Past

Although tensor products and their symmetrisation have appeared in mathematical literature since at least the mid-nineteenth century, they rarely appear in the function-theoretic operator theory literature. In this talk, I will introduce the symmetric and antisymmetric tensor products from an operator theoretic point of view. I will present results concerning some of the most fundamental operator-theoretic questions in this area, such as finding the norm and spectrum of the symmetric tensor products of operators. I will then work through some examples of symmetric tensor products of familiar operators, such as the unilateral shift, the adjoint of the shift, and diagonal operators.

The talk is based on joint work with Uri Bader. We prove that arithmetic lattices in a semisimple Lie group G satisfy a higher-degree version of property T below the rank of G. The proof relies on functional analysis and the polynomiality of higher Dehn functions of arithmetic lattices below the rank and avoids any automorphic machinery. If time permits, we describe applications to the cohomology and stability of arithmetic groups (the latter being joint work with Alex Lubotzky and Shmuel Weinberger).

Let $p \ge 3$ be a prime integer. The density of a non-empty solution set of a system of affine equations $L_i(x) = b_i$, $i=1,\dots,k$ on a vector space over the field $\mathbb{F}_p$ is determined by the dimension of the linear subspace $\langle L_1,\dots,L_k \rangle$, and tends to $0$ with the dimension of that subspace. In particular, if the solution set is dense, then the system of equations contains at most boundedly many pairwise distinct linear forms. In the more general setting of systems of affine conditions $L_i(x) \in E_i$ for some strict subsets $E_i$ of $\mathbb{F}_p$ and where the solution set and its density are taken inside $S^n$ for some non-empty subset $S$ of $\mathbb{F}_p$ (such as $\{0,1\}$), we can however usually find subsets of $S^n$ with density at least $1/2$ but such that the linear subspace $\langle L_1,\dots,L_k \rangle$ has arbitrarily high dimension. We shall nonetheless establish an approximate boundedness result: if the solution set of a system of affine conditions is dense, then it contains the solution set of a system of boundedly many affine conditions and which has approximately the same density as the original solution set. Using a recent generalisation by Gowers and the speaker of a result of Green and Tao on the equidistribution of high-rank polynomials on finite prime fields we shall furthermore prove a weaker analogous result for polynomials of small degree.

Based on joint work with Timothy Gowers (College de France and University of Cambridge).

***CANCELLED*** In this talk, I will discuss recent results related to the mathematical justification of PDEs which model multi-phase flows at the macroscopic level from mesoscopic descriptions with jump conditions at interfaces. We will also present interesting and difficult open problems.

The notion of a pseudocharacter was introduced by A.Wiles for GL_2 and generalized by R.Taylor to GL_n. It is a tool that allows us to deal with the
deformation theory of a residually reducible Galois representation when the usual techniques fail. G.Chenevier gave an alternative theory of "determinants" extending that of pseudocharacters to arbitrary rings. In this talk we will discuss some aspects of this theory and introduce a similar definition in the case of the symplectic group, which is the subject of a forthcoming work joint with J.Quast. 

I will describe the proof of the following classification result for manifolds with positive scalar curvature. Let M be a closed manifold of dimension $n=4$ or $5$ that is "sufficiently connected", i.e. its second fundamental group is trivial (if $n=4$) or second and third fundamental groups are trivial (if $n=5$). Then a finite covering of $M$ is homotopy equivalent to a sphere or a connect sum of $S^{n-1} \times S^1$. The proof uses techniques from minimal surfaces, metric geometry, geometric group theory. This is a joint work with Otis Chodosh and Chao Li.
 

Using rough path techniques, we provide a priori estimates for the output of Deep Residual Neural Networks in terms of both the input data and the (trained) network weights. As trained network weights are typically very rough when seen as functions of the layer, we propose to derive stability bounds in terms of the total p-variation of trained weights for any p∈[1,3]. Unlike the C1-theory underlying the neural ODE literature, our estimates remain bounded even in the limiting case of weights behaving like Brownian motions, as suggested in [Cohen-Cont-Rossier-Xu, "Scaling Properties of Deep Residual Networks”, 2021]. Mathematically, we interpret residual neural network as solutions to (rough) difference equations, and analyse them based on recent results of discrete time signatures and rough path theory. Based on joint work with C. Bayer and P. K. Friz.
 

Glueing construction has been used extensively to construct solutions to nonlinear geometric PDEs. In this talk, I will focus on the glueing construction of solutions to Lagrangian mean curvature flow. Specifically, I will explain the construction of Lagrangian translating solitons by glueing a small special Lagrangian 'Lawlor neck' into the intersection point of suitably rotated Lagrangian Grim Reaper cylinders. I will also discuss an ongoing joint project with Chung-Jun Tsai and Albert Wood, where we investigate the construction of solutions to Lagrangian mean curvature flow with infinite time singularities.

In her recent work, Mina Aganagic proposed novel perspectives on computing knot homologies associated with any simple Lie algebra. One of her proposals relies on counting intersection points between Lagrangians in Landau-Ginsburg models on symmetric powers of Riemann surfaces. In my talk, I am going to present a concrete algebraic algorithm for finding such intersection points, turning the proposal into an actual calculational tool. I am going to illustrate the construction on the example of the sl_2 invariant for the Hopf link. I am also going to comment on the extension of the story to homological invariants associated to gl(m|n) super Lie algebras, solving this long-standing problem. The talk is based on our work in progress with Mina Aganagic and Elise LePage.

What is curiosity? Is it an emotion? A behavior? A cognitive process? Curiosity seems to be an abstract concept—like love, perhaps, or justice—far from the realm of those bits of nature that mathematics can possibly address. However, contrary to intuition, it turns out that the leading theories of curiosity are surprisingly amenable to formalization in the mathematics of network science. In this talk, I will unpack some of those theories, and show how they can be formalized in the mathematics of networks. Then, I will describe relevant data from human behavior and linguistic corpora, and ask which theories that data supports. Throughout, I will make a case for the position that individual and collective curiosity are both network building processes, providing a connective counterpoint to the common acquisitional account of curiosity in humans.

What is curiosity? Is it an emotion? A behavior? A cognitive process? Curiosity seems to be an abstract concept—like love, perhaps, or justice—far from the realm of those bits of nature that mathematics can possibly address. However, contrary to intuition, it turns out that the leading theories of curiosity are surprisingly amenable to formalization in the mathematics of network science. In this talk, I will unpack some of those theories, and show how they can be formalized in the mathematics of networks. Then, I will describe relevant data from human behavior and linguistic corpora, and ask which theories that data supports. Throughout, I will make a case for the position that individual and collective curiosity are both network building processes, providing a connective counterpoint to the common acquisitional account of curiosity in humans.

My talk will attempt to capture the imperfect state of the art in high-resolution ice sheet modelling, aiming to expose the core performance-limiting issues.  The essential equations for modeling ice flow in a changing climate will be presented, assuming no prior knowledge of the problem.  These geophysical/climate problems are of both free-boundary and algebraic-equation-constrained character.  Current-technology models usually solve non-linear Stokes equations, or approximations thereof, at every explicit time-step.  Scale analysis shows why this current design paradigm is expensive, but building significantly faster high-resolution ice sheet models requires new techniques.  I'll survey some recently-arrived tools, some near-term improvements, and sketch some new ideas.

The dynamics of a tissue in development or regeneration arises from the behaviour of its constituent cells and their interactions. We use mathematical models and inference from experimental data to to infer the likely cellular behaviours underlying changing tissue states. In this talk I will show examples of how we apply canonical birth-death process models to novel experimental data, how we are extending such models with volume exclusion and multistate dynamics, and how we attempt to more generally learn cell-cell interaction models directly from data in interpretable ways. The applications range from in vitro models of embryo development to in vivo blood regeneration that is disrupted with ageing.

Let $G$ be a $p$-adic Lie group. From the perspective of $p$-adic manifolds, possibly the most natural $p$-adic representations of $G$ to consider are the locally analytic ones.  Motivated by work of Pan, when $G$ acts on a rigid analytic variety $X$ (e.g., the flag variety), we would like to geometrise locally analytic $G$-representations, via a covariant localisation theory which should intertwine Schneider-Teitelbaum's duality with the $p$-adic Beilinson-Bernstein localisation. I will report some partial progress in the simplified situation when we replace $G$ by its germ at $1$. The main ingredient is an infinite jet bundle $\mathcal{J}^\omega_X$ which is dual to $\widehat{\mathcal{D}}_X$. Our "co"localisation functor is given by a coinduction to $\mathcal{J}^\omega_X$. Work in progress.

I will present results on short sums of arithmetic functions (in particular the von Mangoldt and divisor functions) twisted by polynomial exponential phases or more general nilsequence phases. These results imply the Gowers uniformity of suitably normalised versions of these functions in intervals of length X^c around X for suitable values of c (depending on the function and on whether one considers all or almost all short sums). I will also discuss an application to an averaged form of the Hardy-Littlewood conjecture. This is based on joint works with Kaisa Matomäki, Maksym Radziwiłł, Xuancheng Shao and Terence Tao.

We conduct a systematic large-scale analysis of order book-driven predictability in high-frequency returns by leveraging deep learning techniques. First, we introduce a new and robust representation of the order book, the volume representation. Next, we carry out an extensive empirical experiment to address various questions regarding predictability. We investigate if and how far ahead there is predictability, the importance of a robust data representation, the advantages of multi-horizon modeling, and the presence of universal trading patterns. We use model confidence sets, which provide a formalized statistical inference framework particularly well suited to answer these questions. Our findings show that at high frequencies predictability in mid-price returns is not just present, but ubiquitous. The performance of the deep learning models is strongly dependent on the choice of order book representation, and in this respect, the volume representation appears to have multiple practical advantages.

In work of Haiden-Katzarkov-Konsevich-Pandit (HKKP), a canonical filtration, labeled by sequences of real numbers, of a semistable quiver representation or vector bundle on a curve is defined. The HKKP filtration is a purely algebraic object that depends only on a poset, yet it governs the asymptotic behaviour of a natural gradient flow in the space of metrics of the object. 

In this talk, we show that the HKKP filtration can be recovered from the stack of semistable objects, thus generalising the HKKP filtration to other moduli problems of non-linear origin. In particular, we will make sense of the notion of a filtration labelled by sequence of numbers for a point of an algebraic stack.

There is a deep analogy between Kleinaian groups and subgroups of the mapping class group. Inspired by this, Farb and Mosher defined convex cocompact subgroups of the mapping class group in analogy with convex cocompact Kleinian groups. These subgroups have since seen immense study, producing surprising applications to the geometry of surface group extension and surface bundles.  In particular, Hamenstadt plus Farb and Mosher proved that a subgroup of the mapping class groups is convex cocompact if and only if the corresponding surface group extension is Gromov hyperbolic.

Among Kleinian groups, convex cocompact groups are a special case of the geometrically finite groups. Despite the progress on convex cocompactness, no robust notion of geometric finiteness in the mapping class group has emerged.  Durham, Dowdall, Leininger, and Sisto recently proposed that geometric finiteness in the mapping class group might be characterized by the corresponding surface group extension being hierarchically hyperbolic instead of Gromov hyperbolic. We provide evidence in favor of this hypothesis by proving that the surface group extension of the stabilizer of a multicurve is hierarchically hyperbolic.

I will start with two very brief surveys.  First is a class of problems, namely variational inequalities (VIs), which generalize PDE problems, and second is a class of solver algorithms, namely full approximation storage (FAS) nonlinear multigrid for PDEs.  Motivation for applying FAS to VIs is demonstrated in the standard mathematical model for glacier surface evolution, a very general VI problem relevant to climate modeling.  (Residuals for this nonlinear and non-local VI problem are computed by solving a Stokes model.)  Some existing nonlinear multilevel VI schemes, based on global (Newton) linearization would seem to be less suited to such general VI problems.  From this context I will sketch some work-in-progress toward the scalable solutions of nonlinear and nonlocal VIs by an FAS-type multilevel method.

We explore the dynamics of a compressible fluid bubble surrounded by an incompressible fluid of infinite extent in three-dimensions, constructing bubble solutions with finite time blowup under this framework when the equation of state relating pressure and volume is soft (e.g., with volume singularities that are locally weaker than that in the Boyle-Mariotte law), resulting in a finite time blowup of the surrounding incompressible fluid, as well. We focus on two families of solutions, corresponding to a soft polytropic process (with the bubble decreasing in size until eventual collapse, resulting in velocity and pressure blowup) and a cavitation equation of state (with the bubble expanding until it reaches a critical cavitation volume, at which pressure blows up to negative infinity, indicating a vacuum). Interestingly, the kinetic energy of these solutions remains bounded up to the finite blowup time, making these solutions more physically plausible than those developing infinite energy. For all cases considered, we construct exact solutions for specific parameter sets, as well as analytical and numerical solutions which show the robustness of the qualitative blowup behaviors for more generic parameter sets. Our approach suggests novel -- and perhaps physical -- routes to the finite time blowup of fluid equations.

A fundamental result in 3-manifold topology is the loop theorem: Given a null-homotopic simple closed curve in the boundary of a compact 3-manifold $M$, it bounds an embedded disk in $M$. The standard topological proof of this uses the tower construction due to Papakyriakopoulos. In this talk, I will introduce basic existence and regularity results on minimal surfaces, and show how to use the tower construction to prove a geometric version of the loop theorem due to Meeks--Yau: Given a null-homotopic simple closed curve in the boundary of a compact Riemannian 3-manifold $M$ with convex boundary, it bounds an embedded disk of least area. This also gives an independent proof of the (topological) loop theorem.

Wreath-like product groups were introduced recently and used to construct the first positive examples of rigidity conjectures of Connes and Jones. In this talk, I will review those examples, as well as discuss some ideas to construct examples with other rigidity phenomena by modifying the wreath-like product construction.

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 discuss the difficult problem of proving reasonable bounds in the multidimensional generalization of Szemerédi's theorem and describe a proof of such bounds for sets lacking nontrivial configurations of the form (x,y), (x,y+z), (x,y+2z), (x+z,y) in the finite field model setting.

We consider the problem of solving nonconvex quadratic optimization problems, potentially with additional integrality constraints on the variables.  Gurobi takes a branch-and-bound approach to solve such problems to global optimality, and in this talk we will review the three main algorithmic components that Gurobi uses:  Convex relaxations based on local linearization, globally valid cutting planes, and nonlinear local optimization heuristics.  We will explain how these parts play together, and discuss some of the implementation details.

In this talk, I will review some recent progress on random graph matching problems, that is, to recover the vertex correspondence between a pair of correlated random graphs from the observation of two unlabelled graphs. In this talk, I will touch issues of information threshold, efficient algorithms as well as complexity theory. This is based on joint works with Hang Du, Shuyang Gong and Zhangsong Li.

I will explain how bornological and condensed structures can both be described as algebraic theories. I will also show how this permits the construction of functors between bornological and condensed structures. If time permits I will also briefly describe how to compare condensed derived geometry and bornological derived geometry and sketch how they relate to analytic geometry and Arakelov geometry

The numerical solution of elliptic PDEs is often the most computationally intensive task in large-scale continuum mechanics simulations.  High-order finite element methods can efficiently exploit modern parallel hardware while offering very rapid convergence properties.  As the polynomial degree is increased, the efficient solution of such PDEs becomes difficult. In this talk we introduce preconditioners for high-order discretizations. We build upon the pioneering work of Pavarino, who proved in 1993 that the additive Schwarz method with vertex patches and a low-order coarse space gives a  solver for symmetric and coercive problems that is robust to the polynomial degree. 

However, for very high polynomial degrees it is not feasible to assemble or factorize the matrices for each vertex patch, as the patch matrices contain dense blocks, which couple together all degrees of freedom within a cell. The central novelty of the preconditioners we develop is that they have the same time and space complexity as sum-factorized operator application on unstructured meshes of tensor-product cells, i.e., we can solve $A x=b$ with the same complexity as evaluating $b-A x$. Our solver relies on new finite elements for the de Rham complex that enable the blocks in the stiffness matrix corresponding to the cell interiors to become diagonal for scalar PDEs or block diagonal for vector-valued PDEs.  With these new elements, the patch problems are as sparse as a low-order finite difference discretization, while having a sparser Cholesky factorization. In the non-separable case, themethod can be applied as a preconditioner by approximating the problem with a separable surrogate.  Through the careful use of incomplete factorizations and choice of space decomposition we achieve optimal fill-in in the patch factors, ultimately allowing for optimal-complexity storage and computational cost across the setup and solution stages.

We demonstrate the approach by solving the Riesz maps of $H^1$, $H(\mathrm{curl})$, and $H(\mathrm{div})$ in three dimensions at $p = 15$.

Data sets of self-reported online resumes are a valuable tool to understand workers' career trajectories and how workers may adapt to the changing demands of employers. However, the sample of workers that choose to upload their resumes online may not be representative of a nation's workforce. To understand the advantages and limitations of these datasets, we analyze a data set of more than 1 Million online resumes and compare the findings with a administrative data from the Current Population Survey (CPS).
 

I will report on a research program which uses ideas from stochastic analysis in the context of constructive Euclidean quantum field theory. Stochastic analysis is the study of measures on path spaces via push-forward from Gaussian measures. The foundational example is the map, introduced by Itô, which sends Brownian motion to a diffusion process solution to a stochastic differential equation. Parisi–Wu's stochastic quantisation is the stochastic analysis of an Euclidean quantum field, in the above sense. In this introductory talk, I will put these ideas in context and illustrate various stochastic quantisation procedures and some of the rigorous results one can obtain from them.

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