Past

Bestvina--Brady groups were first introduced by Bestvina and Brady for their interesting finiteness properties. In this talk, we discuss their Dehn functions, that are a notion of isoperimetric inequality for finitely presented groups and can be thought of as a "quantitative version" of finite presentability. A result of Dison shows that the Dehn function of a Bestvina--Brady group is always bounded above by a quartic polynomial.

Our main result is to compute the Dehn function for all finitely presented Bestvina--Brady groups. In particular, we show that the Dehn function of a Bestvina--Brady group grows as a polynomial of integer degree, and we present the combinatorial criteria on the graph that determine whether the Dehn functions of the associated Bestvina--Brady group is linear, quadratic, cubic, or quartic.

This is joint work with Chang and García-Mejía.

We consider quantum electrodynamics in 2+1 dimensions (QED3) with N matter fields and Chern-Simons level k. For small values of k and N, this theory describes various experimentally relevant systems in condensed matter, and is also conjectured to be part of a web of non-supersymmetric dualities. We compute the scaling dimensions of monopole operators in a large N and k expansion, which appears to be extremely accurate even down to the smallest values of N and k, and allows us to find dynamical evidence for these dualities and make predictions about the phase transitions. For instance, we combine these estimates with the conformal bootstrap to predict that the notorious Neel-VBS transition (QED3 with 2 scalars) is tricritical, which was recently confirmed by independent lattice simulations. Lastly, we propose a novel phase diagram for QED3 with 2 fermions, including duality with the O(4) Wilson-Fisher fixed point.

When dealing with PDEs arising in fluid mechanics, bounded-energy weaksolutions are, in many cases, the only type of solutions for which one can guarantee global existence without imposing any restrictions on the size of the initial data or forcing terms. Understanding how to construct such solutions is also crucial for designing stable numerical schemes.

In this talk, we will explain the strategy for contructing weak solutions for the Navier-Stokes system for viscous compressible flows, emphasizing the difficulties encountered in the case of non-isotropic viscous stress tensors. In particular, I will present some results obtained in collaboration with Didier Bresch and Maja Szlenk.

Given a mod $p$ Galois representation, one often wonders whether it arises by reducing a $p$-adic one, and whether these lifts are suitably 'well-behaved'. In this talk, we discuss how ideas from homotopy theory aid the study of Galois deformations, reviewing work of Galatius-Venkatesh.

Kreck’s modified surgery gives an approach to classify 2n-manifolds up to stable diffeomorphism, i.e., up to a connected sum with copies of $S^n \times  S^n$. In dimension 4, we use a combination of modified and classical surgery to compare the stable diffeomorphism classification with other stable equivalence relations. Most importantly, we consider homotopy equivalence up to connected sum with copies of $S^2 \times  S^2$. This talk is based on joint work with John Nicholson and Simona Veselá.

In this seminar, we explore the quantitative convergence of wide deep neural networks with Gaussian weights to Gaussian processes, establishing novel rates for their Gaussian approximation. We show that the Wasserstein distance between the network output and its Gaussian counterpart scales inversely with network width, with bounds apply for any finite input set under specific non-degeneracy conditions of the covariances. Additionally, we extend our analysis to the Bayesian framework, by studying exact posteriors for neural networks, when endowed with Gaussian priors and regular Likelihood functions, but we also provide recent advancements in quantitative approximation of trained networks via gradient descent in the NTK regime. Based on joint works with A. Basteri, and A. Agazzi and E. Mosig.

The accurate semantic segmentation of individual tree crowns within remotely sensed data is crucial for scientific endeavours such as forest management, biodiversity studies, and carbon sequestration quantification. However, precise segmentation remains challenging due to complexities in the forest canopy, including shadows, intricate backgrounds, scale variations, and subtle spectral differences among tree species. While deep learning models improve accuracy by learning hierarchical features, they often fail to effectively capture fine-grained details and long-range dependencies within complex forest canopies.

This seminar introduces PerceptiveNet, a novel model that incorporates a Logarithmic Gabor-implemented convolutional layer alongside a backbone designed to extract salient features while capturing extensive context and spatial information through a wider receptive field. The presentation will explore the impact of Log-Gabor, Gabor, and standard convolutional layers on semantic segmentation performance, providing a comprehensive analysis of experimental findings. An ablation study will assess the contributions of individual layers and their interactions to overall model effectiveness. Furthermore, PerceptiveNet will be evaluated as a backbone within a hybrid CNN-Transformer model, demonstrating how improved feature representation and long-range dependency modelling enhance segmentation accuracy.

In recent meetings of the journal club, two constructions that have been
discussed are Mellin transforms and chord diagrams. In my talk, I will
continue that thread and review  how a Mellin transform describes the
insertion of subgraphs into Feynman integrals. This operation comes up
in various contexts, as a concrete example, I will show how to compute
the infinite sum of rainbow diagrams in phi^3 theory in 6 dimensions. On
a combinatorial level, the procedure can be encoded by chord diagrams,
or by tubings of rooted trees, which I will mention in passing.
The talk is loosely based on doi 10.1112/jlms.70006 .
 

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.

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.

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.

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.

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.

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.