This should give an isotopy invariant of the knot. I will explain what has been proved and what remains to be done. Minimal surfaces correspond to $J$-holomorphic curves in the twistor space $Z\to\mathbb{H}^4$, and so this invariant can be seen as a Gromov-Witten type invariant of $Z$. The big difference with the “standard” situation is that the almost complex structure on $Z$ (equivalently, the metric on $\mathbb{H}^4$) blows up at the boundary. This means the $J$-holomorphic equation, or minimal surface equation, becomes degenerate at the boundary of the domain. As a consequence, both the Fredholm and compactness parts of the story need to be reworked by hand. If there is time I will explain how this can be done, relying on results of Mazzeo-Melrose from the 0-calculus, and also some results from the theory of minimal surfaces.

There are many ways to engage with mathematics, and therefore many different forms of maths engagement aimed at different audiences. A brief survey will present a range of different practices and encourage thought about target audience. We'll consider how to design effective outreach, and how informal engagement with maths before university can influence the decision to study maths at university. From a practical point of view, I will describe a programme of maths engagement for secondary school students implemented in local schools and on campus.

We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense than that usually employed to solve initial boundary value problems (IBVP), we show that it is sufficient to solve IBVP whose spatial parts are described by such operators. We argue that the method described may be viewed as a reimplementation of the Fokas transform method for linear evolution equations on the finite interval. The results are extended to multipoint and interface operators, including operators defined on networks of finite intervals, in which the coefficients of the differential operator may vary between subintervals, and arbitrary interface and boundary conditions may be imposed\DSEMIC differential operators with piecewise constant coefficients are thus included.

How often does a randomly chosen variety have a point? Answering this question depends on the family of varieties in question, how we decide to order them, and what kinds of points we are looking for. Motivated by rational points, we endeavor to explicitly describe how often a randomly chosen variety is everywhere locally soluble. When our family is described by the fibers of a suitable morphism, this likelihood is equal to the product of local probabilities at each place and in some cases may be computed exactly. In particular, in joint work with Lea Beneish we find that for almost 97% of integral binary sextic forms f(x,z), the superelliptic curve y^3 = f(x,z) is everywhere locally soluble, with the local factors described explicitly by rational functions. Time permitting, we will discuss ongoing work on determining how often a cubic hypersurface has a rational point.

Sequential Monte Carlo Samplers (SMCS) constitute a widely used class of SMC algorithms that calculate normalizing constants and simulate complex multi-modal target distributions. Typically, SMCS utilizes a process known as annealing, which propagates solutions from a tractable reference distribution to the intractable target through a continuous path of increasingly complex distributions. SMCS delivers state-of-the-art performance when adequately tuned, although this can pose a challenge for current tuning methods, yielding a random run-time and compromising the normalizing constant's unbiasedness.

In this talk, we intend to describe all the components of an SMCS algorithm and their influence on the variance of the normalizing constant. Specifically, we will demonstrate that SMCS exhibits fundamentally different behaviour in large particle and dense schedule limits. The dense schedule limit reveals the natural geometry induced by annealing, which can pinpoint optimal performance and tune the number of particles, number of annealing distributions, annealing schedules, the resampling schedule, and the path. Lastly, we propose an efficient, black-box algorithm for tuning SMCS that delivers optimal performance within a fixed, user-specified computation budget, all while preserving the unbiasedness of the normalizing constant.