Mathematical Sciences Seminar Abstracts 2022
Stefano Bonaccorsi (University of Trento, Italy)
Title: Some notes on Malliavin calculus, surface integrals and integration by parts formulas in infinite dimensional spaces
Abstract: In the first part, we introduce a basic construction of the Malliavin calculus which aims to unify different perspectives (e.g., Bogachev, Da Prato, Lunardi, Nualart). Then we introduce, for suitable functionals of the Brownian motion, the problem of constructing a surface measure (i.e., the restriction of the reference measure on the infinite dimensional Wiener space) on their level sets. Finally, we give some interpretation of such results in terms of other stochastic processes.
Henk Dijkstra (Utrecht University, the Netherlands)
Title: Tipping of the Atlantic Ocean Circulation
Abstract: The Atlantic Ocean Circulation, in particular its zonally averaged component called the Atlantic Meridional Overturning Circulation (AMOC), is one of the tipping elements in the climate system. The AMOC is sensitive to freshwater perturbations and may undergo a transition to a climate disrupting state within a few decades under continuing greenhouse gas emissions. In this presentation, I will discuss the current state of knowledge on the AMOC behavior and in particular focus on the problem of determining probabilities of AMOC transitions.
Didier Clamond (Université Côte d’Azur, France)
Title: Numerical Simulation of Surface Gravity Waves
Abstract: A fast and accurate numerical model for direct simulations of water waves is presented. The algorithm allows simulations of large domains over a long period of time. It is well suited, in particular, for rogue waves and tsunamis.
Luke Kelly (Université Paris Dauphine-PSL, France)
Title: Coupling Markov chain Monte Carlo phylogenetic inference
Abstract: Phylogenetic inference is an intractable statistical problem in a complex, high-dimensional space. The likelihood function for the tree, branch lengths and other parameters is an integral over unobserved evolutionary events on the tree and is often multimodal. Markov chain Monte Carlo (MCMC) methods are the primary tool for Bayesian phylogenetic inference, but it is challenging to construct efficient schemes to explore the associated posterior distributions or assess their performance. Existing approaches to quantifying behavior of Markov schemes in this setting typically rely on ad hoc comparisons of low dimensional summaries of marginal components of the state and already require convergence to have occurred to be meaningful. As such, they lack power and may fail to diagnose mixing or convergence jointly across all components of the model.
Couplings of MCMC algorithms have recently been used to construct unbiased estimators and estimate bounds on the marginal distribution of chains from their target. We describe a procedure to couple Markov chains targeting a posterior distribution over a space of phylogenetic trees with branch lengths, scalar parameters, and latent variables. Pairs of chains meet exactly at a random, finite time and allow us to estimate a qualitative bound on the total variation distance to their equilibrium distribution. We have assessed the efficacy of our coupling scheme on trees with up to 200 leaves. There are many avenues for future research in this area, such as developing maximal couplings of operations on trees.
Martin Haugh (Imperial College London, UK)
Title: Play Like the Pros? Solving the Game of Darts as a Dynamic Zero-Sum Game
Abstract: The game of darts has enjoyed great growth over the past decade with the perception of darts moving from that of a pub game to a game that is regularly scheduled on prime-time television in many countries such as the United Kingdom, Germany, the Netherlands, and Australia, among others. It involves strategic interactions between two players, but to date, the literature has ignored these interactions. In this paper, we formulate and solve the game of darts as a dynamic zero-sum game (ZSG), and to the best of our knowledge, we are the first to do so. We also estimate individual skill models using a novel data set based on darts matches that were played by the top 16 professional players in the world during the 2019 season. Using the fitted skill models and our ZSG problem formulation, we quantify the importance of playing strategically—that is, taking into account the score and strategy of one’s opponent—when computing an optimal strategy. For top professionals, we find that playing strategically results in an increase in win probability of just 0.2%–0.6% over a single leg but as much as 2.2% over a best-of-31-legs match. (Co-authored with Chun Wang of Tsinghua University)
Jeremiah Buckley (King's College London, UK)
Title: Gaussian complex zeros are not always normal
Abstract: The “hyperbolic Gaussian analytic function” is a family of random holomorphic functions on the unit disc. It is particularly interesting because the distribution of its zero set is invariant under disc automorphisms. I will discuss the limiting behaviour of the zero set. The family is parameterised by the “intensity”, the mean number of zeroes per unit hyperbolic area. It is known that there is a transition in the behaviour of the variance at a certain value of the intensity, “L=1/2”. Our main finding elaborates on this transition. We will show that for L≥1/2 the zeroes satisfy a CLT, while for L<1/2 we find a skewed limiting distribution. We will also discuss the case L=0; in this case the “boundary values” of the random function form a “log-correlated process” on the unit circle. Joint work with Alon Nishry (arXiv:2104.12598 [math.PR]).
Alex Belton (Lancaster University, UK)
Title: An introduction to Hirschman-Widder densities and their preservers
Abstract: Hirschman-Widder densities may be viewed as the probability density functions of positive linear combinations of independent and identically distributed exponential random variables. They also arise naturally in the study of Pólya frequency functions, which are integrable functions that give rise to totally positive Toeplitz kernels. This talk will introduce the class of Hirschman-Widder densities and discuss some of its properties. We will demonstrate connections to Schur polynomials and to orbital integrals. We will conclude by describing the rigidity of this class under composition with polynomial functions.
This is joint work with Dominique Guillot (University of Delaware), Apoorva Khare (Indian Institute of Science, Bangalore) and Mihai Putinar (University of California at Santa Barbara and Newcastle University).
Louis M. Pecora (U.S. Naval Research Laboratory)
Title: Statistics of Attractor Embeddings in Reservoir Computing
Abstract: A recent branch of AI or Neural Networks that can handle time-varying signals often in real time has emerged as a new direction for signal analysis. These dynamical systems are usually referred to as reservoir computers. A central question in the operation of these systems is why a reservoir computer (RC), driven by only one time series from a driving or source system of many time-dependent components, can be trained to recreate all dynamical time series signals from the drive leads to the idea that the RC must be internally recreating all the drive dynamics or attractor. In addition, there have been some speculations that RCs may be a fundamental type of system that describes how neuronal networks in biology process sensory input. This has led to the possibility that the RC is creating an embedding of the drive attractor in the RC dynamics. There have been some mathematical advances that move that argument closer to a general theorem. However, for RCs constructed from actual physical systems like interacting lasers or analog circuits or possibly actual neuronal networks, the RC dynamics may not be known well or known at all. And many of the existing embedding theorems have restrictive assumptions on the dynamics. We first show that the best way to analyze RC behavior is to first treat it properly like a dynamical system, which it is. This will lead to some conflict with existing ideas about RCs, but also a clarification of those ideas. Secondly, in the absence of complete theories on RCs and attractor embeddings, we show several ways to analyze the RC behavior to help understand what underlying processes are in place, especially regarding if there are good embeddings of the drive system in the RC. We show that a statistic we developed for other uses can help test for homeomorphisms between a drive system and the RC by using the time series from both systems. This statistic is called the continuity statistic and it is modeled on the mathematical definition of a continuous function. We show the interplay of dynamical quantities (e.g. Lyapunov exponents, Kaplan-Yorke dimensions, generalized synchronization, etc.) and embeddings as exposed by the continuity statistic and other statistics based on ideas from nonlinear dynamical systems theory. These viewpoints and results lead to a clarification of various currently vague concepts about RCs, such as fading memory, stability, and types of dynamics that are useful.
This is joint work with Thomas L. Carroll.
Mark Holland (University of Exeter, UK)
Title: On record events and extremes for dynamical systems
Abstract: Record events occur in many situations, such as in temperature records within weather, financial asset price records, and in sporting events, e.g. the 100m sprint. Within probability and random processes, the study of records can be formalised and their limit distributions studied. If we take a sequence of random variables X1,..., Xn, a record time corresponds to the time t event where we have Xt>max(X1,..., Xt-1), i.e. Xt exceeds all values occurring before time t. A topic of interest is the distribution of such record times, and corresponding record values. In the talk, we review classical results which are part of a wider extreme value theory. We consider first processes that are independent, and identically distributed. Then we mention recent progress when the process (Xn) is generated by a dynamical system.