Past

This session is aimed at first-year undergraduates preparing for Prelims exams. A panel of lecturers will share key advice on exam technique and revision strategies, and a current student will offer practical tips from their own experience. This event complements the Friday@2 session in Week 1 on Dealing with Exam Anxiety.

Complex representations of p-adic groups are in many ways well-understood. The category has Bernstein's decomposition into blocks, and for many groups each block is known to be equivalent to modules over a Hecke algebra. In particular, the unipotent block of GLn (the block containing the trivial representation) is equivalent to the modules over an extended affine hecke algebra of type A. Over \bar{Fl} the situation is more complicated in the general case: the Bernstein block decomposition can fail (eg for SP8), and there is no longer in general an equivalence with the Hecke algebra. However, some groups, such as GLn and its inner forms, still have a Bernstein decomposition. Furthermore, Vigernas showed that the unipotent block of GLn contains a subcategory that is equivalent to modules over the Schur algebra, a mild extension of the Hecke algebra with much of the same theory, and this subcategory generates the unipotent block under extensions. Building on this work, we show that the derived category of the unipotent block of GLn is triangulated-equivalent to the perfect complexes over a dg-enriched Schur algebra. We prove this by combining general finiteness results about Schur algebras with the well-known structure of the l-modular unipotent blocks of GLn over finite fields.

Circadian clocks generate autonomous daily rhythms of gene expression in anticipation of daily sunlight and temperature cycles in a variety of organisms. The simples and best characterised of all circadian clocks in nature is the cyanobacterial clock, the core of which consists of just 3 proteins - KaiA, KaiB and KaiC - locked in a 24-h phosphorylation-dephosphorylation loop. Substantial progress has been made in understanding how cells generate and sustain this rhythm, but important questions remain: how does the clock maintain resilience in the face of internal and external fluctuations, how is the clock coupled to other cellular processes and what dynamics arise from this coupling? We address these questions using an interdisciplinary approach combining time-lapse microscopy and modelling. In this talk, I will first characterise the clock's free-running robustness and explore how the clock buffers environmental noise and genetic mutations. Our stochastic model predicts how the clock filters out such noise, including fast light fluctuations, to keep time while remaining responsive to environmental shifts, revealing also that the wild-type operates at a noise optimum. Next, I will focus on how the clock interacts with the other major cellular cycle, the cell division cycle. Our single-cell data shows that the clock couples to the division rate and expression of cell cycle-dependent factors using both frequency modulation and amplitude modulation strategies, with implications for cell growth and cell size control. Our findings illustrate how simple systems can exhibit complex dynamics, advancing our understanding of the interdependency between gene circuits and cellular physiology.  
 

Feferman proved in 1962 that any arithmetical theorem is a consequence of a suitable transfinite iteration of uniform reflections. This result is commonly known as Feferman's completeness theorem. The talk aims to give one or two new proofs of Feferman's completeness theorem that, we hope, shed new light on this mysterious and often overlooked result.

Moreover, one of the proofs furnishes sharp bounds on the order types of well-orders necessary to attain completeness.

(This is joint work with Fedor Pakhomov and Dino Rossegger.)

We discuss ongoing work with Joseph Leung in which we obtain estimates for sums of Fourier coefficients of GL(2) and certain GL(3) automorphic forms along the values of irreducible binary cubics.

Cryptocurrencies such as Bitcoin have only recently become a significant part of the financial landscape. Many billions of dollars are now traded daily on limit order-book markets such as Binance, and these are probably among the most open, liquid and transparent markets there are. They therefore make an interesting platform from which to investigate myriad questions to do with market microstructure. I shall talk about a few of these, including live-trading experiments to investigate the difference between on-paper strategy analysis (typical in the academic literature) and actual trading outcomes. I shall also mention very recent work on the new Hyperliquid exchange which runs on a blockchain basis, showing how to use this architecture to obtain datasets of an unprecendented level of granularity. This is joint work with Jakob Albers, Mihai Cucuringu and Alex Shestopaloff.

The Alfvén waves are fundamental wave phenomena in magnetized plasmas. Mathematically, the dynamics of Alfvén waves are governed by a system of nonlinear partial differential equations called the magnetohydrodynamics (MHD) equations. Let us introduce some recent results about inverse scattering of Alfvén waves in ideal MHD, which are intended to establish the relationship between Alfvén waves emanating from the plasma and their scattering fields at infinities.The proof is mainly based on the weighted energy estimates. Moreover, the null structure inherent in MHD equations is thoroughly examined, especially when we estimate the pressure term.

Many applications in electromagnetism, magnetohydrodynamics, and pour media flow are well-posed in spaces from the 3D de Rham complex involving $H^1$, $H(curl)$, $H(div)$, and $L^2$. Discretizing these spaces with the usual conforming finite element spaces typically leads to discrete problems that are both structure-preserving and uniformly stable with respect to the mesh size and polynomial degree. Robust preconditioners/solvers usually require the inversion of subproblems or auxiliary problems on vertex, edge, or face patches of elements. For high-order discretizations, the cost of inverting these patch problems scales like $\mathcal{O}(p^9)$ and is thus prohibitively expensive. We propose a new set of basis functions for each of the spaces in the discrete de Rham complex that reduce the cost of the patch problems to $\mathcal{O}(p^6)$ complexity. By taking advantage of additional properties of the new basis, we propose further computationally cheaper variants of existing preconditioners. Various numerical examples demonstrate the performance of the solvers.

We establish that a criterion based on ring-theoretic amenability is both necessary and sufficient for the abelian version of the Elekes-Szabó theorem to be sharp in the case of positive characteristic. Moreover, the criterion is always sufficient. We provide illustrative examples in the theories ACF_p and DCF_0.

In this talk, we give an accessible introduction to the theory of coarse cohomology of metric spaces in the sense of Margolis, which we present in direct analogy with group cohomology for discrete groups. We explain how this yields the robust notion of coarse cohomological dimension (due to Margolis), which is a genuine quasi-isometry invariant of metric spaces generalising the cohomological dimension of groups when the latter is finite. We then give applications to geometric properties of quasimorphisms and motivate how such considerations might be useful in the setting of non-positively curved groups. This is joint reading/work with Paula Heim.

q-deformations offer a systematic way to generalize familiar mathematical structures, revealing hidden symmetries and richer geometries that collapse back to classical frameworks as the deformation parameter goes to 1. Beyond their mathematical elegance, q-deformations have naturally emerged in diverse areas of theoretical physics, offering fresh perspectives on quantization, regularization, and non-commutative geometry. In this talk, we will explore how q-deformations intersect with the intriguing question of the uniqueness of string scattering amplitudes.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

The Stochastic Burgers Equation (SBE) was introduced in the eighties by van Beijren, Kutner and Spohn as a mesoscopic model for driven diffusive systems with one conserved scalar quantity. In the subcritical dimension d=1, it coincides with the derivative of the KPZ equation whose large-scale behaviour is polynomially superdiffusive and given by the KPZ Fixed Point, and in the super-critical dimensions d>2, it was recently shown to be diffusive and rescale to an anisotropic Stochastic Heat equation. At the critical dimension d=2, the SBE was conjectured to be logarithmically superdiffusive with a precise exponent but this has only been shown up to lower order corrections. This talk is based on the work joint with Giuseppe Cannizzaro and Fabio Toninelli under the same name https://arxiv.org/abs/2501.00344, where we pin down the logarithmic superdiffusivity by identifying exactly the large-time asymptotic behaviour of the so-called diffusion matrix and show that, once the logarithmic corrections to the scaling are taken into account, the solution of the SBE satisfies a central limit theorem. This is the first superdiffusive scaling limit result for a critical SPDE, beyond the weak coupling regime.

The Wasserstein metric originates in the theory of optimal transport, and among many other applications, it provides a natural way to measure how evenly distributed a finite point set is. We give a survey of classical and more recent results that describe the behaviour of some random point processes in Wasserstein metric, including the eigenvalues of some random matrix models, and explain the connection to the logarithm of the characteristic polynomial of a random unitary matrix. We also discuss a simple random walk model on the unit circle defined in terms of a quadratic irrational number, which turns out to be related to surprisingly deep arithmetic properties of real quadratic fields.

Building on the concept of diagonal dimension introduced by Li, Liao, and Winter in 2023, we propose a topological dimension for an inclusion pair of C*-algebras. This new framework allows for finite values in cases of Cartan inclusions that are not diagonal. In this talk, we present calculations for both upper and lower bounds concerning the inclusion of the unitization of c_0(\mathbb{N}) into the Toeplitz algebra. This work is a collaboration with W. Winter.

We use Non-Reductive GIT to construct compactifications of Hilbert schemes of complete intersections. We then study ample line bundles on these compactifications in order to construct moduli spaces of complete intersections for certain degree types.

The mapping class group a solid handlebody of genus g sits between mapping class groups of surfaces and Out(F_n), in the sense there is an injective map to the mapping class group of the boundary and a surjective map to Out(F_g) via the action on the fundamental group. Similar behaviour happens with actions on associated spaces, such curve complexes and Teichmuller space. I’ll give an expository talk on this, partly in the context of our proof with Petersen that handlebody groups are virtual duality groups, and partly in the context of a problem list on handlebody groups written with Andrew, Hensel, and Hughes.

I will construct a fully-faithful functor from the category of co-admissible D-cap modules of Ardakov—Wadsley, to the category of quasi-coherent sheaves on the "analytic de Rham space”, at least in the case when the rigid variety is affinoid and étale over a polydisk. 

A matroid is frame if it can be extended such that it possesses a basis $B$ (a frame) such that every element is spanned by at most two elements of $B$. Frame matroids extend the class of graphic matroids and also have natural graphical representations. We characterise the inequivalent graphical representations of 3-connected frame matroids that have a fixed element $\ell$ in their frame $B$. One consequence is a polynomial time recognition algorithm for frame matroids with a distinguished frame element.

Joint work with Jim Geelen and Cynthia Rodríquez.

I will outline some recent progress in identifying realistic models of particle physics in heterotic string theory, supported by several mathematical and computational advancements which include: analytic expressions for bundle valued cohomology dimensions on complex projective varieties, heuristic methods of discrete optimisation such as reinforcement learning and genetic algorithms, as well as efficient neural-network approaches for the computation of Ricci-flat metrics on Calabi-Yau manifolds, hermitian Yang-Mills connections on holomorphic vector bundles and bundle valued harmonic forms. I will present a proof of concept computation of quark masses in a string model that recovers the exact standard model spectrum and discuss several other models that can accommodate the entire range of flavour parameters observed in the standard model. 

In this talk, we present a mathematical model for tumour growth that incorporates viscoelastic effects. Starting from a basic system of PDEs, we gradually introduce the relevant biological and physical mechanisms and explain how they are integrated into the model. The resulting system features a Cahn--Hilliard type equation for the tumour cells coupled to a convection-reaction-diffusion equation for a nutrient species, and a viscoelastic subsystem for an internal velocity.
Key biological processes such as active transport, apoptosis, and proliferation are modeled via source and sink terms as well as cross-diffusion effects. The viscoelastic behaviour is described using the Oldroyd-B model, which is based on a multiplicative decomposition of the deformation gradient to account for elasticity alongside growth and relaxation effects.
We will highlight several of these effects through numerical simulations.
Moreover, we discuss the main analytical and numerical challenges. Particular focus will be given to the treatment of source and cross-diffusion terms, the elastic energy density, and the difficulties arising from the viscoelastic subsystem. The main analytical result is the global-in-time existence of weak solutions in two spatial dimensions, under the assumption of additional viscoelastic diffusion in the Oldroyd-B equation.
This work is based on joint work with Harald Garcke (University of Regensburg, Germany) and Balázs Kovács (University of Paderborn, Germany).

The arithmetic regularity lemma says that any dense set A in F_p^n can be cut along cosets of some small codimension subspace H <= F_p^n such that on almost all cosets of H, A is either random or structured (in a precise quantitative manner). A standard example shows that one cannot hope to improve "almost all" to "all", nor to have a good quantitative dependency between the constants involved. Adding a further combinatorial assumption on A to the arithmetic regularity lemma makes its conclusion so strong that one can essentially classify such sets A. In this talk, I will use use the analogous problem with F_p^n replaced with R^n as a way the motivate the funny title.