Past

Anomalies characterize the breaking of a classical symmetry at the quantum level. They play an important role in quantum field theories, and constitute robust observables which appear in various contexts from phenomenological particle physics to black hole microstates, or to classify phases of matter. The anomalies of a d-dimensional QFT are naturally encoded via descent equations into the so-called anomaly polynomial in (d+2)-dimensions. The aim of this seminar is to review the descent procedure, anomaly polynomial, anomaly inflow, and in particular their realisation in M-theory. While this is quite an old story, there has been some more recent developments involving holography that I'll describe if time permits. 

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 Martinet flat structure is one of the simplest sub-Riemannian manifolds that has many non-Riemannian features: it is not equiregular, it has abnormal geodesics, and the Carnot-Carathéodory sphere is not sub-analytic. I will review how the geometry of the Martinet flat structure is tied to the equations of the pendulum. Surprisingly, the Measure Contraction Property (a weak synthetic formulation of Ricci curvature bounds in non-smooth spaces) fails, and we will try to understand why. If time permits, I will also discuss how this can be generalised to some Carnot groups that have abnormal extremals. This is a joint work in progress with Luca Rizzi.

Tensor methods are methods for unconstrained continuous optimization that can incorporate derivative information of up to order p > 2 by computing a step based on the pth-order Taylor expansion at each iteration. The most important among them are regularization-based tensor methods which have been shown to have optimal worst-case iteration complexity of finding an approximate minimizer. Moreover, as one might expect, this worst-case complexity improves as p increases, highlighting the potential advantage of tensor methods. Still, the global complexity results only guarantee pessimistic sublinear rates, so it is natural to ask how local rates depend on the order of the Taylor expansion p. In the case of strongly convex functions and a fixed regularization parameter, the answer is given in a paper by Doikov and Nesterov from 2022: we get pth-order local convergence of function values and gradient norms. 
The value of the regularization parameter in their analysis depends on the Lipschitz constant of the pth derivative. Since this constant is not usually known in advance, adaptive regularization methods are more practical. We extend the local convergence results to locally strongly convex functions and fully adaptive methods. 
We discuss how for p > 2 it becomes crucial to select the "right" minimizer of the regularized local model in each iteration to ensure all iterations are eventually successful. Counterexamples show that in particular the global minimizer of the subproblem is not suitable in general. If the right minimizer is used, the pth-order local convergence is preserved, otherwise the rate stays superlinear but with an exponent arbitrarily close to one depending on the algorithm parameters.

Tension-induced giant actuation in elastic sheets

Dr. Marc Suñé

Buckling is normally associated with a compressive load applied to a slender structure; from railway tracks in extreme heat to microtubules in cytoplasm, axial compression is relieved by out-of-plane buckling. However, recent studies have demonstrated that tension applied to structured thin sheets leads to transverse motion that may be harnessed for novel applications, such as kirigami grippers, multi-stable `groovy-sheets', and elastic ribbed sheets that close into tubes. Qualitatively similar behaviour has also been observed in simulations of thermalized graphene sheets, where clamping along one edge leads to tilting in the transverse direction. I will discuss how this counter-intuitive behaviour is, in fact, generic for thin sheets that have a relatively low stretching modulus compared to the bending modulus, which allows `giant actuation' with moderate strain.

Embedding the group algebra into a division ring has proven to be a powerful tool for detecting structural properties of the group, especially in relation to its homology. In this talk, we will show how division rings can be used to identify residual properties of groups, one-ended groups, and coherent groups. We will place special emphasis on the class of free-by-cyclic groups to provide a clear, explicit exposition.

A C*-algebra is said to be residually finite-dimensional (RFD) when it has `sufficiently many' finite-dimensional representations. The RFD property is an important, and still somewhat mysterious notion, with subtle connections to residual finiteness properties of groups. In this talk I will present certain characterisations of the RFD property for C*-algebras of amenable étale groupoids and for C*-algebraic crossed products by amenable actions of discrete groups, extending (and inspired by) earlier results of Bekka, Exel, and Loring. I will also explain the role of the amenability assumption and describe several consequences of our main theorems. Finally, I will discuss some examples, notably these related to semidirect products of groups.

We study networks of firms in which inputs for production are not easily substitutable, as in several real-world supply chains. Building on Robert May's original argument for large ecosystems, we argue that such networks generically become dysfunctional when their size increases, when the heterogeneity between firms becomes too strong, or when substitutability of their production inputs is reduced. At marginal stability and for large heterogeneities, crises can be triggered by small idiosyncratic shocks, which lead to “avalanches” of defaults. This scenario would naturally explain the well-known “small shocks, large business cycles” puzzle, as anticipated long ago by Bak, Chen, Scheinkman, and Woodford. However, an out-of-equilibrium version of the model suggests that other scenarios are possible, in particular that of `turbulent economies’.

I’ll discuss joint work of mine with with Ascencio-Martin, Britnell, Duncan, Francoeurs and Koutny to set up and study algebraic models of concurrent computation. 

Trace monoids were introduced by Mazurkiewicz as algebraic models of Petri nets, where some pairs of actions can be applied in either of two orders and have the same effect. Abstractly, a trace monoid is simply a right-angled Artin monoid. More recently Koutny et al. introduced the concept of a step trace monoid, which allows the additional possibility that a pair of actions may have the same effect performed simultaneously as sequentially. 

I shall introduce these monoids, discuss some problems we’d like to be able to solve, and the methods with which we are trying to solve them. In particular I’ll discuss normal forms for traces, comtraces and step traces, and generalisations of Stallings folding techniques for finitely presented groups and monoids.

This talk is rescheduled and will take place on 21 January 2025

A zero-one matrix $M$ is said to contain another zero-one matrix $A$ if we can delete some rows and columns of $M$ and replace some 1-entries with 0-entries such that the resulting matrix is $A$. The extremal number of $A$, denoted $\operatorname{ex}(n,A)$, is the maximum number of 1-entries that an $n\times n$ zero-one matrix can have without containing $A$. The systematic study of this function for various patterns $A$ goes back to the work of Furedi and Hajnal from 1992, and the field has many connections to other areas of mathematics and theoretical computer science. The problem has been particularly extensively studied for so-called acyclic matrices, but very little is known about the general case (that is, the case where $A$ is not necessarily acyclic). We prove the first asymptotically tight general result by showing that if $A$ has at most $t$ 1-entries in every row, then $\operatorname{ex}(n,A)\leq n^{2-1/t+o(1)}$. This verifies a conjecture of Methuku and Tomon.

Our result also provides the first tight general bound for the extremal number of vertex-ordered graphs with interval chromatic number two, generalizing a celebrated result of Furedi, and Alon, Krivelevich and Sudakov about the (unordered) extremal number of bipartite graphs with maximum degree $t$ in one of the vertex classes.

Joint work with Barnabas Janzer, Van Magnan and Abhishek Methuku.

A modern perspective on symmetry in quantum theories identifies the topological invariance of a symmetry operator within correlation functions as its defining property. Within this paradigm, a framework has emerged enabling a calculus of topological defects in terms of a higher-dimensional topological quantum field theory. In this seminar, I will discuss aspects of this construction for Euclidean lattice field theories. Exploiting this framework, I will present generalisations of the celebrated Kramers-Wannier duality of the Ising model, as combinations of gauging procedures and generalised Fourier transforms of the local weights encoding the dynamics. If time permits, I will discuss implications of this framework for the real-space renormalisation group flow of these theories.

The "index of hypocoercivity" is defined via a coercivity-type estimate for the self-adjoint/skew-adjoint parts of the generator, and it quantifies `how degenerate' a hypocoercive evolution equation is, both for ODEs and for evolutions equations in a Hilbert space. We show that this index characterizes the polynomial decay of the propagator norm for short time and illustrate these concepts for the Lorentz kinetic equation on a torus. Discrete time analogues of the above systems (obtained via the mid-point rule) are contractive, but typically not strictly contractive. For this setting we introduce "hypocontractivity" and an "index of hypocontractivity" and discuss their close connection to the continuous time evolution equations.

This talk is based on joint work with F. Achleitner, E. Carlen, E. Nigsch, and V. Mehrmann.

References:
1) F. Achleitner, A. Arnold, E. Carlen, The Hypocoercivity Index for the short time behavior of linear time-invariant ODE systems, J. of Differential Equations (2023).
2) A. Arnold, B. Signorello, Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium, Kinetic and Related Models (2022).
3) F. Achleitner, A. Arnold, V. Mehrmann, E. Nigsch, Hypocoercivity in Hilbert spaces, J. of Functional Analysis (2025).
 

Heegner points are a powerful tool for understanding the structure of the group of rational points on elliptic curves. In this talk, I will describe these points and the ideas surrounding their generalisation to other situations.

June Huh proved in 2012 that the Betti numbers of the complement of a complex hyperplane arrangement form a log concave sequence.  But what if the arrangement has symmetries, and we regard the cohomology as a representation of the symmetry group?  The motivating example is the braid arrangement, where the complement is the configuration space of n points in the plane, and the symmetric group acts by permuting the points.  I will present an equivariant log concavity conjecture, and show that one can use representation stability to prove infinitely many cases of this conjecture for configuration spaces.
 

The talk will focus on recent developments regarding the (near-)critical behaviour of certain statistical physics models with long-range dependence in dimensions larger than 2, but smaller than 6, above which mean-field behaviour is known to set in. This “intermediate” regime remains a great challenge for mathematicians. The models revolve around a certain percolation phase transition that brings into play very natural probabilistic objects, such as random walk traces and the Gaussian free field. 

I will discuss recent and ongoing work with Davesh Maulik that explains how Gromov-Witten invariants behave under simple normal crossings degenerations. The main outcome of the study is that if a projective manifold $X$ undergoes a simple normal crossings degeneration, the Gromov-Witten theory of $X$ is determined, via universal formulas, by the Gromov-Witten theory of the strata of the degeneration. Although the proof proceeds via logarithmic geometry, the statement involves only traditional Gromov-Witten cycles. Indeed, one consequence is a folklore conjecture of Abramovich-Wise, that logarithmic Gromov-Witten theory “does not contain new invariants”. I will also discuss applications of this to a conjecture of Levine and Pandharipande, concerning the relationship between Gromov-Witten theory and the cohomology of the moduli space of curves.

In recent years, a new class of deep neural networks has emerged, which finds its roots at model-based iterative algorithms solving inverse problems. We call these model-based neural networks deep unfolding networks (DUNs). The term is coined due to their formulation: the iterations of optimization algorithms are “unfolded” as layers of neural networks, which solve the inverse problem at hand. Ever since their advent, DUNs have been employed for tackling assorted problems, e.g., compressed sensing (CS), denoising, super-resolution, pansharpening. 

In this talk, we will revisit the application of DUNs on the CS problem, which pertains to reconstructing data from incomplete observations. We will present recent trends regarding the broader family of DUNs for CS and dive into their theory, which mainly revolves around their generalization performance; the latter is important, because it informs us about the behaviour of a neural network on examples it has never been trained on before. 
Particularly, we will focus our interest on overparameterized DUNs, which exhibit remarkable performance in terms of reconstruction and generalization error. As supported by our theoretical and empirical findings, the generalization performance of overparameterized DUNs depends on their structural properties. Our analysis sets a solid mathematical ground for developing more stable, robust, and efficient DUNs, boosting their real-world performance.

I will discuss the preprint arXiv:2409.18130 describing an interesting connection between 4d N=2 SCFTs and 2d chiral CFTs (alias: Vertex Operator Algebras) by way of 3d EFTs.

This session is particularly aimed at fourth-year and OMMS students who are completing a dissertation this year. For many of you this will be the first time you have written such an extended piece on mathematics. The talk will include advice on planning a timetable, managing the workload, presenting mathematics, structuring the dissertation and creating a narrative, and avoiding plagiarism.

After recalling how Hecke algebras occur in the representation theory of reductive groups, we will introduce affine Hecke algebras through a combinatorial object called a root datum. Through a worked example we will construct a filtration on the affine Hecke algebra from which we obtain the graded Hecke algebra. This has a role analogous to the Lie algebra of an algebraic group.

We will discuss star operations on these rings, with a view towards the classical problem of studying unitary representations of reductive groups.