Applied Mathematics Seminars in 2023

Abstract: I will talk about the multivariate linear statistics (also known as U-statistics) of determinantal point processes on unit spheres. I will first present a graphical representation for the cumulants of the multivariate linear statistics, extending the famous Soshnikov's formula for the univariate case. Then I will explain how we derive the 1st and 2nd Wiener chaos using this graphical representation. We computed sphere cases as introductory examples, but the method can be applied to any other determinantal point processes. This is based on the joint work with F. Goetze and D. Yao.

Abstract: Generative adversarial networks (GANs) are an approach to fitting generative models over complex structured spaces. Within this framework, the fitting problem is posed as a zero-sum game between two competing neural networks which are trained simultaneously. Mathematically, this problem takes the form of a saddle-point problem; a well-known example of the type of problem where the usual (stochastic) gradient descent-type approaches used for training neural networks fail. In this talk, we rectify this shortcoming by proposing a new method for training GANs that has: (i) a sounds theoretical foundation, and (ii) does not increase the algorithm's per iteration complexity (as compared to gradient descent). The theoretical analysis is performed within the framework of monotone operator splitting.

Abstract: Amorphous solids – glasses, plastics, ceramics and ‘glassy’ metal alloys – make up an important part of the material world but they remain poorly understood at the microscopic level. The difficulties arise from the non-uniqueness of the solid state. This talk will address the issues of the complexity of the energy landscape associated with disordered configurations, the diversity of geometric structures, the localized response to strain (non-affine displacements) and thermal fluctuations (dynamic heterogeneities). Related phenomenon such as jamming will also be discussed.

Abstract: Coarse-graining is the procedure of approximating large and complex systems by simpler and lower-dimensional ones. It is typically characterised by a mapping which projects the full state of the system onto a smaller set; this mapping captures the relevant (often slow) features of the system. Starting from a continuous-time Markov chain and such a mapping, I will discuss an effective dynamics which approximates the true projected Markov chain and present error estimates in relative entropy on the approximation error. This talk is based on joint work with Bastian Hilder.

Abstract: The formation of rogue waves is of interest, from North sea waves and waves in tanks to waves in nonlinear optics. Most common models used to investigate rogue bursts use the nonlinear Schroedinger (NLS) equation and its variants. However, such integrable settings and analytical solutions are rare in higher dimensions. So we propose to use the model of a dissipative system: which describes interaction between standing waves in domains of moderate aspect ratio. When spatial reflection symmetry is broken, the left and right running waves can interact strongly producing a temporally localised extremely large amplitude event.

Abstract: It is often handy in trajectory planning problems to have a linear control system or linearizable control system. In the case that a control system is ``intrinsically nonlinear" the next best thing is flat outputs or a dynamic feedback linearization. It is also often the case that control systems inspired by nature/engineering have symmetries that may provide insight into many questions about a given control system. It turns out we can use symmetries to probe the existence of dynamic feedback linearizations and even construct them explicitly! I'll present this procedure via an explicit example. If time permits I'll mention a possible application to Darboux integrable PDE. This approach is known as cascade feedback linearizability and is joint work with Peter J. Vassiliou and Jeanne N. Clelland.

Abstract: Motivated by a range of problems in embryology and ecology, I will present recent extensions to Turing's classical reaction-diffusion paradigm for pattern formation. This will start by reviewing reaction-diffusion systems and their analysis via classical linear instability theory, followed by a range of generalizations to more realistic scenarios of reaction-transport models in complex domains. Such extensions are motivated by the evolving and heterogeneous landscapes of pattern formation in nature. Throughout this discussion, numerical simulations will play key roles in validating and extending the near-equilibrium theory. To drive home this last point, I will present VisualPDE, a new web-based simulator for lightning-fast interactive explorations of these systems. Such accessible numerical tools are invaluable for rapidly prototyping models of complex biological phenomena. Importantly, accessible simulations underscore the need for sound theory which goes beyond phenomenological modelling in biology.

Title: Transitions in Bayesian model selection problems: link prediction on complex networks and symbolic regression

Abstract: In this talk, I want to show you the effects of the prior and the likelihood in Bayesian inference problems applied to model selection problems and how it can help us understand some aspects of the results that we get. To do that, first I am going to show you the importance of both terms in the Bayesian inference process by showing you how the balance in the likelihood and the prior are relevant in the models that we choose. After that I am going to show you a couple of applications of this framework: a link prediction problem in complex networks and a symbolic regression problem. In the link prediction, I show how nodes metadata (extra information such as the age, ethnicity, gender...) produce a crossover of the types of models that we get with the observed links, affecting accuracy. In symbolic regression, I show that the noise of the data and its size, produce a transition of the learnability of the ground true model that generated our dataset.

Abstract: In the early 1950s, Fermi, Pasta, Ulam and Tzingou decided to investigate the dynamics of large degree of freedom physical systems, by performing some of the first numerical simulations in history. Their study of a conservative chain of nonlinearly interacting oscillators revealed quite a surprise: instead of the expected evolution to a "thermal equilibrium", they observed recurrent behavior. Since then, different approaches were able to explain features of their observations, using for instance integrable PDE approximations and perturbation theory. In this talk I will give an overview of some cornerstone results on the analysis of the problem, including their shortcomings, and I will present some recent work of Antonio Ponno (Padova), Matteo Gallone (SISSA Trieste) and myself on the near-integrable evolution of unidirectional waves.

Abstract: In this talk I will delve into the behaviour of sea ice, examining both its microstructure and its effective properties. Firstly, I will introduce a thermodynamically consistent model for the liquid-solid phase change in sea ice that incorporates the effects of salt, using multiscale analysis to derive a quasi-equilibrium Stefan-type problem. Secondly, I will investigate the thermal conduction in sea ice in the presence of fluid flow, using a new Stieltjes integral representation for the effective conductivity and present rigorous bounds on the conductivity obtained through Padé approximates.

Abstract: Elliptic orthogonal polynomials are a family of special functions that satisfy a certain orthogonality condition with respect to a weight function on an elliptic curve. Building up on several recent works on the topic, we establish a framework using Riemann-Hilbert problems to study such polynomials. When the weight function is constant, these polynomials relate to the elliptic form of the sixth Painleve equation. This talk is based on an ongoing work with Tomas Latimer and Pieter Roffelsen.

Abstract: What forces drive populations to extinction? Mathematical biologists have explored a number of mechanisms that increase the probability of extinction in small populations - especially those facing environmental change. Populations can undergo adaptation to escape extinction, in a process known as evolutionary rescue. By studying population models we show that Darwinian evolution can, ironically, reduce the chance of surviving environmental change. The extinction risk is unexpectedly more pronounced in moderate to large populations. Genetic linkage is at the core of these effects. Our results therefore have implications for the evolution of sex and recombination. They also raise questions about whether and how asexual species - e.g. bacterial species - can go extinct. We have begun to explore these questions with population models.

Abstract: An outstanding problem in mathematical biology is using laboratory and field observations to tune a model’s functional form and parameter values. In this talk I will discuss an ongoing project developing models of the Australian plague locust for which excellent field and experimental data is available. Under favorable environmental conditions flightless locust juveniles may aggregate into coherent, aligned swarms referred to as hopper bands. We will develop two models of hopper bands in tandem; an agent-based model that tracks the position of individuals and a partial differential equation model that describes locust and resource density. By examining 4.4 million parameter combinations, we identify a set of the problem’s ten parameters that reproduce field observations.

I will then discuss two ongoing efforts to improve this model. The first uses ideas from dynamical systems and continuum mechanics to extend this model into two dimensions by modeling the known tendency of locusts to align using ideas from the Kuramoto model of oscillator synchronization. The second, firmly based in data science, uses motion tracking of tens of thousands of locusts to shed light on how locust movement is informed by interactions with other individuals.

Abstract: We discuss solitonic field configurations on a spherical magnet. Exploiting the Hamiltonian structure and concepts of angular momentum, we present a new family of localized solutions to the Landau–Lifshitz equation that are topologically distinct from the ground state and break rotational symmetry. The approach illustrates emergent spin-orbit coupling arising from the loss of individual rotational invariance in spin and coordinate space – a common feature of condensed matter systems with topological phases.

Abstract: Classical modulation theory can be viewed as a weaker alternative to center manifold theory which can be used to study instabilities associated with the crossing of continuous spectra into the right-half plane. This approach is often applied to the study of pattern formation close to linear instabilities. In this talk, we propose and apply an extension to the case of dynamic bifurcations, where the control parameters are allowed to evolve slowy in time. The key analytical method is a novel extension of the so-called geometric blow-up technique, which has been successfully applied to the study of dynamic bifurcations in ODEs for many years now, to the PDE setting. We show that the classical multi-scale ansatz in modulation theory can be reformulated as a geometric blow-up transformation, after which modulation equations can be derived in the dynamic setting using an adaptation of the formal method of multiple scales. We conclude by demonstrating the method for model problems featuring dynamic Turing and dynamic Hopf bifurcations.

Abstract: High-dimensional problems appear in various mathematical models. Their numerical approximation involves the well-known curse of dimension, which renders any direct discretization obsolete. One approach to circumvent this issue, at least to some extent, is the use of generalized sparse grid methods, which can exploit additional smoothness properties if present in the underlying problem.

In this talk, we will discuss the main principles and basic features of generalized sparse grids and show their application in such diverse areas as econometrics, fluid dynamics, quantum chemistry, uncertainty quantification and machine learning.

Abstract: The exact WKB analysis is a method to analyze differential equations with a small parameter \(h\). The main ingredient of the exact WKB analysis is a formal solution for \(h\), called a WKB solution. When we study differential equations by using the exact WKB analysis, Voros coefficients provide important quantities for describing global behavior of solutions of differential equations. The Voros coefficient is defined as a contour integral of the logarithmic derivative of WKB solutions.

On the other hand, the topological recursion introduced by B. Eynard and N. Orantin is a recursive algorithm to construct a formal solution to the loop equations that the correlation functions of the matrix model satisfy.

The quantization scheme connects WKB solutions with the topological recursion. It is found that WKB solutions can be constructed via the topological recursion.

In this talk, we prove that the Voros coefficients for hypergeometric differential equations are described by the generating functions of free energies defined in terms of the topological recursion. Furthermore, as its applications we show the following objects can be explicitly computed for hypergeometric equations: (i) three-term difference equations that the generating function of free energies satisfies, (ii) explicit forms of the free energies, and (iii) explicit forms of Voros coefficients.

Abstract: A positive basis is a set that non-negatively spans $R^n$ and contains no proper subsets with the same property. These attributes make positive bases a useful tool in derivative-free algorithms and an interesting concept in mathematics. In this talk, we examine some properties of positive bases, including how to check if something is a positive basis and how to construct positive bases with nice structures.