Past

The tangle hypothesis is a variant of the cobordism hypothesis that considers cobordisms embedded in some finite-dimensional Euclidean space (together with framings). Such tangles of codimension d can be organized into an E_d-monoidal n-category, where n is the maximal dimension of the tangles. The tangle hypothesis then asserts that this category of tangles is the free E_d-monoidal n-category with duals generated by a single object.

In this talk, based on joint work in progress with Yonatan Harpaz, I will describe an infinitesimal version of the tangle hypothesis: Instead of showing that the E_d-monoidal category of tangles is freely generated by an object, we show that its cotangent complex is free of rank 1. This provides supporting evidence for the tangle hypothesis, but can also be used to reduce the tangle hypothesis to a statement at the level of E_d-monoidal (n+1, n)-categories by means of obstruction theory.

In this talk I will present several new stochastic representations for
solutions of the Navier-Stokes equations in a wall-bounded region,
in the spirit of mean field theory. These new representations are
obtained by using the duality of conditional laws associated with the Taylor diffusion family.
By using these representation, Monte-Carlo simulations for boundary fluid flows, including
boundary turbulence, may be implemented. Numerical experiments are given to demonstrate the usefulness of this approach.

According to the Yau-Tian-Donaldson conjecture, the existence of a constant scalar curvature Kähler (cscK) metric in the cohomology class of an ample line bundle $L$ on a compact complex manifold $X$ should be equivalent to an algebro-geometric "stability condition" satisfied by the pair $(X,L)$. The cscK metrics are the critical points of Mabuchi's $K$-energy functional $M$, defined on the space of Kähler potentials, and an important result of Chen-Cheng shows that cscK metrics exist iff $M$ satisfies a standard growth condition (coercivity/properness). Recently the speaker has shown that the $K$-energy is indeed proper if and only if the polarized manifold is stable. The stability condition is closely related to the classical notion of Hilbert-Mumford stability. The speaker will give a non-technical account of the many areas of mathematics that are involved in the proof. In particular, he hopes to discuss the surprising role played by arithmetic geometry ​in the spirit of Arakelov, Faltings, and Bismut-Gillet-Soule.

I will describe algorithms for regularizing and training deep neural networks. Soft constraints, which add a penalty term to the loss, are typically used as a form ofexplicit regularization for neural network training. In this talk I describe a method for efficiently incorporating constraints into a stochastic gradient Langevin framework for the training of deep neural networks. In contrast to soft constraints, our constraints offer direct control of the parameter space, which allows us to study their effect on generalization. In the second part of the talk, I illustrate the presence of latent multiple time scales in deep learning applications.

Different features present in the data can be learned by training a neural network on different time scales simultaneously. By choosing appropriate partitionings of the network parameters into fast and slow parts I show that our multirate techniques can be used to train deep neural networks for transfer learning applications in vision and natural language processing in half the time, without reducing the generalization performance of the model.

When do two quantum field theories describe the same physics? I will discuss some approaches to this question in the context of superconformal field theories in four and six dimensions. First, I will discuss the construction of 6d (1,0) SCFTs from the perspective of the "atomic classification", focussing on an oft-overlooked subtlety whereby distinct SCFTs in fact share an effective description on the generic point of the tensor branch. We will see how to determine the difference in the Higgs branch operator spectrum from the atomic perspective, and how that agrees with a dual class S perspective. I will explain how other 4d N=2 SCFTs, which a priori look like distinct theories, can be shown to describe the same physics, as they arise as torus-compactifications of identical 6d theories.

I will discuss a line of work on estimating causal effects from observational data. In the first part of the talk, I will discuss identification and estimation of causal effects when the underlying causal graph is known, using adjustment. In the second part, I will discuss what one can do when the causal graph is unknown. Throughout, examples will be used to illustrate the concepts and no background in causality is assumed.

Mathematical models of childhood diseases are often fitted using deterministic methods under the assumption of homogeneous contact rates within populations. Such models can provide good agreement with data in the absence of significant changes in population demography or transmission, such as in the case of pre-vaccine era measles. However, accurate modeling and forecasting after the start of mass vaccination has proved more challenging. This is true even in the case of measles which has a well understood natural history and a very effective vaccine. We demonstrate how the dynamics of homogeneous and age-structured models can be similar in the absence of vaccination, but diverge after vaccine roll-out. We also present some fundamental differences in deterministic and stochastic methods to fit models to data, and propose techniques to fit long term time series with imperfect covariate information. The methods we develop can be applied to many types of complex systems beyond those in disease ecology.
 

I'll briefly explain quantum groups and $R$-matrices and why they're the same thing. Then we'll see how to construct various $R$-matrices from Nakajima quiver varieties and some possible applications.

The equivalence of NSOP${}_1$ and NSOP${}_3$, two model-theoretic complexity properties, remains open, and both the classes NSOP${}_1$ and NSOP${}_3$ are more complex than even the simple unstable theories. And yet, it turns out that classical geometric stability theory, in particular the group configuration theorem of Hrushovski (1992), is capable of controlling classification theory on either side of the NSOP${}_1$-SOP${}_3$ dichotomy, via the expansion of stable theories by generic predicates and equivalence relations. This allows us to construct new examples of strictly NSOP${}_1$ theories. We introduce generic expansions corresponding, though universal axioms, to definable relations in the underlying theory, and discuss the existence of model companions for some of these expansions. In the case where the defining relation in the underlying theory $T$ is a ternary relation $R(x, y, z)$ coming from a surface in 3-space, we give a surprising application of the group configuration theorem to classifying the corresponding generic expansion $T^R$. Namely, when $T$ is weakly minimal and eliminates the quantifier $\exists^{\infty}$, $T^R$ is strictly NSOP${}_4$ and TP${}_2$ exactly when $R$ comes from the graph of a type-definable group operation; otherwise, depending on whether the expansion is by a generic predicate or a generic equivalence relation, it is simple or NSOP${}_1$.

Wiles' modularity lifting theorem was the central argument in his proof of modularity of (semistable) elliptic curves over Q, and hence of Fermat's Last Theorem. His proof relied on two key components: his "patching" argument (developed in collaboration with Taylor) and his numerical isomorphism criterion.

In the time since Wiles' proof, the patching argument has been generalized extensively to prove a wide variety of modularity lifting results. In particular Calegari and Geraghty have found a way to generalize it to prove potential modularity of elliptic curves over imaginary quadratic fields (contingent on some standard conjectures). The numerical criterion on the other hand has proved far more difficult to generalize, although in situations where it can be used it can prove stronger results than what can be proven purely via patching.

In this talk I will present joint work with Srikanth Iyengar and Chandrashekhar Khare which proves a generalization of the numerical criterion to the context considered by Calegari and Geraghty (and contingent on the same conjectures). This allows us to prove integral "R=T" theorems at non-minimal levels over imaginary quadratic fields, which are inaccessible by Calegari and Geraghty's method. The results provide new evidence in favor of a torsion analog of the classical Langlands correspondence.

We study the impact of transition scenario uncertainty, and in particular, the uncertainty about future carbon price and electricity demand, on the pace of decarbonization of the electricity industry. To this end, we build a discrete time mean-field game model for the long-term dynamics of the electricity market subject to common random shocks affecting the carbon price and the electricity demand. These shocks depend on a macroeconomic scenario, which is not observed by the agents, but can be partially deduced from the frequency of the shocks. Due to this partial observation feature, the common noise is non-Markovian. We consider two classes of agents: conventional producers and renewable producers. The former choose an optimal moment to exit the market and the latter choose an optimal moment to enter the market by investing into renewable generation. The agents interact through the market price determined by a merit order mechanism with an exogenous stochastic demand. We prove the existence of Nash equilibria in the resulting mean-field game of optimal stopping with common noise, developing a novel linear programming approach for these problems. We illustrate our model by an example inspired by the UK electricity market, and show that scenario uncertainty leads to significant changes in the speed of replacement of conventional generators by renewable production.

We give an introduction to the subject of higher geometry, by giving many examples of higher geometric objects, and looking at their properties. These include examples of 2-rings, 2-vector spaces, and 2-vector bundles. We show how these concepts help solve problems in ordinary geometry, as one of the many motivations of the subject. We assume no prerequisites on the subject, and the talk should be applicable to both differential and algebraic geometry.

Whilst we all enjoy travelling to exciting and far-off locations, the current climate crisis is making flights less and less attractive. But is there anything we can do about this? By plotting courses that make best use of atmospheric data to minimise aircraft fuel burn, airlines can not only save money on fuel, but also reduce emissions, whilst not significantly increasing flight times. In each case the route between London Heathrow Airport and John F Kennedy Airport in New York is considered.  Atmospheric data is taken from a re-analysis dataset based on daily averages from 1st December, 2019 to 29th February, 2020.

Initially Pontryagin’s minimum principle is used to find time minimal routes between the airports and these are compared with flight times along the organised track structure routes prepared by the air navigation service providers NATS and NAV CANADA for each day.  Efficiency of tracks is measured using air distance, revealing that potential savings of between 0.7% and 16.4% can be made depending on the track flown. This amounts to a reduction of 6.7 million kg of CO2 across the whole winter period considered.

In a second formulation, fixed time flights are considered, thus reducing landing delays.  Here a direct method involving a reduced gradient approach is applied to find fuel minimal flight routes either by controlling just heading angle or both heading angle and airspeed. By comparing fuel burn for each of these scenarios, the importance of airspeed in the control formulation is established.  

Finally dynamic programming is applied to the problem to minimise fuel use and the resulting flight routes are compared with those actually flown by 9 different models of aircraft during the winter of 2019 to 2020. Results show that savings of 4.6% can be made flying east and 3.9% flying west, amounting to 16.6 million kg of CO2 savings in total.

Thus large reductions in fuel consumption and emissions are possible immediately, by planning time or fuel minimal trajectories, without waiting decades for incremental improvements in fuel-efficiency through technological advances.
 

Biological systems achieve their complexity by processing and exploiting information stored in molecular copolymers such as DNA, RNA and proteins. Despite the ubiquity and power of this approach in natural systems, our ability to implement similar functionality in synthetic systems is very limited. In this talk, we will first outline a new mathematical framework for analysing general models of colymerisation for infinitely long polymers. For a given model of copolymerisation, this approach allows for the extraction of key quantities such as the sequence distribution, speed of polymerisation and the rate of molecular fuel consumption without resorting to simulation. Subsequently, we will explore mechanisms that allow for reliable copying of the information stored in finite-length template copolymers, before touching on recent experimental work in which these ideas are put into practice.  

Many classes of finitely generated groups have been studied using formal language theory techniques. One historical example is the study of geodesics, which gives rise to the strict growth series of a group. Properties of languages associated to groups can provide insight into the nature of the growth series.

In this talk we will introduce languages associated to conjugacy classes, rather than elements of the group. This will lead us to define an analogous series, namely the conjugacy growth series of a group, which has become a popular topic in recent years. After discussing the necessary group theoretic and language tools needed, we will focus on how these conjugacy languages behave in graph products. We will finish with some new results which look at when these properties can extend to virtual graph products.

We will be joined for lunch by the undergraduate society for women and non-binary students in Maths. Sign up here: https://forms.gle/q3uywCN3Dn78Kvkq6. Note: this event is only open to women and non-binary students.

This is a continuation of https://www.maths.ox.ac.uk/node/61240

An example of a uniformly proper action is the action of a group (or any of its subgroups) on its Cayley graph. A natural question appearing in a paper of Coulon and Osin, is whether the class of groups acting uniformly properly on hyperbolic spaces coincides with the class of subgroups of hyperbolic groups. In joint work with Vladimir Vankov we construct an uncountable family of finitely generated groups which act uniformly properly on hyperbolic spaces. This gives the first examples of finitely generated groups acting uniformly properly on hyperbolic spaces that are not subgroups of hyperbolic groups. We also give examples that are not virtually torsion-free.

We present two ways of obtaining hypercontractive inequalities for low-degree functions on compact Lie groups: one based on Ricci curvature bounds, the Bakry-Emery criterion and the representation theory of compact Lie groups, and another based on a (very different) probabilistic coupling approach. As applications we make progress on a question of Gowers concerning product-free subsets of the special unitary groups, and we also obtain 'mixing' inequalities for the special unitary groups, the special orthogonal groups, the spin groups and the compact symplectic groups. We expect that the latter inequalities will have applications in physics.

Based on joint work with Guy Kindler (HUJI), Noam Lifshitz (HUJI) and Dor Minzer (MIT).

The local Langlands conjectures connect representations of p-adic groups to certain representations of Galois groups of local fields called Langlands parameters.  Recently, there has been a shift towards studying representations over more general coefficient rings and towards certain categorical enhancements of the original conjectures.  In this talk, we will focus on representations over coefficient rings with p invertible and how the corresponding category of representations of the p-adic group decomposes.  

We use entropy methods to show that the heat equation with Dirichlet boundary conditions on the complement of a compact set in R^d shows a self-similar behaviour much like the usual heat equation on R^d, once we account for the loss of mass due to the boundary. Giving good lower bounds for the fundamental solution on these sets is surprisingly a relatively recent result, and we find some improvements using some advances in logarithmic Sobolev inequalities. In particular, we are able to give optimal asymptotic bounds for large times for the fundamental solution with an explicit approach rate in dimensions larger than 2, and some new bounds in dimension 2.

This is a work in collaboration with Alejandro Gárriz and Fernando Quirós.

In this talk we will study the graph structure of supersingular isogeny graphs. These graphs are known to have very few loops and multi-edges. We formalize this idea by studying and finding bounds for their number of loops and multi-edges. We also find conditions under which these graphs are simple. To do so, we introduce a method of counting the total number of collisions (which are special endomorphisms) based on a trace formula of Gross and a known formula of Kronecker, Gierster and Hurwitz. 

The method presented in this talk can be used to study many kinds of collisions in supersingular isogeny graphs. As an application, we will see how this method was used to estimate a certain number of collisions and then show that isogeny graphs do not satisfy a certain cryptographic property that was falsely believed (and proven!) to hold.

I shall begin with a brief history of the problem of trying to understand infinite groups knowing only their finite quotients. I'll then focus on 3-manifold groups, describing the prominent role that they have played in advancing our understanding of this problem in recent years. The story for 3-manifold groups involves a rich interplay of algebra, geometry, and arithmetic. I shall describe arithmetic Kleinian groups that are profinitely rigid in the absolute sense -- ie they can be distinguished from all other finitely generated, residually finite groups by their set of finite quotients. I shall then explain more recent work involving products of Seifert fibered manifolds -- here we find groups that are profinitely rigid in the class of finitely presented groups but not in the class of finitely generated groups. This is joint work with McReynolds, Reid, and Spitler.