An informal session for DPhil students, ECRs and undergraduates with an interest in probability. The aim is to gain exposure to areas outside of your own research interests in an informal and accessible way.

# Past Forthcoming Seminars

Authors:

Anne Balter and Antoon Pelsser

Models can be wrong and recognising their limitations is important in financial and economic decision making under uncertainty. Robust strategies, which are least sensitive to perturbations of the underlying model, take uncertainty into account. Interpreting

the explicit set of alternative models surrounding the baseline model has been difficult so far. We specify alternative models by a stochastic change of probability measure and derive a quantitative bound on the uncertainty set. We find an explicit ex ante relation

between the choice parameter k, which is the radius of the uncertainty set, and the Type I and II error probabilities on the statistical test that is hypothetically performed to investigate whether the model specification could be rejected at the future test horizon.

The hypothetical test is constructed to obtain all alternative models that cannot be distinguished from the baseline model with sufficient power. Moreover, we also link the ambiguity bound, which is now a function of interpretable variables, to numerical

values on several divergence measures. Finally, we illustrate the methodology on a robust investment problem and identify how the robustness multiplier can be numerically interpreted by ascribing meaning to the amount of ambiguity.

Following recent papers by Nima Arkani-Hamed.

The Oberwolfach Research Institute for Mathematics (Mathematisches Forschungsinstitut Oberwolfach/MFO) was founded in late 1944 by the Freiburg mathematician Wilhelm Süss (1895-1958) as the „National Institute for Mathematics“. In the 1950s and 1960s the MFO developed into an increasingly international conference centre.

The aim of my project is to analyse the history of the MFO as it institutionally changed from the National Institute for Mathematics with a wide, but standard range of responsibilities, to an international social infrastructure for research completely new in the framework of German academia. The project focusses on the evolvement of the institutional identity of the MFO between 1944 and the early 1960s, namely the development and importance of the MFO’s scientific programme (workshops, team work, Bourbaki) and the instruments of research employed (library, workshops) as well as the corresponding strategies to safeguard the MFO’s existence (for instance under the wings of the Max-Planck-Society). In particular, three aspects are key to the project, namely the analyses of the historical processes of (1) the development and shaping of the MFO’s workshop activities, (2) the (complex) institutional safeguarding of the MFO, and (3) the role the MFO played for the re-internationalisation of mathematics in Germany. Thus the project opens a window on topics of more general relevance in the history of science such as the complexity of science funding and the re-internationalisation of the sciences in the early years of the Federal Republic of Germany.

Joint work with Zsolt Vizi (Bolyai Institute, University of Szeged, Hungary), Istvan Kiss (Department

of Mathematics, University of Sussex, United Kingdom)

Pairwise models have been proven to be a flexible framework for analytical approximations

of stochastic epidemic processes on networks that are in many situations much more accurate

than mean field compartmental models. The non-Markovian aspects of disease transmission

are undoubtedly important, but very challenging to incorporate them into both numerical

stochastic simulations and analytical investigations. Here we present a generalization of

pairwise models to non-Markovian epidemics on networks. For the case of infectious periods

of fixed length, the resulting pairwise model is a system of delay differential equations, which

shows excellent agreement with results based on the explicit stochastic simulations. For more

general distribution classes (uniform, gamma, lognormal etc.) the resulting models are PDEs

that can be transformed into systems of integro-differential equations. We derive pairwise

reproduction numbers and relations for the final epidemic size, and initiate a systematic

study of the impact of the shape of the particular distributions of recovery times on how

the time evolution of the disease dynamics play out.

The geometry of the moduli space of 4d N=2 moduli spaces, and in particular of their Coulomb branches (CBs), is very constrained. In this talk I will show that through its careful study, we can learn general and somewhat surprising lessons about the properties of N=2 super conformal field theories (SCFTs). Specifically I will show that we can prove that the scaling dimension of CB coordinates, and thus of the corresponding operator at the SCFT fixed point, has to be rational and it has a rank-dependent maximum value and that in general the moduli spaces of N=2 SCFTs can have metric singularities as well as complex structure singularities.

Finally I will outline how we can explicitly perform a classification of geometries of N>=3 SCFTs and carry out the program up to rank-2. The results are surprising and exciting in many ways.

We discuss shock reflection problem for compressible gas dynamics, von Neumann conjectures on transition between regular and Mach reflections, and existence of regular reflection solutions for potential flow equation. Then we will talk about recent results on uniqueness and stability of regular reflection solutions for potential flow equation in a natural class of self-similar solutions. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear elliptic equation, and prove uniqueness by a version of method of continuity. A property of solutions important for the proof of uniqueness is convexity of the free boundary.

This talk is based on joint works with G.-Q. Chen and W. Xiang.

Massless Quantum Field Theories can be described perturbatively by chiral worldsheet models - the so-called Ambitwistor Strings. In contrast to conventional string theory, where loop amplitudes are calculated from higher genus Riemann surfaces, loop amplitudes in the ambitwistor string localise on the non-separating boundary of the moduli space. I will describe the resulting framework for QFT amplitudes from (nodal) Riemann spheres, building up from tree-level to two-loop amplitudes.

**Robert Timms**

Title: Multiscale modelling of lithium-ion batteries

Lithium-ion batteries are one of the most widely used technologies for energy storage, with applications ranging from portable electronics to electric vehicles. Due to their popularity, there is a continued interest in the development of mathematical models of lithium-ion batteries. These models encompass various levels of complexity, which may be suitable to aid with design, or for real-time monitoring of performance. After a brief introduction to lithium-ion batteries, I will discuss some of the modelling efforts undertaken here at Oxford and within the wider battery modelling community.

**Jan Vonk**

Title: Singular moduli for real quadratic fields

At the 1900 ICM, David Hilbert posed a series of problems, of which the 12th remains completely open today. I will discuss how to solve this problem in the simplest open case, by considering certain exotic (so called p-adic) metrics on the set of numbers, and using its concomitant theories of analysis and geometry.