Past Fridays@4

How much do you know actually about the research that is going on across the department? The SIAM Student Chapter brings you a 3 minute thesis competition challenging a group of DPhil students to go head to head to explain their research in just 3 minutes with the aid of a single slide. This is the perfect opportunity to hear about a wide range of topics within applied mathematics, and to gain insight into the impact that mathematical research can have. The winner will be decided by a judging panel comprising Professors Helen Byrne, Jon Chapman, Patrick Farrell, and Christina Goldschmidt.
 

Cristina Palmer-Anghel: Quantum invariants via topological intersection pairings
The world of quantum invariants for knots started in 1984 with the discovery of a strong link invariant, namely the Jones polynomial. Then, Reshetikhin and Turaev developed a conceptual algebraic method that, starting with any quantum group, produces invariants for knots. The question that we have in mind is to find topological models for certain types of quantum invariants. On the topological side, in 2000, Bigelow, building on earlier work of Lawrence,
interpreted the original Jones polynomial in a homological manner- as a graded intersection pairing in a covering of a configuration space of the punctured disc. On the quantum side of the story, the coloured Jones polynomials are a sequence of quantum invariants constructed through the Reshetikhin-Turaev recipe from the quantum group Uq(sl(2)). The first invariant of this sequence is the original Jones polynomial. In this talk we will present how one can use topological intersection pairings in order to describe a topological model for all coloured Jones polynomials.

Francis Woodhouse: Autonomous mechanisms inspired by biology

Unlike the air around us, biological systems are not in equilibrium: cells consume chemical energy to keep growing and moving, forming a clear arrow of time. The recent creation of artificial versions of these ‘active’ materials suggests that these concepts can be harnessed to power new soft robotic systems fuelled by as simple a source as oxygen. After an introduction to the physics of natural and artificial active systems, we will see how endowing a mechanical network with activity can create an intricate self-actuating mechanism.

In this session we discuss various different routes for promoting your research through a panel discussion with Dawn Gordon (Project Manager, Oxford University Innovation), Dyrol Lumbard (External Relations Manager, Mathematical Institute), James Maynard (Academic Faculty, Mathematical Institute) and Ian Griffiths, and chaired by Frances Kirwan. The panel discussion will include the topics of outreach, impact, and strategies for promoting aspects of mathematics that are less amenable to public engagement. 

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.

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?

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.
 

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.
 

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.
 

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.
 

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
   

A panel discussion and Q&A, looking at some of the challenges and opportunities available for mathematicians outside universities. Featuring:

Madeleine Copin – North London Collegiate School
Josephine French – Health Data Insight, working in partnership with Public Health England
Martin Gould – Spotify
Dan Jones – Quadrature Capital
Adam Sardar – e-therapeutics

Wondering about how to organise your DPhil? How to make the most of your supervision meetings?

In this session we will explore these and other questions related to what makes a successful DPhil with help from faculty members, postdocs and DPhil students.

• In the first half of the session Dan Ciubotaru and Philip Maini will give short talks on their experiences as PhD students and supervisors.
• The second part of the session will be a panel discussion with final-year Dphil students and early postdocs.

The panel will consist of Thomas Wasserman, Renee Hoekzema, Jaroslav Fowkes and Carolina Matte Gregory. Senior faculty members will be kindly asked to leave the lecture theatre to ensure that students feel comfortable discussing their experiences with other students and postdocs without any senior faculty present.

Richard Wade:   Classifying spaces, automorphisms, and right-angled Artin groups 

Right-angled Artin groups (otherwise known as partially commutative groups, or graph groups), interpolate between free abelian groups and free groups. These groups have seen a lot of attention recently, much of this due to some surprising links to the world of hyperbolic 3-manifolds.We will look at classifying spaces for such groups and their associated automorphism groups. These spaces are useful as they give a topological way to understand algebraic invariants of groups. This leads us to study some beautiful mathematical objects: deformation spaces of tori and trees. We will look at some recent results that aim to bridge the gap between these two families of spaces.
 
Andrey Kormilitzin:   Learning from electronic health records using the theory of rough paths

In this talk, we bring the theory of rough paths to the study of non-parametric statistics on streamed data and particularly to the problem of regression and classification, where the input variable is a stream of information, and the dependent response is also (potentially) a path or a stream.  We informally explain how a certain graded feature set of a stream, known in the rough path literature as the signature of the path, has a universality that allows one to characterise the functional relationship summarising the conditional distribution of the dependent response. At the same time this feature set allows explicit computational approaches through machine learning algorithms.

Finally, the signature-based modelling can be applied to some real-world problems in medicine, in particular in mental health and gastro-enterology.

Laura Capuano's talk 'Pell equations and continued fractions in number theory'

The classical Pell equation has an extraordinary long history and it is very useful in many different areas of number theory. For example, they given a way to write a prime congruent to 1 modulo 4 as a sum of two squares, or they can also be used to break RSA excryption when the decription key is too small. In this talk, I will present some properties of this wonderful equation and its relation with continued fractions. I will also treat the case of Pell equations in other contexts, such as the ring of polynomials, showing the differences with the classical case. 

Noemi Picco's talk 'Cortical neurogenesis: how humans (and mathematicians) can do more than macaque, with less'

The cerebral cortex is perhaps the crowning achievement of evolution and is the region of the brain that distinguishes us from other species. Studying the developmental programmes that generate cortices of different sizes and neuron counts, is the key to understanding both brain evolution and disease. I will show what mathematical modeling has to say about cortex evolution, when data resolution is poor. I will then discuss why humans are so special in the way they create their cortex, and how we are just like everybody else in many other aspects of brain development.

Lisa Lamberti

Geometric models in algebra and beyond

Many phenomena in mathematics and related sciences are described by geometrical models.

In this talk, we will see how triangulations in polytopes can be used to uncover combinatorial structures in algebras. We will also glimpse at possible generalizations and open questions, as well as some applications of geometric models in other disciplines.

Jaroslav Fowkes

[[{"fid":"47972","view_mode":"media_square","fields":{"format":"media_square","field_file_image_alt_text[und][0][value]":"Jaroslav Fowkes","field_file_image_title_text[und][0][value]":"Jaroslav Fowkes"},"type":"media","attributes":{"alt":"Jaroslav Fowkes","title":"Jaroslav Fowkes","height":"258","width":"258","class":"media-element file-media-square"}}]]

Optimization Challenges in the Commercial Aviation Sector

The commercial aviation sector is a low-margin business with high fixed costs, namely operating the aircraft themselves. It is therefore of great importance for an airline to maximize passenger capacity on its route network. The majority of existing full-service airlines use largely outdated capacity allocation models based on customer segmentation and fixed, pre-determined price levels. Low-cost airlines, on the other hand, mostly fly single-leg routes and have been using dynamic pricing models to control demand by setting prices in real-time. In this talk, I will review our recent research on dynamic pricing models for the Emirates route network which, unlike that of most low-cost airlines, has multiple routes traversing (and therefore competing for) the same leg.

In this session we will refresh our understanding of the purpose of an interview, review some top tips, and practise answering some typical interview questions. Rachel will also signpost further resources on interview preparation available at the Careers Service.

Erik Panzer

Feynman integrals, graph polynomials and zeta values

Where do particle physicists, algebraic geometers and number theorists meet?

Feynman integrals compute how elementary particles interact and they are fundamental for our understanding of collider experiments. At the same time, they provide a rich family of special functions that are defined as period integrals, including special values of certain L functions.

In the talk I will give the definition of Feynman integrals via graph polynomials and discuss some examples that evaluate to values of the Riemann zeta function. Then I will discuss some of the interesting questions in this field and mention some of the techniques that are used to study these.

[[{"fid":"47855","view_mode":"media_square","fields":{"format":"media_square","field_file_image_alt_text[und][0][value]":"Yuji Nakatsukasa","field_file_image_title_text[und][0][value]":"Yuji Nakatsukasa"},"type":"media","attributes":{"alt":"Yuji Nakatsukasa","title":"Yuji Nakatsukasa","height":"258","width":"258","class":"media-element file-media-square"}}]]

Yuji Nakatsukasa

Computing matrix eigenvalues

The numerical linear algebra community solves two main problems: linear systems, and eigenvalue problems. They are both vastly important; it would not be too far-fetched to say that most (continuous) problems in scientific computing eventually boil down to one or both of these.

This talk focuses on eigenvalue problems. I will first describe some of their applications, such as Google's PageRank, PCA, finding zeros and poles of functions, and global optimization. I will then turn to algorithms for computing eigenvalues, namely the classical QR algorithm---which is still the basis for state-of-the-art. I will emphasize that the underlying mathematics is (together with the power method and numerical stability analysis) rational approximation theory.

Professor Uta Frith FRS is a distinguished developmental psychologist who is well known for her pioneering research on autism spectrum disorders. She also has a long-standing interest in matters relating to diversity in science, and is the Chair of the Royal Society's Diversity Committee. Oxford Mathematician Dr Maria Bruna is a Junior Research Fellow in Mathematics at St John's College, and has won prizes such as the L'Oréal-UNESCO UK and Ireland For Women in Science Fellowship and the Olga Taussky Pauli Fellowship, Wolfgang Pauli Institute. This informal discussion will no doubt include a range of topics -- but it is hard to say in advance where the conversation might go!

Alan is the Head of Counselling at the University of Oxford.  He will talk about the importance of managing expectations and not having rigid expectations, about challenging perfectionism, and about building emotional resilience through adaptability and compassion.