I will present a gentle introduction to the theory of conformal dimension, focusing on its applications to the boundaries of hyperbolic groups, and the difficulty of classifying groups whose boundaries have conformal dimension 1.

# Forthcoming Seminars

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

Prime numbers have intrigued, inspired and infuriated mathematicians for millennia and yet mathematicians' difficulty with answering simple questions about them reveals their depth and subtlety.

Join Vicky to learn about recent progress towards proving the famous Twin Primes Conjecture and to hear the very different ways in which these breakthroughs have been made - a solo mathematician working in isolation, a young mathematician displaying creativity at the start of a career, a large collaboration that reveals much about how mathematicians go about their work.

Vicky Neale is Whitehead Lecturer at the Mathematical Institute, University of Oxford and Supernumerary Fellow at Balliol College.

Please email external-relations@maths.ox.ac.uk to register.

We construct maximal surfaces with Neumann boundary conditions in Minkowski space using mean curvature flow. In particular we find curvature conditions on a boundary manifold so that mean curvature flow may be shown to exist for all time, and give conditions under which the maximal hypersurfaces are stable under the flow.

The mathematical analysis and numerical simulation of acoustic and electromagnetic wave scattering by planar screens is a classical topic. The standard technique involves reformulating the problem as a boundary integral equation on the screen, which can be solved numerically using a boundary element method. Theory and computation are both well-developed for the case where the screen is an open subset of the plane with smooth (e.g. Lipschitz or smoother) boundary. In this talk I will explore the case where the screen is an arbitrary subset of the plane; in particular, the screen could have fractal boundary, or itself be a fractal. Such problems are of interest in the study of fractal antennas in electrical engineering, light scattering by snowflakes/ice crystals in atmospheric physics, and in certain diffraction problems in laser optics. The roughness of the screen presents challenging questions concerning how boundary conditions should be enforced, and the appropriate function space setting. But progress is possible and there is interesting behaviour to be discovered: for example, a sound-soft screen with zero area (planar measure zero) can scatter waves provided the fractal dimension of the set is large enough. Accurate computations are also challenging because of the need to adapt the mesh to the fine structure of the fractal. As well as presenting numerical results, I will outline some of the outstanding open questions from the point of view of numerical analysis. This is joint work with Simon Chandler-Wilde (Reading) and Andrea Moiola (Pavia).

Compactifications of 6D Superconformal Field Theories (SCFTs) on four-manidolfds lead to novel interacting 2D SCFTs. I will describe the various Lagrangian and non-Lagrangian sectors of the resulting 2D theories, as well as their interactions. In general this construction can be embedded in compactifications of the physical superstring, providing a general template for realizing 2D conformal field theories coupled to worldsheet gravity, i.e. a UV completion for non-critical string theories.

Discontinuous solutions, such as cracks or cavities, can suddenly appear in elastic solids when a limiting condition is reached. Similarly, self-contacting folds can nucleate at a free surface of a soft material subjected to a critical compression. Unlike other elastic instabilities, such as buckling and wrinkling, creasing is still poorly understood. Being invisible to linearization techniques, crease nucleation is a problem of high mathematical complexity.

In this talk, I will discuss some recent theoretical insights solving the quest for both the nucleation threshold and the emerging crease morphology. The analytic predictions are in agreement with experimental and numerical data. They prove a fundamental insight either for understanding the creasing onset in living matter, e.g. brain convolutions, or for guiding engineering applications, e.g. morphable meta-materials.

Recall that an integer n is called y-smooth when each of its prime divisors is less than or equal to y. It is conjectured that, for any a>0, any polynomial of positive degree having integral coefficients should possess infinitely many values at integral arguments n that are n^a-smooth. One could consider this problem to be morally “dual” to the cognate problem of establishing that irreducible polynomials assume prime values infinitely often, unless local conditions preclude this possibility. This smooth values conjecture is known to be true in several different ways for linear polynomials, but in general remains unproven for any degree exceeding 1. We will describe some limited progress in the direction of the conjecture, highlighting along the way analogous conclusions for polynomial smoothness. Despite being motivated by a problem in analytic number theory, most of the methods make use of little more than pre-Galois theory. A guest appearance will be made by several hyperelliptic curves. [This talk is based on work joint with Jonathan Bober, Dan Fretwell and Greg Martin].

We derive sharp bounds for the prices of VIX futures using the full information of S&P 500 smiles. To that end, we formulate the model-free sub/superreplication of the VIX by trading in the S&P 500 and its vanilla options as well as the forward-starting log-contracts. A dual problem of minimizing/maximizing certain risk-neutral expectations is introduced and shown to yield the same value. The classical bounds for VIX futures given the smiles only use a calendar spread of log-contracts on the S&P 500. We analyze for which smiles the classical bounds are sharp and how they can be improved when they are not. In particular, we introduce a tractable family of functionally generated portfolios which often improves the classical spread while still being tractable, more precisely, determined by a single concave/convex function on the line. Numerical experiments on market data and SABR smiles show that the classical lower bound can be improved dramatically, whereas the upper bound is often close to optimal.

This talk will be a gentle introduction to braided fusion categories, with the eventual aim to explain a result from my thesis about symmetric fusion categories.

Fusion categories are certain kinds of monoidal categories. They can be viewed as a categorification of the finite dimensional algebras, and appear in low-dimensional topological quantum field theories, as well as being studied in their own right. A braided fusion category is additionally commutative up to a natural isomorphism, symmetry is an additional condition on this natural isomorphism. Computations in these categories can be done pictorially, using so-called string diagrams (also known as ``those cool pictures'').

In this talk I will introduce fusion categories using these string diagrams. I will then discuss the Drinfeld centre construction that takes a fusion category and returns a braided fusion category. We then show, if the input is a symmetric fusion category, that this Drinfeld centre carries an additional tensor product. All of this also serves as a good excuse to draw lots of pictures.