The theory of connections on curves and Hitchin systems is something like a “global theory of Lie groups”, where one works over a Riemann surface rather than just at a point. We’ll describe how one can take this analogy a few steps further by attempting to make precise the class of rich geometric objects that appear in this story (including the non-compact case), and discuss their classification, outlining a theory of “Dynkin diagrams” as a step towards classifying some examples of such objects.

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.

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

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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 The majority of existing full-service airlines use largely outdated capacity allocation models based on customer segmentation and fixed, pre-determined price levels.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.

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.

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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.

Understanding the spatial distribution of organisms throughout an environment is an important topic in population ecology. We briefly review ecological questions underpinning certain mathematical work that has been done in this area, before presenting a few examples of spatially structured population models. As a first example, we consider a model of two species aggregation and clustering in two-dimensional domains in the presence of heterogeneity, and demonstrate novel aggregation mechanisms in this setting. We next consider a second example consisting of a predator-prey-subsidy model in a spatially continuous domain where the spatial distribution of the subsidy influences the stability and spatial structure of steady states of the system. Finally, we discuss ongoing work on extending such results to network-structured domains, and discuss how and when the presence of a subsidy can stabilize predator-prey dynamics."

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Compaction is a primary process in the evolution of a sedimentary basin. Various 1D models exist to model a basin compacting due to overburden load. We explore a multi-dimensional model for a basin undergoing mechanical and chemical compaction. We discuss some properties of our model. Some test cases in the presence of geological features are considered, with appropriate numerical techniques presented.

Research takes a long time while the attention span of the world is apparently decreasing, so today's researchers need to be able to get their message across quickly and succinctly. In this session we'll share some tips on how to communicate the key messages of your work in just a few minutes, and give you a chance to have a go yourself.  This will be helpful for job and funding applications and interviews, and also for public engagement. In September there will be an opportunity to do it for real, for our alumni, when we'll showcase Oxford Mathematics at the Alumni Weekend.

I will described how ideas from constructive quantum field theory can be adapted to produce a systematic approach for analytic renormalization in the theory of regularity structures.

In 1933, lattice theory was a new subject, put forth by Garrett Birkhoff. In contrast, in 1940, it was already a mature subject, worth publishing a book on. Indeed, the first monograph, written by the same G. Birkhoff, was the result of these 7 years of working on a lattice theory. In my talk, I would like to focus on this fast development. I will present the notion of a theory not only as an actors' category but as an historical category. Relying on that definition, I would like to focus on some collaborations around the notion of lattices. In particular, we will study lattice theory as a meeting point between the works of G. Birkhoff and two other mathematicians: John von Neumann and Marshall Stone.

Magnus expansion based methods are an efficient class of integrators for solving Schrödinger equations that feature time dependent potentials such as lasers. These methods have been found to be highly effective in computational quantum chemistry since the pioneering work of Tal Ezer and Kosloff in the early 90s. The convergence of the Magnus expansion, however, is usually understood only for ODEs and traditional analysis suggests a much poorer performance of these methods than observed experimentally. It was not till the work of Hochbruck and Lubich in 2003 that a rigorous analysis justifying the application to PDEs with unbounded operators, such as the Schrödinger equation, was presented. In this talk we will extend this analysis to the semiclassical regime, where the highly oscillatory solution conventionally suggests large errors and a requirement for very small time steps.