Landscapes in Mathematical Sciences

This is our Department Colloquium, with distinguished speakers giving an overview of a topic of general mathematical interest. The talks are aimed at a level to be accessible to all postgraduate students and staff in the department. Advanced undergraduates and members of other departments are also welcome to attend.

Talks last a full hour and, unless otherwise stated, take place in the Wolfson Theatre (4W1.7), beginning at 3:15pm. Tea is available from 30 minutes before the talk in the foyer outside the lecture theatre.


Next Landscapes Seminar

Date Speaker Title/Abstract

Spring 2018

Date Speaker Title/Abstract
19 January 2018 Jonathan Partington
Poncelet's theorem, Blaschke products, and numerical ranges of matrices
This is a talk combining geometry, algebra and analysis, in which Poncelet's theorem from geometry is reformulated and generalized using Blaschke products from complex analysis, together with the concept of the numerical range of a matrix. It comes from joint work with Pamela Gorkin and Isabelle Chalendar. Some interpolation and distance problems studied in control theory also turn out to play a role.
9 February 2018 Suchitra Sebastien
Exploring Materials Universes
Materials comprise trillions of electrons that interact with each other to create a diversity of physical behaviours. A particularly exciting quest is to discover new types of exotic quantum phases arising from collective electron behaviours in materials. Such discoveries, however, are often serendipitous, given that materials can be thought of as complex universes teeming with vast numbers of electrons, making their behaviours challenging to understand or predict. A question we are often confronted with is how to make progress in discovering novel collective electron behaviours. I will discuss possible approaches to increasing the odds of making discoveries, with examples from cases such as new superconductors and new types of dual metal-insulating materials.
16 February 2018 Arezki Boudaoud
Modelling stochasticity and robustness in morphogenesis at multiple scales video link
What determines the form of living beings? How do organs know when to stop growing? How do organs and organisms reach reproducible sizes and shapes despite substantial variability at the cellular level?

During this talk, I will show how to address such questions by using stochastic models for plant growth and architecture. I will consider discrete dynamical systems and discrete or continuous mechanical models. I will show how these models allow us to relate biological observations at different levels, yielding an understanding of how variability may be enhanced or filtered out according to levels and systems.
9 March 2018 Reidun Twarock
Viruses and Geometry: New Insights into Virus Structure, Assembly, Evolution & Therapy
Viruses are remarkable examples of order at the nanoscale. The capsids of many viruses, enclosing and protecting their genomes, are organised in lattice-like arrangements with overall icosahedral symmetry. Mathematical techniques from group, graph and tiling theory can therefore be used to characterise their architectures. In this talk, I will introduce our mathematical approach to the modelling of viral capsids and its applications in vaccine design. I will then present our Hamiltonian path approach to the modelling of genome packing and virus assembly that underpins the discovery of an assembly instruction manual in a wide range of viruses, including Picornaviruses, Hepatitis C and Hepatitis B virus. Based on these results, I will then construct fitness landscapes in order to model viral evolution and compare different forms of anti-viral therapy.
27 April 2018 Thanasis Fokas
From the Wiener-Hopf technique to the Lindelöf Hypothesis video link
For many years, the Wiener-Hopf techique was the only manifestation in applications of the Riemann-Hilbert problem. However, in the last 50 years the Riemann-Hilbert formalism, and its natural generalization called the d-bar formalism, have appeared in a large number of problems in mathematics and mathematical physics. In this lecture, I will review the impact of the above formalisms in the following: the development of a novel, hybrid numerical-analytics method for solving boundary value problems (the Fokas method), the introduction of new algorithms in nuclear medical imaging, and most importantly, the formal proof of the Lindelöf Hypothesis which is a close relative of the Riemann Hypothesis.
11 May 2018 Shi Jin
Madison, Wisconsin
Uncertainty Quantification in Kinetic Theory video link
Kinetic equations describe dynamics of probability density distributions of large number of particles, and have classical applications in rarified gas, plasma, nuclear engineering and emerging ones in biological and social sciences. Since they are not first principle equations, rather are usually derived from mean field approximations of Newton's second law, thus contain inevitably uncertianties in collision kernels, scattering coefficients, initial and boundary data, forcing and source terms, etc.

In this talk we will review a few recent results for kinetic equations with random uncertainties. We will extend hypocoercivity theory, developed for deterministic kinetic equations, to study local sensitivity, regularity, local time behavior of the solutions in the random space, and also establish corresponding theory for their numerical approximations.
25 May 2018 Nalini Anantharaman
Quantum ergodicity and delocalization of Schrödinger eigenfunctions
The question of "quantum ergodicity" is to understand how the ergodic properties of a classical dynamical system are translated into spectral properties of the associated quantum dynamics. This question can be traced back to a paper by Einstein written in 1917. It takes its full meaning after the introduction of the Schrödinger equation in 1926, and even more after the numerical simulations of the 80s that seem to indicate that, for "chaotic" classical dynamics, the spectrum of the associated Schrödinger operator resembles that of a class of large random matrices. Proving this is still fully open. However, we start to understand quite well how the chaotic properties of classical dynamics lead to delocalization properties of the wave functions. We will review the results on the subject and some examples.

Autumn 2017

Date Speaker Title/Abstract
6 October 2017 Alain Goriely
The mathematics and mechanics of brain morphogenesis video link
The human brain is an organ of extreme complexity, the object of ultimate intellectual egocentrism, and a source of endless scientific challenges. Its intricate folded shape has fascinated generations of scientists and has, so far, defied a complete description. How does it emerge? How is its shape related to its function?

In this talk, I will review our current understanding of brain morphogenesis and its unique place within a general mathematical theory of biological growth. In particular, I will present simple models for basic pattern formation and show how they help us understand axonal growth, brain folding, and skull formation.
13 October 2017 Radek Erban
Molecular Dynamics, Random Walks and PDEs
I will introduce several deterministic and stochastic dynamical systems which have been used for mathematical modelling in biology, describing processes at different spatial and temporal scales. Using simple illustrative examples, I will discuss connections between (detailed) molecular dynamics simulations, (less detailed) Brownian dynamics approaches and (even coarser) models written as partial differential equations. I will use this discussion to highlight some open mathematical problems in the field of mathematical biology.
27 October 2017 John Parker
Surfaces from some viewpoints in algebra, geometry and number theory video link
A classical circle of ideas describes a torus as the quotient of the complex plane by a lattice; as the domain of an elliptic function or through the cubic polynomial satisfied by this elliptic function. In turn, the moduli space of such tori is the quotient of the hyperbolic plane by a lattice which is defined arithmetically -- namely the modular group. In this seminar I want to explore several ways these ideas can be extended, thereby linking ideas from geometry (both hyperbolic and algebraic), group theory and number theory.
3 November 2017 Christoph Ortner
Atomistic Simulation of Crystalline Defects - [A Case Study from the Perspective of Numerical Analysis] video link
Atomistic simulation is an indispensible tool of modern materials science as it allows scientists to study individual atoms and molecules in a way that is impossible in the laboratory. A key problem of atomistic materials modelling is to determine properties of crystalline defects, such as geometries, formation energies, or mobility, from which meso-scopic material properties or coarse-grained models (e.g., Kinetic Monte-Carlo, Discrete Dislocation Dynamics, Griffith-type fracture laws) can be derived.

In this lecture I will focus on the most basic task: computing the equilibrium configuration of a crystalline defect (time permitting I will comment on other properties). Typical numerical schemes can be interpreted as Galerkin approximations of an infinite-dimensional nonlinear variational problem. This point of view can be effectively employed to study properties of exact solutions and derive rigorous error bounds. It also serves as an excellent test-bed for classifying and benchmarking multi-scale methods such as atomistic/continuum and QM/MM schemes. A key observation is that all these different numerical schemes (classical or multi-scale) simply provide different boundary conditions for the defect core. After this realisation I will conclude the talk by deriving a new boundary condition (work in progress) that promises significantly improved accuracy at negligible increase in computational cost or complexity of algorithms.
17 November 2017 Peter Benner
MPI Magdeburg
Low-rank Techniques: from Matrix Equations to Molecular Physics video link
The use of low-rank approximations to matrices and tensors has become one of the most successful techniques in Computational Mathematics, Sciences, and Engineering during the last decade. In some, if not many, areas, it has replaced sparsity as the main paradigm to follow for solving extremal large-scale problems. We discuss some of the underlying principles, why and when one can expect low-rank approaches to work. We illustrate the idea of low-rank numerics by examples from the work of the author, ranging from matrix equations to PDE-constrained optimization to molecular physics.

A list of talks in recent past years can be found here.