School of Mathematics Colloquium

Mine Çetinkaya-Rundel

Title: The art and science of teaching data science

Abstract: Modern statistics is fundamentally a computational discipline, but too often this fact is not reflected in our statistics curricula. With the rise of data science it has become increasingly clear that students want, expect, and need explicit training in this area of the discipline. Additionally, recent curricular guidelines clearly state that working with data requires extensive computing skills and that statistics students should be fluent in accessing, manipulating, analyzing, and modeling with professional statistical analysis software. In this talk we introduce the design philosophy behind an introductory data science course, discuss in progress and future research on student learning as well as new directions in assessment and tooling as we scale up the course.


Lars Schewe

Title: Mixed-Integer Nonlinear Programming to improve energy networks and markets

Abstract: Mixed-Integer Nonlinear Programs (MINLP) are a class of optimization problems that are well-suited to model systems that contain both physical and economical components, like energy networks. The resulting models are, however, too complex to be solved by off-the-shelf software. We will use an example from gas network optimization to show how practical problems can motivate new algorithmic approaches for much more general optimization problems.

The Moment-SOS hierarchy initially introduced in optimization in 2000, is based on the theory of the K-moment problem and its dual counterpart, polynomials that are positive on K. It turns out that this methodology can be also applied to solve problems with positivity constraints $f (x) \geq 0$ for all $x \in K$ and/or linear constraints on Borel measures. Such problems can be viewed as specific instances of the "Generalized Problem of Moments" (GPM) whose list of important applications in various domains is endless. We describe this methodology and outline some of its applications in various domain of science & engineering.

So-called ADE-type Dynkin diagrams provide a classification of surprisingly diverse structures in mathematics, including the 'simple' complex Lie algebras and the 'simple' singularities. For these singularities, Diaconescu, Donagi and Pantev discovered in 2007 that the deformation spaces are governed by so-called ADE-type integrable systems, which had been introduced previously by Hitchin in the rather different context of Higgs bundles. Many ADE-type Dynkin diagrams have automorphisms by which they can be 'folded' to non-ADE-diagrams. It is therefore natural to develop a folding procedure for ADE-type integrable systems, as was initiated by Beck in his 2016 thesis. In joint work with Beck and Donagi we have now developed folding procedures for the different incarnations of ADE-type integrable systems, allowing a comparison to orbifolding. The talk will give a lightning introduction to ADE-classifications and folding, including our geometric applications to integrable systems.

Understanding the adaptation and evolution of populations is a huge challenge, in particular for microorganisms since it plays a main role in the virulence evolution or in bacterial antibiotics resistances. We propose a general stochastic model of population dynamics with clonal reproduction and mutations. Moreover the individuals compete for resources and exchange genes. We study different asymptotics of this general birth and death process depending on the respective demographic, ecological and transfer time-scales and on the population size. We show that the horizontal gene transfer can have a major impact on the distribution of the successive mutational fixations, leading to dramatically different behaviors, from expected evolution scenarios to evolutionary suicide. Simulations are given to illustrate these phenomena.

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Jelle Hartong

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Desmond Higham

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Cell migration is a fundamentally important phenomenon underlying wound healing, tissue development, immune response and cancer metastasis. Understanding basic physics of the cell migration presented a great challenge until, in the last three decades, a combination of biological, biophysical and mathematical approaches shed light on basic mechanisms of the cell migration. I will first focus on the simplest mode of cell locomotion, lamellipodial motility. I will describe models, based on nonlinear partial differential equations and free boundary problems, which predicted that individual keratocyte cells do not linger in a symmetric stationary state, but rather spontaneously break symmetry and initiate motility. The cells can either crawl straight, or turn, depending on mechanical parameters. I will show how experimental data supported the models. Most cells, however, migrate collectively, not individually, and in 3D. I will introduce experimental data on collective migration of two heart progenitor cells in Ciona embryo. These cells crawl cohesively squeezing between stiff ectoderm and elastic endoderm with persistent leader-trailer polarity. Most active and passive forces are concentrated in the 2D cortex of these cells, with hydrostatic pressure of the 3D cytoplasm assisting the cortex forces in generating stress balances optimizing the cell migration. I will present a computational model that sheds light on design principles of this motile system.

This talk will cover a variety of topics which hopefully shall offer a flavour of applied mathematical methods to characterise the effective properties of multi-phase materials. Such composites appear everywhere, from electromagnetics through modern engineering laminates to granular materials. Determining the wave propagation characteristics for such composites may be required for evaluation, inspection or design purposes, and this is the principal objective of the present research. In order to increase the interest and accessibility of the talk, we shall focus on the use of complex variable methods for tackling problems in this area, and focus on simple one-dimensional and two-dimensional models that exhibit behaviour similar to that found in more complicated cases. We shall commence the talk with a look at the Wiener-Hopf technique, its solution procedure and limitations, and then discuss how this may be useful for certain simple wave models. We sketch the approach for materials with periodically arranged microstructure, and then to the scaterring properties at the surface of random composites. The latter shall be approached by use of ensemble averaging and the quasi-crystalline approximation. The main conclusion of the talk will be the recently-found result that random composites permit more than a single effective wave, and that all such waves are required to accurately predict the scattering coefficients in any given case.

Jonathan Hickman

Title: Kakeya sets

Abstract: A Kakeya set is a compact subset of $\mathbb{R}^n$ which contains a unit line segment in every direction. Understanding the geometry of such sets is crucial to many problems in analysis, including basic questions regarding convergence of Fourier series and regularity of solutions to the wave equation. In this short talk I will sketch some of these connections.

Daniel Paulin

Title: Dual space preconditioning for gradient descent

Abstract: The conditions of relative convexity and smoothness were recently introduced by Bauschke, Bolte, and Teboulle and Lu, Freund, and Nesterov for the analysis of first-order methods optimizing a convex function. Those papers considered conditions over the primal space. We introduce a fully explicit descent scheme with relative smoothness in the dual space between the convex conjugate of the function and a designed dual reference function. Our method is a form of non-linear preconditioning of gradient descent that extends the class of convex functions on which explicit first-order methods can be well-conditioned.

Alper Yildirim

Title: Convex Relaxations of Nonconvex Optimization Problems

Abstract: In this talk, we focus on the interplay between convex and nonconvex optimization. For a large class of nonconvex optimization problems, we describe how to obtain alternative convex relaxations and compare these relaxations in terms of their strengths and their computational costs. We discuss current challenges and point out some future research directions.