Past

Congruence Subgroup Property is a characterisation of finite-index subgroups of automorphism groups. It first arose from the study of subgroups of linear groups. In this talk, I will show a few examples where it holds and where it fails, and give an overview of what is known about the family $SL_n\mathbb{Z}$, $Out(F_n)$, $MCG(\Sigma)$. Then I will describe some related results in the case of Mapping Class Groups, and explain their relation to profinite rigidity of 3-manifolds.

For over 150 years, the study of phase transitions—such as water freezing into ice or magnets losing their magnetism—has been a cornerstone of statistical physics. In this talk, we explore the critical behavior of two-dimensional percolation models, which use random graphs to model the behavior of porous media. At the critical point, remarkable symmetries and emergent properties arise, providing precise insights into the nature of these systems and enriching our understanding of phase transitions. The presentation is designed to be accessible and does not assume any prior background in percolation theory.

About the Speaker: 

Hugo Duminil-Copin is is a French mathematician recognised for his groundbreaking work in probability theory and mathematical physics. He was appointed full professor at the University of Geneva in 2014 and since 2016 has also been a permanent professor at the Institut des Hautes Études Scientifiques (IHES) in France. In 2022 he was awarded the Fields Medal, the highest distinction in mathematics.

In this talk, we discuss the condition for the marginal distribution of the solution to singular SPDEs on the d-dimensional torus to be singular with respect to the law of the Gaussian measure induced by the linearized equation. As applications of our result, we see the singularity of the Phi^4_3-measure with respect to the Gaussian free field measure and the border of parameters for the fractional Phi^4-measure to be singular with respect to the base Gaussian measure. This talk is based on a joint work with Martin Hairer and Seiichiro Kusuoka.

It is a remarkable property of random matrices, that their resolvents tend to concentrate around a deterministic matrix as the dimension of the matrix tends to infinity, even for a small imaginary part of the involved spectral parameter.
These estimates are called local laws and they are the cornerstone in most of the recent results in random matrix theory. 
In this talk, I will present a novel method of proving single-resolvent and multi-resolvent local laws for random matrices, the Zigzag strategy, which is a recursive tandem of the characteristic flow method and a Green function comparison argument. Novel results, which we obtained via the Zigzag strategy, include the optimal Eigenstate Thermalization Hypothesis (ETH) for Wigner matrices, uniformly in the spectrum, and universality of eigenvalue statistics at cusp singularities for correlated random matrices. 
 

Based on joint works with G. Cipolloni, L. Erdös, O. Kolupaiev, and V. Riabov.

Two-dimensional chiral conformal field theories (CFTs) admit three distinct mathematical formulations: vertex operator algebras (VOAs), conformal nets, and Segal (functorial) chiral CFTs. With the broader aim to build fully extended Segal chiral CFTs, we start with the input of a conformal net. 

In this talk, we focus on presenting three equivalent constructions of the category of solitons, i.e. the category of solitonic representations of the net, which we propose is what theory (chiral CFT) assigns to a point. Solitonic representations of the net are one of the primary class of examples of bicommutant categories (a categorified analogue of a von Neumann algebras). The Drinfel’d centre of solitonic representations is the representation category of the conformal net which has been studied before, particularly in the context of rational CFTs (finite-index nets). If time permits, we will briefly outline ongoing work on bicommutant category modules (which are the structures assigned by the Segal Chiral CFT at the level of 1-manifolds), hinting towards a categorified analogue of Connes fusion of von Neumann algebra modules.

(Bicommutant categories act on W*-categories analogous to von Neumann algebras acting on Hilbert spaces)

A trace on a group is a positive-definite conjugation-invariant function on it. In the past couple of decades, the study of traces has led to exciting connections to the rigidity, stability, and dynamics of groups. In this talk, I will explain these connections and focus on the topological structure of the space of traces of some groups. We will see the different behaviours of these spaces for free groups vs. higher-rank lattices. This is based on joint works with Arie Levit, Joav Orovitz and Itamar Vigdorovich.

Valulations (of polytopes or matroids) are very useful and very mysterious. After taking some time to explain this concept, I will categorify it, with the aim of making it both more useful and less mysterious.

"The question of whether an impurity can be screened by bulk degrees of freedom is central to the study of defects and to (variations of) the Kondo problem. In this talk I discuss how symmetry, generalized or not, can give serious constraints on the possible scenarios at long distances. These can be quantified in the UV where the defect is weakly coupled. I will give some examples of interesting symmetric defect RG flows in (1+1) and (2+1)d.

Based on https://arxiv.org/pdf/2412.18652 and work in progress."

Minimal discrete energy problems arise in a variety of scientific contexts – such as crystallography, nanotechnology, information theory, and viral morphology, to name but a few.     Our goal is to analyze the structure of configurations generated by optimal (and near optimal)-point configurations that minimize the Riesz s-energy over a sphere in Euclidean space R^d and, more generally, over a bounded manifold. The Riesz s-energy potential, which is a generalization of the Coulomb potential, is simply given by 1/r^s, where r denotes the distance between pairs of points. We show how such potentials for s>d and their minimizing point configurations are ideal for use in sampling surfaces.

Connections to the results by Field's medalist M. Viazovska and her collaborators on best-packing and universal optimality in 8 and 24 dimensions will be discussed. Finally we analyze the minimization of a "k-nearest neighbor" truncated version of Riesz energy that reduces the order N^2 computation for energy minimization to order N log N , while preserving global and local properties.

The security of many widely used communication systems hinges on the presumed difficulty of factoring integers or computing discrete logarithms. However, Shor's celebrated algorithm from 1994 demonstrated that quantum computers can perform these tasks in polynomial time. In 2023, Regev proposed an even faster quantum algorithm for factoring integers. Unfortunately, the correctness of his new method is conditional on an ad hoc number-theoretic conjecture. Using tools from analytic number theory, we establish a result in the direction of Regev's conjecture. This enables us to design a provably correct quantum algorithm for factoring and solving the discrete logarithm problem, whose efficiency is comparable to Regev's approach. In this talk, we will give an accessible account of these developments.

 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.

Discrete and continuous energy problems that arise in a variety of scientific contexts are introduced, along with their fundamental existence and uniqueness results. Particular emphasis will be on Riesz and Gaussian pair potentials and their connections with best-packing and the discretization of manifolds. The latter application leads to the asymptotic theory (as N → ∞) for N-point configurations that minimize energy when the potential is hypersingular (short-range). For fixed N, the determination of such minimizing configurations on the d-dimensional unit sphere S d is especially significant in a range of contexts that include coding theory, discrete geometry, and physics. We will review linear programming methods for proving the optimality of configurations on S d , including Cohn and Kumar’s theory of universal optimality. The following reference will be made available during the short course: Discrete Energy on Rectifiable Sets, by S. Borodachov, D.P. Hardin and E.B. Saff, Springer Monographs in Mathematics, 2019.

Sessions:

Friday, 24 January 14:00-16:00

Friday, 31 January 14:00-16:00

The Junior Algebra and Representation Theory Seminar will kick-off the start of Hilary Term with a social event in the common room. Come to catch up with your fellow students and maybe play a board game or two. Afterwards we'll have lunch together.

Deep learning algorithms provide unprecedented opportunities to characterise complex structure in large data, but typically in a manner that cannot easily be interpreted beyond the 'black box'. We are developing methods to leverage the benefits of deep generative learning and computational modelling (e.g. fluid dynamics, solid mechanics, biochemistry), particularly in conjunction with biomedical imaging, to enable new insights into disease to be made. In this talk, I will describe our applications in several areas, including modelling drug delivery in cancer and retinal blood vessel loss in diabetes, and how this is leading us into the development of personalised digital twins.

I will describe the contents of a joint project with Pablo Destic and Nuno Hultberg. In the paper we confirm a conjecture of Roberto Gualdi regarding a formula for the average height of the intersection of twisted (by roots of unity) hyperplanes in a toric variety. I will introduce the 'GVF analytification' of a variety, which is defined similarly as the Berkovich analytification, but with norms replaced by heights. Moreover, I will discuss some motivations coming from (continuous) model theory and Arakelov geometry.

We present an exponential-integrator-based multi-index Monte Carlo (MIMC) method for the weak approximation of mild solutions to semilinear stochastic partial differential equations (SPDEs). Theoretical results on multi-index coupled solutions of the SPDE are provided, demonstrating their stability and the satisfaction of multiplicative error estimates. Leveraging this theory, we develop a tractable MIMC algorithm. Numerical experiments illustrate that MIMC outperforms alternative approaches, such as multilevel Monte Carlo, particularly in low-regularity settings.

Anomalies in quantum systems are present when a classical symmetry is broken by quantum effects. They give rise to physical predictions and constraints. This talk will focus on the mathematical features of anomalies of continuous, ordinary, symmetries. In the first part, we will review the topological nature of anomalies, in particular the connection to the Atiyah-Singer index theorem and its non-perturbative path-integral computation by Fujikawa. We will then discuss how anomalies and their associated (topological) Chern-Simons polynomials are related to BRST cohomology via the Stora-Zumino chain of descent equations, explaining the connection to the two-step descent procedure reviewed in the talk by Alice Lüscher last term.

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.

Active solids consume energy to allow for actuation and shape change not possible in equilibrium. I will first introduce active solids in comparison with their active fluid counterparts. I will then focus on active solids composed of non-reciprocal springs and show how so-called odd elastic moduli arise in these materials. Odd active solids have counter-intuitive elastic properties and require new design principles for optimal response. For example, in floppy lattices, zero modes couple to microscopic non-reciprocity, which destroys odd moduli entirely in a phenomenon reminiscent of rigidity percolation. Instead, an optimal odd lattice will be sufficiently soft to activate elastic deformations, but not too soft. These results provide a theoretical underpinning for recent experiments and point to the design of novel soft machines.

High-order tensor methods employing local Taylor approximations have attracted considerable attention for convex and nonconvex optimisation. The pth-order adaptive regularisation (ARp) approach builds a local model comprising a pth-order Taylor expansion and a (p+1)th-order regularisation term, delivering optimal worst-case global and local convergence rates. However, for p≥2, subproblem minimisation can yield multiple local minima, and while a global minimiser is recommended for p=2, effectively identifying a suitable local minimum for p≥3 remains elusive.
This work extends interpolation-based updating strategies, originally proposed for p=2, to cases where p≥3, allowing the regularisation parameter to adapt in response to interpolation models. Additionally, it introduces a new prerejection mechanism to discard unfavourable subproblem minimisers before function evaluations, thus reducing computational costs for p≥3.
Numerical experiments, particularly on Chebyshev-Rosenbrock problems with p=3, indicate that the proper use of different minimisers can significantly improve practical performance, offering a promising direction for designing more efficient high-order methods.

The class of henselian valued fields with non-discrete value group is not well-understood. In 2018, Koenigsmann conjectured that a list of seven natural axioms describes a complete axiomatisation of Q_p^ab, the maximal extension of the p-adic numbers Q_p with abelian Galois group, which is an example of such a valued field. Informed by the recent work of Jahnke-Kartas on the model theory of perfectoid fields, we formulate an eighth axiom (the discriminant property) that is not a consequence of the other seven. Revisiting work by Koenigsmann (the Galois characterisation of Q_p) and Jahnke-Kartas, we give a uniform treatment of their underlying method. In particular, we highlight how this method yields short, non-standard model-theoretic proofs of known results (e.g. finite extensions of perfectoid fields are perfectoid).

Donaldson proved that there are pairs of 4-manifolds that are homeomorphic but not diffeomorphic, a phenomenon that does not appear for any lower dimensional manifolds. Until recently, proving this for compact manifolds has required smooth 4-manifold invariants coming from gauge theory. In this talk, we will give an introduction to an exciting new smooth 4-manifold invariant of Morrison Walker and Wedich, called a skein lasagna module that does not rely on gauge theory. Further, this talk will not assume any knowledge of 4-manifold topology.

Paraphrasing the title of Riemann’s famous lecture of 1854 I ask: What is the most rudimentary notion of a geometry? A possible answer is a path system: Consider a finite set of “points” $x_1,…,x_n$ and provide a recipe how to walk between $x_i$ and $x_j$ for all $i\neq j$, namely decide on a path $P_{ij}$, i.e., a sequence of points that starts at $x_i$ and ends at $x_j$, where $P_{ji}$ is $P_{ij}$, in reverse order. The main property that we consider is consistency. A path system is called consistent if it is closed under taking subpaths. What do such systems look like? How to generate all of them? We still do not know. One way to generate a consistent path system is to associate a positive number $w_{ij}>0$ with every pair and let $P_{ij}$ be the corresponding $w$-shortest path between $x_i$ and $x_j$. Such a path system is called metrical. It turns out that the class of consistent path systems is way richer than the metrical ones.

My main emphasis in this lecture is on what we don’t know and wish to know, yet there is already a considerable body of work that we have done on the subject.

The new results that I will present are joint with my student Daniel Cizma as well as with him and with Maria Chudnovsky.

In subfactor theory, it has been observed that operator algebras often admit symmetries beyond mere groups, sometimes called quantum symmetries. Besides recent substantial progress on the classification programs of simple amenable C*-algebras and group actions on them, there has been increasing interest in their quantum symmetries. This talk is devoted to an attempt to ensure the existence of various quantum symmetries on simple amenable C*-algebras, at least in the purely infinite case, by providing a systematic way to produce them. As a technical ingredient, a simplicity criterion for certain Pimsner algebras is given.

Derived algebraic geometry allows us to study formal moduli problems via their tangent Lie algebras. After briefly reviewing this general paradigm, I will explain how it sheds light on deformations of Calabi-Yau varieties. 
In joint work with Taelman, we prove a mixed characteristic analogue of the Bogomolov–Tian–Todorov theorem, which asserts that Calabi-Yau varieties in characteristic $0$ are unobstructed. Moreover, we show that ordinary Calabi–Yau varieties in characteristic $p$ admit canonical (and algebraisable) lifts to characteristic $0$, generalising results of Serre-Tate for abelian varieties and Deligne-Nygaard for K3 surfaces. 
If time permits, I will conclude by discussing some intriguing questions related to our canonical lifts.  
 

I'll discuss how a certain notion of symmetry captures the computational complexity of approximating homomorphism problems between relational structures, also known as constraint satisfaction problems. I'll present recent results on inapproximability of conflict-free and linearly-ordered hypergraph colourings and solvability of systems of equations.

Let X --> S be a family of algebraic varieties parametrized by an infinitesimal disk S, possibly of mixed characteristic. The Bloch conductor conjecture expresses the difference of the Euler characteristics of the special and generic fibers in algebraic and arithmetic terms. I'll describe a proof of some new cases of this conjecture, including the case of isolated singularities. The latter was a conjecture of Deligne generalizing Milnor's formula on vanishing cycles. 

This is joint work with Massimo Pippi; our methods use derived and non-commutative algebraic geometry. 

Celestial Holography posits the existence of a holographic description of gravitational theories in asymptotically flat space-times. To date, top-down constructions of such dualities involve a combination of twisted holography and twistor theory. The gravitational theory is the closed string B model living in a suitable twistor space, while the dual is a chiral 2d gauge theory living on a stack of D1 branes wrapping a twistor line. I’ll talk about a variant of these models that yields a theory of self-dual Einstein gravity (via the Plebanski equations) in four dimensions. This is based on work in progress with Roland Bittleston, Kevin Costello & Atul Sharma.

In this talk I will give a short introduction to moderately interacting particle systems and the general notion of fluctuations around the mean-field limit. We will see how a central limit theorem can be shown for moderately interacting particles on the whole space for certain types of interaction potentials. The interaction potential approximates singular attractive potentials of sub-Coulomb type and we can show that the fluctuations become asymptotically Gaussians. The methodology is inspired by the classical work of Oelschläger in the 1980s on fluctuations for the porous-medium equation. To allow for attractive potentials we use a new approach of quantitative mean-field convergence in probability in order to include aggregation effects. 

The Farrell--Jones conjecture predicts that the algebraic K-theory of a group ring is isomorphic to a certain equivariant homology theory, and there are also versions for L-theory and Waldhausen's A-theory. In principle, this provides a way to calculate these K-groups, and has many applications. These include classifying manifolds admitting a given fundamental group and a positive resolution of the Borel conjecture.

I will discuss work with Yassine Guerch and Sam Hughes on the Farrell--Jones conjecture for extensions of relatively hyperbolic groups, as well as an application to their automorphism groups in the one-ended case. The methods are from geometric group theory: we go via the theory of JSJ decompositions to produce acylindrical actions on trees.

In this talk, we will make an explicit link between self-dual Yang-Mills instantons on the Taub-NUT space, and G2-instantons on the BGGG space, by displaying the latter space as a fibration by the former. In doing so, we will discuss analysis on non-compact manifolds, circle symmetries, and a new method of constructing solutions to quadratically singular ODE systems. This talk is based on joint work with Matt Turner: https://arxiv.org/pdf/2409.03886. 

Abelian gauge theories in 2+1 dimensions are very interesting QFTs: they are strongly coupled and exhibit non-trivial dynamics. However, they are somewhat more tractable than non-Abelian theories in 3+1 dimensions. In this talk, I will first review the known properties of fermions in 2+1 dimensions and some conjectures about QED_3 with a single Dirac fermion. I will then present the recent proposal from [arXiv:2409.17913] regarding the phase diagram of QED_3 with two fermions. The findings reveal surprising (yet compelling) features: while semiclassical analysis would suggest two trivially gapped phases and a single phase transition, the actual dynamics indicate the presence of two distinct phase transitions separated by a "quantum phase." This intermediate phase exists over a finite range of parameters in the strong coupling regime and is not visible semiclassically. Moreover, these phase transitions are second-order and exhibit symmetry enhancement. The proposal is supported by several non-trivial checks and is consistent with results from numerical bootstrap, lattice simulations, and extrapolations from the large-Nf expansion.

The geodesic cycles (resp. Eisenstein classes) for SL(2,Z) are special classes in the homology (resp. cohomology) of modular curve (for SL(2,Z)) defined by the closed geodesics (resp. Eisenstein series). It is known that the pairing between these geodesic cycles and Eisenstein classes gives the special values of partial zeta functions of real quadratic fields, and this has many applications. In this talk, I would like to report on some recent observations on the size of the homology subgroup generated by geodesic cycles and their applications. This is a joint work with Ryotaro Sakamoto.

 I will first give a short review of the concepts of Asymptotically Flat Spacetimes, IR triangle and Noether's theorems. I will then present what Asymptotic Higher Spin Symmetries are and how they were introduced as a candidate for an approximate symmetry of General Relativity and the S-matrix. Next, I'll move on to the recent developments of establishing these symmetries as Noether symmetries and describing how they are canonically and non-linearly realized on the asymptotic gravitational phase space. I will discuss how the introduction of dual equations of motion encapsulates the non-perturbativity of the analysis. Finally I'll emphasize the relation to twistor, especially with 2407.04028. Based on 2409.12178 and 2410.15219

I want to give a survey about the rational cohomology of SL_n
Z. This includes recent developments of finding Hopf algebras in the
direct sum of all cohomology groups of SL_n Z for all n. I will give a
quick overview about Hopf algebras and what this structure implies for
the cohomology of SL_n Z.

Are you thinking of launching your own start-up or considering joining an early-stage company? Navigating the entrepreneurial landscape can be both exciting and challenging. Join Pete for an interactive exploration of the unwritten rules and hidden insights that can make or break a start-up journey.

Drawing from personal experience, Pete's talk will offer practical wisdom for aspiring founders and team members, revealing the challenges and opportunities of building a new business from the ground up.

Whether you're an aspiring entrepreneur, a potential start-up team member, or simply curious about innovative businesses, you'll gain valuable perspectives on the realities of creating something from scratch.

This isn't a traditional lecture – it will be a lively conversation that invites participants to learn, share, and reflect on the world of start-ups. Come prepared to challenge your assumptions and discover practical insights that aren't found in standard business guides.
 

Speaker: Professor Pete Grindrod

Topological data analysis, and in particular persistent homology, has provided robust results for numerous applications, such as protein structure, cancer detection, and material science. In the field of neuroscience, the applications of TDA are abundant, ranging from the analysis of single cells to the analysis of neuronal networks. The topological representation of branching trees has been successfully used for a variety of classification and clustering problems of neurons and microglia, demonstrating a successful path of applications that go from the space of trees to the space of barcodes. In this talk, I will present some recent results on topological representation of brain cells, with a focus on neurons. I will also describe our solution for solving the inverse TDA problem on neurons: how can we efficiently go from persistence barcodes back to the space of neuronal trees and what can we learn in the process about these spaces. Finally, I will demonstrate how algebraic topology can be used to understand the links between single neurons and networks and start understanding the brain differences between species. The organizational principles that distinguish the human brain from other species have been a long-standing enigma in neuroscience. Human pyramidal cells form highly complex networks, demonstrated by the increased number and simplex dimension compared to mice. This is unexpected because human pyramidal cells are much sparser in the cortex. The number and size of neurons fail to account for this increased network complexity, suggesting that another morphological property is a key determinant of network connectivity. By comparing the topology of dendrites, I will show that human pyramidal cells have much higher perisomatic (basal and oblique) branching density. Therefore greater dendritic complexity, a defining attribute of human L2 and 3 neurons, may provide the human cortex with enhanced computational capacity and cognitive flexibility.

Would you like to meet some of your fellow students, and some graduate students and postdocs, in an informal and relaxed atmosphere, while building your communication skills? In this Friday@2 session, you'll be able to play a selection of board games, meet new people, and practise working together. What better way to spend the final Friday afternoon of term?! We'll play the games in the south Mezzanine area of the Andrew Wiles Building.

In "Higher Operations in Perturbation Theory", Gaiotto, Kulp, and Wu discussed Feynman integrals that control certain deformations in quantum field theory. The corresponding integrands are differential forms in Schwinger parameters. Specifically, the integrand $\alpha$ is associated to a single topological direction of the theory.
I will show how the combinatorial properties of graph polynomials lead to a relatively simple, explicit formula for $\alpha$, that can be evaluated quickly with a computer. This is interesting for two reasons. Firstly, knowing the explicit formula leads to an elementary proof of the fact that $\alpha$ squares to zero, which asserts the absence of quantum corrections in topological field theories of two (or more) dimensions, known as Kontsevich's formality theorem. Secondly, the underlying constructions and proofs are not intrinsically limited to topological theories. In this sense, they serve as a particularly instructive example for simplifications that can occur in Feynman integrals with numerators.