Past Fridays@4

25 May 2018
Claudia Scheimbauer and Alberto Paganini

Claudia Scheimbauer

Title: Quantum field theory meets higher categories

Abstract: Studying physics has always been a driving force in the development of many beautiful pieces of mathematics in many different areas. In the last century, quantum field theory has been a central such force and there have been several fundamentally different approaches using and developing vastly different mathematical tools. One of them, Atiyah and Segal's axiomatic approach to topological and conformal quantum field theories, provides a beautiful link between the geometry of "spacetimes” (mathematically described as cobordisms) and algebraic structures. Combining this approach with the physical notion of "locality" led to the introduction of the language of higher categories into the topic. The Cobordism Hypothesis classifies "fully local" topological field theories and gives us a recipe to construct examples thereof by checking certain algebraic conditions generalizing the existence of the dual of a vector space. I will give an introduction to the topic and very briefly mention on my own work on these "extended" topological field theories.

Alberto Paganini

Title: Shape Optimization with Finite Elements

Abstract: Shape optimization means looking for a domain that minimizes a target cost functional. Such problems are commonly solved iteratively by constructing a minimizing sequence of domains. Often, the target cost functional depends on the solution to a boundary value problem stated on the domain to be optimized. This introduces the difficulty of solving a boundary value problem on a domain that changes at each iteration. I will suggest how to address this issue using finite elements and conclude with an application from optics.

11 May 2018
Vicky Neale

Research suggests that students with a 'growth mindset' may do better than those with a 'fixed mindset'.

  • What does that mean for our teaching?
  • How can we support students to develop a growth mindset?
  • What sorts of mindsets do we ourselves have?
  • And how does that affect our teaching and indeed the rest of our work?
27 April 2018
Jan Sbierski and Andrew Krause

Jan Sbierski

Title: On the unique evolution of solutions to wave equations

Abstract: An important aspect of any physical theory is the ability to predict the future of a system in terms of an initial configuration. This talk focuses on wave equations, which underlie many physical theories. We first present an example of a quasilinear wave equation for which unique predictability in fact fails and then turn to conditions which guarantee predictability. The talk is based on joint work with Felicity Eperon and Harvey Reall.

Andrew Krause

Title: Surprising Dynamics due to Spatial Heterogeneity in Reaction-Diffusion Systems

Abstract: Since Turing's original work, Reaction-Diffusion systems have been used to understand patterning processes during the development of a variety of organisms, as well as emergent patterns in other situations (e.g. chemical oscillators). Motivated by understanding hair follicle formation in the developing mouse, we explore the use of spatial heterogeneity as a form of developmental tuning of a Turing pattern to match experimental observations of size and wavelength modulation in embryonic hair placodes. While spatial heterogeneity was nascent in Turing's original work, much work remains to understand its effects in Reaction-Diffusion processes. We demonstrate novel effects due to heterogeneity in two-component Reaction-Diffusion systems and explore how this affects typical spatial and temporal patterning. We find a novel instability which gives rise to periodic creation, translation, and destruction of spikes in several classical reaction-diffusion systems and demonstrate that this periodic spatiotemporal behaviour appears robustly away from Hopf regimes or other oscillatory instabilities. We provide some evidence for the universal nature of this phenomenon and use it as an exemplar of the mostly unexplored territory of explicit heterogeneity in pattern formation.

9 March 2018
Radu Cimpeanu and Liana Yepremyan

Speaker: Radu Cimpeanu
Title: Crash testing mathematical models in fluid dynamics

Abstract: In the past decades, the broad area of multi-fluid flows (systems in which at least two fluids, be they liquids or mixtures of liquid and gas, co-exist) has benefited from simultaneous innovations in experimental equipment, concentrated efforts on analytical approaches, as well as the rise of high performance computing tools. This provides a wonderful wealth of techniques to approach a given challenge, however it also introduces questions as to which path(s) to take. In this talk I will explore the symbiotic relationship between reduced order modelling and fully nonlinear direct computations, each of their strengths and weaknesses and ultimately how to use a hybrid strategy in order to gain an understanding over larger subsets of often vast solution spaces. The discussion will take us through a number of interesting topics in fluid mechanics on a wide range of scales, from electrohydrodynamic control in microfluidics, to nonlinear waves in channel flows and violent drop impact scenarios.

Speaker: Liana Yepremyan
Title: Turan-type problems for hypergraphs

Abstract: One of the earliest results in extremal graph theory is Mantel's Theorem  from 1907, which says that for given number of vertices, the largest triangle-free graph on these vertices is the complete bipartite graph with (almost) equal sizes. Turan's Theorem from 1941 generalizes this result to all complete graphs. In general, the Tur'\an number of a graph G (or more generally, of  a hypergraph) is the largest number of edges in a graph (hypergraph) on given number of vertices containing no copy of G as a subgraph. For graphs a lot is known about these numbers,  a result by Erd\Hos, Stone and Simonovits determines the correct order of magnitude of Tur\'an numbers  for all non-bipartite graphs. However, these numbers are known only for few  hypergraphs. We don't even know what is the Tur\'an number of the complete 3-uniform hypergraph on 4 vertices. In this talk I will give some  introduction  to these problems and brielfly describe some of the methods used, such as the stability method and the Lagrangian  function, which are interesting on their own.

23 February 2018
Dave Hewett and Alison Trinder

Who are you? What motivates you? What's important to you? How do you react to challenges and adversities? In this session we will explore the power of self-awareness (understanding our own characters, values and motivations) and introduce assertiveness skills in the context of building positive and productive relationships with colleagues, collaborators, students and others.

9 February 2018
Yalong Cao and Doireann O'Kiely

Yaolong Cao: Gauge Theories on Geometric Spaces
In this talk, I will very briefly discuss gauge theories on various geometric spaces, including Riemann surfaces, 4-manifolds and manifolds with special or exceptional holonomy. More details on Calabi-Yau 4-folds will be mentioned, which are related to my research interests.

Doireann O'Kiely: Dynamic Wrinkling of Elastic Sheets
Our lives contain many scenarios in which thin structures wrinkle: a piece of tin foil or cling film crumples in our hand, and creases form in our skin as we age. In this talk I will discuss experimental and theoretical work by researchers in the Mathematical Institute on wrinkling of elastic sheets.
We study the impact of a solid onto an elastic sheet floating at a liquid-air interface. We observe a wave that is reminiscent of the ripples caused by dropping a stone in a pond, as well as spoke-like wrinkles, whose wavelength evolves in time. We describe these phenomena using a combination of asymptotic analysis, numerical simulations and scaling arguments.

2 February 2018
Mike Giles

In this talk I will discuss the upcoming REF2021 and its significance for early career researchers (research fellows and postdocs) including

  • why it is so important to all UK maths departments
  • why the timing of it could have important career consequences for ECRs
  • publication issues such as quality versus quantity, and choice of journal
  • the importance of Impact Case Studies