Want a quick flavour of who we are and what we do?
Our YouTube shorts are, unsurprisingly, short films featuring Oxford Mathematicians talking about, among other things, their research, their toys and their toothpaste. And there are also clips from our popular public lectures to whet the appetite for the full thing.
Nichols algebras, also known as small shuffle algebras, are a family of graded bialgebras including the symmetric algebras, the exterior algebras, the positive parts of quantized enveloping algebras, and, conjecturally, Fomin-Kirillov algebras. As the case of Fomin-Kirillov algebra shows, it can be very
difficult to determine the maximum degree of a minimal generating set of relations of a Nichols algebra.
Building upon Kapranov and Schechtman’s equivalence between the category of perverse sheaves on Sym(C) and the category of graded connected bialgebras, we describe the geometric counterpart of the maximum degree of a generating set of relations of a graded connected bialgebra, and we show how this specialises to the case o Nichols algebras.
The talk is based on joint work with Francesco Esposito and Lleonard Rubio y Degrassi.
Let X be a smooth rigid-analytic variety. Ardakov and Wadsley introduced the sheaf D-cap of infinite order differential operators on X, along with the category of coadmissible D-cap-modules. In this talk, we present a Riemann–Hilbert correspondence for these coadmissible D-cap-modules. Specifically, we interpret a coadmissible D-cap-module as a p-adic differential equation, explain what it means to solve such an equation, and describe how to reconstruct the module from its solutions.
When it comes to the nonlinear heat equation u_t - \Delta u = u^p, a sharp condition for the stability of positive radial steady states was derived in the classical paper by Gui, Ni and Wang. In this talk, I will present some recent joint work with Daniel Devine that focuses on a more general system of reaction-diffusion equations (which is also also known as the parabolic Henon-Lane-Emden system). We obtain a sharp condition that determines the stability of positive radial steady states, and we also study the separation property of these solutions along with their asymptotic behaviour at infinity.
Wednesday 12th March.
It would be great to have as many teams as possible racing in this idyllic 4-leg relay around Oxford, encompassing the River Thames and Christ Church Meadow and beginning and ending at Iffley Road track where Roger Bannister ran the first ever sub-4-minute mile. Each leg is approximately 7km in length. Further details can be found on the Facebook event page.
In recent years, a novel approach to studying integrable models has emerged which leverages a higher-dimensional gauge theory, specifically a holomorphic-topological theory. This new framework provides alternative methods for investigating quantum aspects of integrability and for constructing integrable models in more than two dimensions. This talk will review the foundations of this approach, its applications, and the exciting possibilities it opens up for future research in the field of integrable systems.
While extremal black hole microstates are reproduced by index calculations, the study of near-BPS black holes requires special care to account for quantum fluctuations. A semiclassical analysis indicates that the spectrum of such black holes has a large extremal degeneracy followed by a mass gap up to a continuum of non-BPS states. The inclusion of a theta angle term alters the properties of the spectrum (Witten effect shifting the mass gap and mixed 't Hooft anomaly). This journal club will study two papers by Toldo and Heydeman, [2412.03695] and [2412.03697] where they study 4d near-BPS black holes. As we shall see, a key point of their derivation is the reduction to 2d JT gravity. The dual CFTs are ABJM and some class R (non lagrangian) theories. Since these theories are strongly coupled, the gravity analysis offers a powerful tool to describe their specturm at finite temperature.
