Past events

From genetic studies and astrophysics simulations to statistical modelling and AI, scientific computing has enabled amazing discoveries and there is no doubt it will continue to do so. However, the corresponding environmental impact is a growing concern in light of the urgency of the climate crisis, so what can we all do about it? Tackling this issue and making it easier for scientists to engage with sustainable computing is what motivated the Green Algorithms project. Through the prism of the GREENER principles for environmentally sustainable science, we will discuss what we learned along the way, how to estimate the impact of our work and what levers scientists and institutions have to make their research more sustainable. We will also debate what hurdles exist and what is still needed moving forward.

The Junior Algebra and Representation Theory Seminar will kick-off the start of Trinity 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.

We study supersymmetric Berry connections of 2d N = (2,2) gauged linear sigma models (GLSMs) quantized on a circle, which are periodic monopoles, with the aim to provide a fruitful physical arena for recent mathematical constructions related to the latter. These are difference modules encoding monopole solutions via a Hitchin-Kobayashi correspondence established by Mochizuki. We demonstrate how the difference modules arises naturally by studying the ground states as the cohomology of a one-parameter family of supercharges. In particular, we show how they are related to one kind of monopole spectral data, a deformation of the Cherkis–Kapustin spectral curve, and relate them to the physics of the GLSM. By considering states generated by D-branes and leveraging the difference modules, we derive novel difference equations for brane amplitudes. We then show that in the conformal limit, these degenerate into novel difference equations for hemisphere partition functions, which are exactly calculable. When the GLSM flows to a nonlinear sigma model with Kähler target X, we show that the difference modules are related to deformations of the equivariant quantum cohomology of X.

We prove that any adjoint Chevalley group over an arbitrary commutative ring is regularly bi-interpretable with this ring. The same results hold for central quotients of arbitrary Chevalley groups and for Chevalley groups with bounded generation.
Also, we show that the corresponding classes of Chevalley groups (or their central quotients) are elementarily definable and even finitely axiomatizable.

We consider the reinforcement learning problem of controlling an unknown dynamical system to maximise the long-term average reward along a single trajectory. Most of the literature considers system interactions that occur in discrete time and discrete state-action spaces. Although this standpoint is suitable for games, it is often inadequate for systems in which interactions occur at a high frequency, if not in continuous time, or those whose state spaces are large if not inherently continuous. Perhaps the only exception is the linear quadratic framework for which results exist both in discrete and continuous time. However, its ability to handle continuous states comes with the drawback of a rigid dynamic and reward structure.

        This work aims to overcome these shortcomings by modelling interaction times with a Poisson clock of frequency $\varepsilon^{-1}$ which captures arbitrary time scales from discrete ($\varepsilon=1$) to continuous time ($\varepsilon\downarrow0$). In addition, we consider a generic reward function and model the state dynamics according to a jump process with an arbitrary transition kernel on $\mathbb{R}^d$. We show that the celebrated optimism protocol applies when the sub-tasks (learning and planning) can be performed effectively. We tackle learning by extending the eluder dimension framework and propose an approximate planning method based on a diffusive limit ($\varepsilon\downarrow0$) approximation of the jump process.

        Overall, our algorithm enjoys a regret of order $\tilde{\mathcal{O}}(\sqrt{T})$ or $\tilde{\mathcal{O}}(\varepsilon^{1/2} T+\sqrt{T})$ with the approximate planning. As the frequency of interactions blows up, the approximation error $\varepsilon^{1/2} T$ vanishes, showing that $\tilde{\mathcal{O}}(\sqrt{T})$ is attainable in near-continuous time.

A conjecture of Malle predicts an asymptotic formula for the number of number fields with given Galois group and bounded discriminant. Malle conjectured the shape of the formula but not the leading constant. We present a new conjecture on the leading constant motivated by a version for algebraic stacks of Peyre's constant from Manin's conjecture. This is joint work with Tim Santens.

We introduce a new method - ESPIRA (Estimation of Signal Parameters via Iterative Rational Approximation) \cite{DP22,  DPP21} - for the recovery of complex exponential  sums
$$
f(t)=\sum_{j=1}^{M} \gamma_j \mathrm{e}^{\lambda_j t},
$$
that are determined by a finite number of parameters: the order $M$, weights $\gamma_j \in \mathbb{C} \setminus \{0\}$ and nodes  $\mathrm{e}^{\lambda_j} \in \mathbb{C}$ for $j=1,...,M$.  Our new recovery procedure is based on the observation that Fourier coefficients (or DFT coefficients) of exponential sums have a special rational structure.  To  reconstruct this structure in a stable way we use the AAA algorithm  proposed by Nakatsukasa et al.   We show that ESPIRA can be interpreted as a matrix pencil method applied to Loewner matrices. 

During the talk we will demonstrate that ESPIRA outperforms Prony-like methods such as ESPRIT and MPM for noisy data and for signal approximation by short exponential sums.  

Bibliography
N. Derevianko,  G.  Plonka, 
Exact reconstruction of extended exponential sums using rational approximation of their Fourier coefficients, Anal.  Appl.,  20(3),  2022,  543-577.

N. Derevianko,  G. Plonka,  M. Petz, 
From ESPRIT to ESPIRA: Estimation of signal parameters by iterative rational approximation,   IMA J. Numer. Anal.,  43(2),  2023, 789--827.  

Y. Nakatsukasa, O. Sète,   L.N. Trefethen,  The AAA algorithm for rational approximation.
SIAM J. Sci. Comput., 40(3),   2018,  A1494–A1522.  

We develop a comprehensive geometrically-exact theory for an end-loaded elastic rod constrained to deform on a cylindrical surface. By viewing the rod-cylinder system as a special case of an elastic braid, we are able to obtain all forces and moments imparted by the deforming rod to the cylinder as well as all contact reactions. This framework allows us to give a complete treatment of static friction consistent with force and moment balance. In addition to the commonly considered model of hard frictionless contact, we analyse two friction models in which the rod, possibly with intrinsic curvature, experiences either lateral or tangential friction. As applications of the theory we study buckling of the constrained rod under compressive and torsional loads, finding critical loads to depend on Coulomb-like friction parameters, as well as the tendency of the rod to lift off the cylinder under further loading. The cylinder can also have arbitrary orientation relative to the direction of gravity. The cases of a horizontal and vertical cylinder, with gravity having only a lateral or axial component, are amenable to exact analysis, while numerical results map out the transition in buckling mechanism between the two extremes. Weight has a stabilising effect for near-horizontal cylinders, while for near-vertical cylinders it introduces the possibility of buckling purely due to self-weight. Our results are relevant for many engineering and medical applications in which a slender structure winds inside or outside a cylindrical boundary.

We will decide on speakers for Trinity term 2024.

It is a standard result that mapping class groups of high genus do not surject the integers. This is easily shown by computing the abelianization of the mapping class group using a presentation. Once we pass to finite index subgroups, this becomes a conjecture of Ivanov. More generally, we can ask which groups admit epimorphisms from finite index subgroups of the mapping class group. In this talk, I will present a geometric approach to this question, using harmonic maps, and explain some recent results.

Product systems represent powerful contemporary tools in the study of mathematical structures. A major success in the theory came from Katsura (2007), who provided a complete description of the gauge-invariant ideals of many important C*-algebras arising from product systems over Z+. This result recaptures existing results from the literature, illustrating the versatility of product system theory. The question now becomes whether or not Katsura's result can be bolstered to product systems over semigroups other than Z+ and, if so, what applications do we obtain? An answer has been elusive, owing to the more pathological nature of product systems over general semigroups. However, recent strides by Dor-On and Kakariadis (2018) supply a more tractable subclass of product systems that still includes the important cases of C*-dynamics, row-finite higher-rank graphs, and regular product systems. 

In this talk we will build a parametrisation of the gauge-invariant ideals, starting from first principles and gradually increasing in complexity. We will pay particular attention to the higher-rank subtleties that are not witnessed in Katsura's theorem, and comment on the applications.
 

Approximate lattices are aperiodic generalisations of lattices in locally compact groups. They were first introduced in abelian groups by Yves Meyer before being studied as mathematical models for quasi-crystals. Since then their structure has been thoroughly investigated in both abelian and non-abelian settings.

In this talk I will survey what is known of the structure of approximate lattices. I will highlight some objects - such as a notion of cohomology sitting between group cohomology and bounded cohomology - that appear in their study. I will also formulate open problems and conjectures related to approximate lattices. 

The main component of flow machines is the rotor; however, there may also be stationary parts surrounding the rotor, which are the diffuser blades. In order to consider these two parts simultaneously, the most intuitive approach is to perform a transient flow simulation; however, the computational cost is relatively high. Therefore, one possible approach is the Multiple Reference Frame (MRF) approach, which considers two directly coupled zones: one for the rotating reference frame (for the rotor blades) and one for the stationary reference frame (for the diffuser blades). When taking into account topology optimisation, some changes are required in order to take both rotating and stationary parts simultaneously in the design, which also leads to changes in the composition of the multi-objective function. Therefore, the topology optimisation method is formulated for MRF while also proposing this new multi-objective function. An integer variable-based optimisation algorithm is considered, with some adjustments for the MRF case. Some numerical examples are presented.

We show that there exists $C>1$, such that if $A$ is a subset of a non-alternating finite simple group $G$ of density $|A|/|G|= \alpha$, then $AA^{-1}AA^{-1}$ contains a subgroup of density at least $\alpha^{C}$. We will also give a corresponding (slightly weaker) statement for alternating groups.

To prove our results we introduce new hypercontractive inequalities for simple groups. These allow us to show that the (non-abelian) Fourier spectrum of indicators of 'global' sets are concentrated on the high-dimensional irreducible representations. Here globalness is a pseudorandomness notion reminiscent of the notion of spreadness.

The talk is based on joint works with David Ellis, Shai Evra, Guy Kindler, Nathan Lindzey, and Peter Keevash, and Dor Minzer. No prior knowledge of representation theory will be assumed.

Our research explores the application of reinforcement learning (RL) strategies to solve complex combinatorial research problems, specifically the Job-shop Scheduling Problem (JSP) and the Stochastic Vehicle Routing Problem with Time Windows (SVRP). For JSP, we utilize Curriculum Learning (CL) to enhance the performance of dispatching policies. This approach addresses the significant optimality gap in existing end-to-end solutions by structuring the training process into a sequence of increasingly complex tasks, thus facilitating the handling of larger, more intricate instances. Our study introduces a size-agnostic model and a novel strategy, the Reinforced Adaptive Staircase Curriculum Learning (RASCL), which dynamically adjusts difficulty levels during training, focusing on the most challenging instances. Experimental results on Taillard and Demirkol datasets show that our approach reduces the average optimality gap to 10.46% and 18.85%, respectively.

For SVRP, we propose an end-to-end framework employing an attention-based neural network trained through RL to minimize routing costs while addressing uncertain travel costs and demands, alongside specific customer delivery time windows. This model outperforms the state-of-the-art Ant-Colony Optimization algorithm by achieving a 1.73% reduction in travel costs and demonstrates robustness across diverse environmental settings, making it a valuable baseline for future research. Both studies mark advancements in the application of machine learning techniques to operational research.

We will use Representation Theory to calculate systematically and efficiently the topological invariants of compact Lie groups and homogeneous spaces.
 

Most of the talk is covered by our second paper on ArXiv with John Jones and Adam Thomas, who are both at Warwick. The paper is part of the ongoing project to study the topological invariants of the four exceptional Rosenfeld projective planes.

The curvature-dimension condition, CD(K,N) for short, and the (weaker) measure contraction property, or MCP(K,N), are two synthetic notions for a metric measure space to have Ricci curvature bounded from below by K and dimension bounded from above by N. In this talk, we investigate the validity of these conditions in sub-Finsler geometry, which is a wide generalization of Finsler and sub-Riemannian geometry. Firstly, we show that sub-Finsler manifolds equipped with a smooth strongly convex norm and with a positive smooth measure can not satisfy the CD(K,N) condition for any K and N. Secondly, we focus on the sub-Finsler Heisenberg group, where we show that, on the one hand, the CD(K,N) condition can not hold for any reference norm and, on the other hand, the MCP(K,N) may hold or fail depending on the regularity of the reference norm. 

In this talk, we will study the problem in additive combinatorics of determining for a finite Abelian group $G$ the size of its smallest subset $A\subset G$ that has no unique sum, meaning that for every two $a_1,a_2\in A$ we can write $a_1+a_2=a’_1+a’_2$ for different $a’_1,a’_2\in A$. We begin by using classical rectification methods to obtain the previous best lower bounds of the form $|A|\gg \log p(G)$, which stood for 50 years. Our main aim is to outline the proof of a recent improvement and discuss some of its key notions such as additive dimension and the density increment method. This talk is based on Bedert, B. On Unique Sums in Abelian Groups. Combinatorica (2023).

I will discuss three different constructions of smooth tori in S^4 whose complements have fundamental group Z: turned 1-twist-spun tori due to Boyle, the union of a ribbon disc with a genus one Seifert surface constructed by Cochran and Davis, and certain tori with four critical points. They are all topologically unknotted, but it is not known whether they are smoothly standard, except for tori with four critical points whose middle level set is a split link. The branched double cover of S^4 along any of these surfaces is a potentially exotic copy of S^2 x S^2, though, in the case of Boyle's example, it cannot be distinguished from the standard S^2 x S^2 using Seiberg-Witten invariants. This is joint work with Mark Powell.

We will introduce the Quintic Ornstein-Uhlenbeck model that jointly calibrates SPX-VIX options with a particular focus on its mathematical tractability namely for fast pricing SPX options using Fourier techniques. Then, we will consider the more general class of  stochastic volatility models where the dynamics of the volatility are given by a possibly infinite linear combination of the elements of the time extended signature of a Brownian motion. First, we show that the model is remarkably universal, as it includes, but is not limited to, the celebrated Stein-Stein, Bergomi, and Heston models, together with some path-dependent variants. Second, we derive the joint characteristic functional of the log-price and integrated variance provided that some infinite-dimensional extended tensor algebra valued Riccati equation admits a solution. This allows us to price and (quadratically) hedge certain European and path-dependent options using Fourier inversion techniques. We highlight the efficiency and accuracy of these Fourier techniques in a comprehensive numerical study.