Past

We develop a method to investigate the geometric quantisation of a hypertoric variety from an equivariant viewpoint, in analogy with the equivariant Verlinde for Higgs bundles. We do this by first using the residual circle action on a hypertoric variety to construct its symplectic cut, resulting in a compact cut space which is needed for localisation. We introduce the notion of a moment polyptych associated to a hypertoric variety and prove that the necessary isotropy data can be read off from it. Finally, the equivariant Hirzebruch-Riemann-Roch formula is applied to the cut spaces and expresses the dimension of the equivariant quantisation space as a finite sum over the fixed-points. This is joint work with Johan Martens.

We prove a positive characteristic analogue of the classical result that the centralizer of a nonconstant differential operator in one variable is commutative. This leads to a new, short proof of that classical characteristic zero result, by reduction modulo p. This is joint work with Justin Desrochers available at https://arxiv.org/abs/2201.04606.

In this talk we will discuss the correlations of the Riemann Zeta in various ranges, and prove a new result for correlations of squares. This problem is closely related to correlations of the characteristic polynomial of CUE with a very subtle difference. We will explain where this difference comes from, and what it means for the moments of moments of the Riemann Zeta, and its maximum in short intervals.

The size-Ramsey number $\hat{r}(H)$ of a graph $H$ is the smallest number of edges a (host) graph $G$ can have, such that for any red/blue coloring of $G$, there is a monochromatic copy of $H$ in $G$. Recently, Conlon, Nenadov and Trujić showed that if $H$ is a graph on $n$ vertices and maximum degree three, then $\hat{r}(H) = O(n^{8/5})$, improving upon the bound of $n^{5/3 + o(1)}$ by Kohayakawa, Rödl, Schacht and Szemerédi. In our paper, we show that $\hat{r}(H)\leq n^{3/2+o(1)}$. While the previously used host graphs were vanilla binomial random graphs, we prove our result by using a novel host graph construction.
We also discuss why our bound is a natural barrier for the existing methods.
This is joint work with Kalina Petrova.

Since their introduction in 1928 by Paul A. Dirac, Dirac operators have been playing essential roles in many areas of Physics and Mathematics. In particular, they provide powerful and efficient tools to clarify (and sometimes solve) important problems in representation theory of real Lie groups, p-adic groups or Hecke algebras, such as classification, unitarity and geometric realization. In this representation theoretic context, the Dirac index of a Harish-Chandra module is a virtual module induced by Vogan’s Dirac cohomology. Once we observe that Dirac index commutes with translation functors, we will associate a polynomial (on a Cartan subalgebra) with a coherent family of Harish-Chandra modules. Then we shall explain how this polynomial can be used to connect nilpotent orbits, associated cycles and the leading term of the Taylor expansion of the characters of Harish-Chandra modules. This is joint wok with P. Pandzic, D. Vogan and R. Zierau.
 

Chimera states have attracted significant attention as symmetry-broken states exhibiting the coexistence of coherence and incoherence. Despite the valuable insights gained by analyzing specific systems, the understanding of the physical mechanism underlying the emergence of chimeras has been incomplete. In this presentation, I will argue that an important class of stable chimeras arise because coherence in part of the system is sustained by incoherence in the rest of the system. This mechanism may be regarded as a deterministic analog of noise-induced synchronization and is shown to underlie the emergence of so-called strong chimeras. These are chimera states whose coherent domain is formed by identically synchronized oscillators. The link between coherence and incoherence revealed by this mechanism also offers insights into the dynamics of a broader class of states, including switching chimera states and incoherence-mediated remote synchronization.

The aim of this course is to give an introduction to the regularity theory of non-smooth spaces with lower bounds on the Ricci Curvature. This is a quickly developing field with motivations coming from classical questions in Riemannian and differential geometry and with connections to Probability, Geometric Measure Theory and Partial Differential Equations.

In the lectures we will focus on the non collapsed case, where much sharper results are available, mainly adopting the synthetic approach of the RCD theory, rather than following the original proofs.

The techniques used in this setting have been applied successfully in the study of Minimal surfaces, Elliptic PDEs, Mean curvature flow and Ricci flow and the course might be of interest also for people working in these subjects.

I will review the series of recent results with Merle (IHES), Rodnianski (Princeton) and Szeftel (Paris Sorbonne) concerning the description of implosion mechanisms for viscous three dimensional compressible fluids. I will explain how the problem is connected to the description of blow up mechanisms for classical super critical defocusing models. 

 "In this talk, we will discuss invariant measures techniques to establish probabilistic global well-posedness for PDEs. We will go over the limitations that the Gibbs measures and the so-called fluctuation-dissipation measures encounter in the context of energy-supercritical PDEs. Then, we will present a new approach combining the two aforementioned methods and apply it to the energy supercritical Schrödinger equations. We will point out other applications as well."

A standard question in group theory is to ask if we can categorise the subgroups of a group in terms of their growth. In this talk we will be asking this question for uniform product set growth, a property that is stronger than the more widely understood notion of uniform exponential growth. We will see how considering acylindrical actions on hyperbolic spaces can help us, and give a particular application to mapping class groups.

For almost calibrated Lagrangian mean curvature flow it is known that all singularities are of Type II. To understand the finer structure of the singularities forming, it is thus necessary to understand the structure of general ancient solutions arising as potential limit flows at such singularities. We will discuss recent progress showing that ancient solutions with a blow-down a pair of static planes meeting along a 1-dimensional line are translators. This is joint work with J. Lotay and G. Szekelyhidi.

We study global 1- and (d−2)-form symmetries for gauge theories based on disconnected gauge groups which include charge conjugation. For pure gauge theories, the 1-form symmetries are shown to be non-invertible. In addition, being the gauge groups disconnected, the theories automatically have a Z2
global (d−2)-form symmetry. We propose String Theory embeddings for gauge theories based on these groups. Remarkably, they all automatically come with twist vortices which break the (d−2)-form global symmetry. 

Title: Topics on regularity theory for fully non-linear integro-differential equations

Abstract: We will focus on local and non-local Monge Ampere type equations, equations with deforming kernels and convex envelopes of functions with optimal special conditions. We discuss global solutions and their regularity properties.

Title: Quasilinear Conservative Collisional Transport in Kinetic Mean Field models

Abstract: We shall focus the on the interplay of nonlinear analysis  and numerical approximations to mean field models in particle physics where kinetic transport flows in momentum are strongly nonlinearly  modified by macroscopic quantities in classical or spectral density spaces. Two noteworthy models arise: the classical Fokker-Plank Landau dynamics as a low magnetized plasma regimes in the modeling of perturbative non-local high order terms. The other one corresponds to perturbation under strongly magnetized dynamics for fast electrons  in momentum space  give raise to a coupled system of classical kinetic diffusion processes described by the balance equations for electron probability density functions (electron pdf) coupled to the time dynamics on spectral energy waves  (quasi-particles) in a quantum process of their resonant interaction. Both models are rather different, yet there are derived form the Liouville-Maxwell system under different scaling. Analytical tools and some numerical  simulations show a presence of  strong hot tail anisotropy  formation taking the stationary states away from Classical equilibrium solutions stabilization for the iteration in a three dimensional cylindrical model. The semi-discrete schemes preserves the total system mass, momentum and energy, which are enforced by the numerical scheme. Error estimates can be obtained as well.

Work in collaboration with Clark Pennie and Kun Huang

Persistent homology is an important methodology from topological data analysis which adapts theory from algebraic topology to data settings and has been successfully implemented in many applications. It produces a statistical summary in the form of a persistence diagram, which captures the shape and size of the data. Despite its widespread use, persistent homology is simply impossible to implement when a dataset is very large. In this talk, I will address the problem of finding a representative persistence diagram for prohibitively large datasets. We adapt the classical statistical method of bootstrapping, namely, drawing and studying smaller multiple subsamples from the large dataset. We show that the mean of the persistence diagrams of subsamples—taken as a mean persistence measure computed from the subsamples—is a valid approximation of the true persistent homology of the larger dataset. We give the rate of convergence of the mean persistence diagram to the true persistence diagram in terms of the number of subsamples and size of each subsample. Given the complex algebraic and geometric nature of persistent homology, we adapt the convexity and stability properties in the space of persistence diagrams together with random set theory to achieve our theoretical results for the general setting of point cloud data. We demonstrate our approach on simulated and real data, including an application of shape clustering on complex large-scale point cloud data.

This is joint work with Yueqi Cao (Imperial College London).

We rely on soil to support the crops on which we depend. Less obviously we also rely on soil for a host of 'free services' from which we benefit. For example, soil buffers the hydrological system greatly reducing the risk of flooding after heavy rain; soil contains very large quantities of carbon, which would otherwise be released into the atmosphere where it would contribute to climate change. Given its importance it is not surprising that soil, especially its interaction with plant roots, has been a focus of many researchers. However the complex and opaque nature of soil has always made it a difficult medium to study. 

In this talk I will show how we can build a state of the art image based model of the physical and chemical properties of soil and soil-root interactions, i.e., a quantitative, model of the rhizosphere based on fundamental scientific laws.
This will be realised by a combination of innovative, data rich fusion of structural and chemical imaging methods, integration of experimental efforts to both support and challenge modelling capabilities at the scale of underpinning bio-physical processes, and application of mathematically sound homogenisation/scale-up techniques to translate knowledge from rhizosphere to field scale. The specific science questions I will address with these techniques are: (1) how does the soil around the root, the rhizosphere, function and influence the soil ecosystems at multiple scales, (2) what is the role of root- soil interface micro morphology on plant nutrient uptake, (3) what is the effect of plant exuded mucilage on the soil morphology, mechanics and resulting field and ecosystem scale soil function and (4) how to translate this knowledge from the single root scale to root system, field and ecosystem scale in order to predict how the climate change, different soil management strategies and plant breeding will influence the soil fertility. 

Effectiveness of COVID-19 vaccines was first demonstrated in randomised trials, but many questions of vital importance to vaccination policies could only be addressed in subsequent observational studies. The pandemic led to a step change in the availability of population-level linked electronic health record data, analysed in privacy-protecting Trusted Research Environments, across the UK. I will discuss methodological approaches to estimating causal effects of COVID-19 vaccines, and their application in estimating vaccine effectiveness and quantifying waning vaccine effectiveness. I will present results from recent analyses using detailed linked data on up to 24 million people in the OpenSAFELY Trusted Research Environment, which was developed by the University of Oxford's Bennett Institute for Applied Data Science.

We review old and new properties of systems of linear partial differential equations with constant coefficients. We discuss solvability in different function classes, to observe very different solution spaces. We examine the existence of vector potentials in the different spaces, by which we mean systems Av=0 with the property v=Bu, where A and B are linear PDE operators with constant coefficients. Properties of the systems and their solutions are examined both from linear algebra and algebraic geometry angles. A special class of operators that are examined is that of constant rank operators, which are prevalent in the nonlinear analysis of compensated compactness theory. We will discuss some of the challenges of extending this theory to non-constant rank operators.

MHD equilibrium is an important topic for fusion (and other MHD applications). A tokamak, in principle, is a toroidally symmetric fusion device and so MHD equilibrium can be reduced to solving the time independent MHD equations in axisymmetry. This produces the Grad-Shafranov equation (a two dimensional, nonlinear PDE) which has been solved using various techniques in the fusion community including finite difference, finite elements and spectral methods. A similar PDE exists if there is a plasma column with helical symmetry. Non-axisymmetric plasmas do occur in tokamaks as a result of instabilities and applied fields. However, if there is no symmetry angle there is no PDE to be solved. The current workhorse for finding non-axisymmetric equilibria uses energy minimization to find the equilibrium. New approaches to this problem that can use state of the art techniques are desirable. The speaker has formulated a coupled set of PDEs for the non-axisymmetric MHD equilibrium problem assuming that flux surfaces are nested (i.e. there are no magnetic islands) and has written this in weak form to use finite element method to solve the equations. The questions are around whether there is an optimal way to try to formulate the problem for FEM and to couple the equations, what sort of elements to use, if other solution techniques would be better suited and so on.

The Hardy-Littlewood generalised twin prime conjecture states an asymptotic formula for the number of primes $p\le X$ such that $p+h$ is prime for any non-zero even integer $h$. While this conjecture remains wide open, Matom\"{a}ki, Radziwi{\l}{\l} and Tao proved that it holds on average over $h$, improving on a previous result of Mikawa. In this talk we will discuss an almost prime analogue of the Hardy-Littlewood conjecture for which we can go beyond what is known for primes. We will describe some recent work in which we prove an asymptotic formula for the number of almost primes $n=p_1p_2 \le X$ such that $n+h$ has exactly two prime factors which holds for a very short average over $h$.

The possibility of `tacit collusion', in which interactions across market-making algorithms lead to an outcome similar to collusion among market makers, has increasingly received regulatory scrutiny. 
    We model the interaction of market makers in a dealer market as a stochastic differential game of intensity control with partial information and study the resulting dynamics of bid-ask spreads. Competition among dealers is modeled as a Nash equilibrium, which we characterise in terms of a system of coupled Hamilton-Jacobi-Bellman (HJB) equations, while Pareto optima correspond to collusion. 
    Using a decentralized multi-agent deep reinforcement learning algorithm to model how competing market makers learn to adjust their quotes, we show how the interaction of market-making algorithms may lead to tacit collusion with spread levels strictly above the competitive equilibrium level, without any explicit sharing of information.
 

A theorem of Chatzidakis, van den Dries and Macintyre, stemming ultimately from the Lang-Weil estimates, asserts, roughly, that if $\phi(x,y)$ is a formula in the language of rings (where $x,y$ are tuples) then the size of the solution set of $\phi(x,a)$ in any finite field $F_q $(where $a$ is a parameter tuple from $F_q$) takes one of finitely many dimension-measure pairs as $F_q$ and $a$ vary: for a finite set $E$ of pairs $(\mu,d)$ ($\mu$ rational, $d$ integer) dependent on $\phi$, any set $\phi(F_q,a)$ has size roughly $\mu q^d$ for some $(\mu,d) \in E$.

This led in work of Elwes, Steinhorn and myself to the notion of 'asymptotic class’ of finite structures (a class satisfying essentially the conclusion of Chatzidakis-van den Dries-Macintyre). As an example, by a theorem of Ryten, any family of finite simple groups of fixed Lie type forms an asymptotic class. There is a corresponding notion for infinite structures of  'measurable structure’ (e.g. a pseudofinite field, by the Chatzidakis-van den Dries-Macintyre theorem, or certain pseudofinite difference fields).

I will discuss a body of work with Sylvy Anscombe, Charles Steinhorn and Daniel Wolf which generalises this, incorporating a richer range of examples with fewer model-theoretic constraints; for example, the corresponding infinite 'generalised measurable’ structures, for which the definable sets are assigned values in some ordered semiring, need no longer have simple theory. I will also discuss a variant in which sizes of definable sets in finite structures are given exactly rather than asymptotically.

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

Motivated by the problem of reconstructing the electron density of a molecule from pulsed X-ray diffraction images (about 10e+9 per reconstruction), we develop a framework for analyzing the convergence to invariant measures of random fixed point iterations built from mappings that, while expansive, nevertheless possess attractive fixed points.  Building on techniques that we have established for determining rates of convergence of numerical methods for inconsistent nonconvex
feasibility, we lift the relevant regularities to the setting of probability spaces to arrive at a convergence analysis for noncontractive Markov operators.  This approach has many other applications, for instance the analysis of distributed randomized algorithms.
We illustrate the approach on the problem of solving linear systems with finite precision arithmetic.

 

On a smooth Riemannian manifold, the uniqueness of a geodesic given initial conditions follows from standard ODE theory. This is known to fail in the setting of RCD(K,N) spaces (metric measure spaces satisfying a synthetic notion of Ricci curvature bounded below) through an example of Cheeger-Colding. Strengthening the assumption a little, one may ask if two geodesics which agree for a definite amount of time must continue on the same trajectory. In this talk, I will show that this is true for RCD(K,N) spaces. In doing so, I will generalize a well-known result of Colding-Naber concerning the Hölder continuity of small balls along geodesics to this setting.

Bacteria represent the major component of the world’s biomass. A number of these bacteria are motile and swim with the use of flagellar filaments, which are slender helical appendages attached to a cell body by a flexible hook. Low Reynolds number hydrodynamics is the key for flagella to generate propulsion at a microscale [1]. In this talk I will discuss two projects related to swimming of a model bacterium Escherichia coli (E. coli).

E. coli has many flagellar filaments that are wrapped in a bundle and rotate in a counterclockwise fashion (if viewed from behind the cell) during the so-called ‘runs’, wherein the cell moves steadily forward. In between runs, the cell undergoes quick ‘tumble’ events, during which at least one flagellum reverses its rotation direction and separates from the bundle, resulting in erratic motion in place. Alternating between runs and tumbles allows cells to sample space by stochastically changing their propulsion direction after each tumble. In the first part of the talk, I will discuss how cells reorient during tumble and the mechanical forces at play and show the predominant role of hydrodynamics in setting the reorientation angle [2].

In the second part, I will talk about hydrodynamics of bacteria near walls in visco-elastic fluids. Flagellar motility next to surfaces in such fluids is crucial for bacterial transport and biofilm formation. In Newtonian fluids, bacteria are known to accumulate near walls where they swim in circles [3,4], while experimental results from our collaborators at the Wu Lab (Chinese University of Hong Kong) show that in polymeric liquids this accumulation is significantly reduced. We use a combination of analytical and numerical models to propose that this reduction is due to a viscoelastic lift directed away from the plane wall induced by flagellar rotation. This viscoelastic lift force weakens hydrodynamic interaction between flagellated swimmers and nearby surfaces, which results in a decrease in surface accumulation for the cells. 

References

[1] Lauga, Eric. "Bacterial hydrodynamics." Annual Review of Fluid Mechanics 48 (2016): 105-130.

[2] Dvoriashyna, Mariia, and Eric Lauga. "Hydrodynamics and direction change of tumbling bacteria." Plos one 16.7 (2021): e0254551.

[3] Berke, Allison P., et al. "Hydrodynamic attraction of swimming microorganisms by surfaces." Physical Review Letters 101.3 (2008): 038102.

[4] Lauga, Eric, et al. "Swimming in circles: motion of bacteria near solid boundaries." Biophysical journal 90.2 (2006): 400-412.

3d N=4 gauge theories can be studied on a circle times a closed Riemann surface. Their partition functions on this geometry, known as twisted indices, were computed some time ago using supersymmetric localisation on the Coulomb branch. An alternative perspective is to consider the theory as a supersymmetric quantum mechanics on S^1. In this talk I will review this point of view, which unveils interesting connection to topics in geometry such as wall-crossing and symplectic duality of quasi-maps.

We review old and new properties of systems of linear partial differential equations with constant coefficients. We discuss solvability in different function classes, to observe very different solution spaces. We examine the existence of vector potentials in the different spaces, by which we mean systems Av=0 with the property v=Bu, where A and B are linear PDE operators with constant coefficients. Properties of the systems and their solutions are examined both from linear algebra and algebraic geometry angles. A special class of operators that are examined is that of constant rank operators, which are prevalent in the nonlinear analysis of compensated compactness theory. We will discuss some of the challenges of extending this theory to non-constant rank operators.

We prove that with high probability $\mathbb{G}^{(3)}(n,n^{-1+o(1)})$ contains a spanning Steiner triple system for $n\equiv 1,3\pmod{6}$, establishing the exponent for the threshold probability for existence of a Steiner triple system. We also prove the analogous theorem for Latin squares. Our result follows from a novel bootstrapping scheme that utilizes iterative absorption as well as the connection between thresholds and fractional expectation-thresholds established by Frankston, Kahn, Narayanan, and Park.
This is joint work with Ashwin Sah and Michael Simkin. 

Characterising the statistical properties of high dimensional random functions has been one of the central focus of the theory of disordered systems, and notably spin glasses, over the last decades. Applications to machine learning via deep neural network has seen a resurgence of interest towards this problem in recent years. The simplest yet non-trivial quantity to characterise these landscapes is the annealed total complexity, i.e. the rate of exponential growth of the average number of stationary points (or equilibria) with the dimension of the underlying space. A paradigmatic model for such random landscape in the $N$-dimensional Euclidean space consists of an isotropic harmonic confinement and a Gaussian random function, with rotationally and translationally invariant covariance [1]. The total annealed complexity in this model has been shown to display a ”topology trivialisation transition”: for weak confinement, the number of stationary points is exponentially large (positive complexity) while for strong confinement there is typically a single stationary point (zero complexity).

In this talk, I will present recent results obtained for a distinct exactly solvable model of random lanscape in the $N$-dimensional Euclidean space where the random Gaussian function is replaced by a superposition of $M > N$ random plane waves [2]. In this model, we compute the total annealed complexity in the limit $N\rightarrow\infty$ with $\alpha = M/N$ fixed and find, in contrast to the scenario exposed above, that the complexity remains strictly positive for any finite value of the confinement strength. Hence, there is no ”topology trivialisation transition” for this model, which seems to be a representative of a distinct class of universality.

References:

[1] Y. V. Fyodorov, Complexity of Random Energy Landscapes, Glass Transition, and Absolute Value of the Spectral Determinant of Random Matrices, Phys. Rev. Lett. 92, 240601 (2004) Erratum: Phys. Rev. Lett. 93, 149901(E) (2004).

[2] B. Lacroix-A-Chez-Toine, S. Belga-Fedeli, Y. V. Fyodorov, Superposition of Random Plane Waves in High Spatial Dimensions: Random Matrix Approach to Landscape Complexity, arXiv preprint arXiv:2202.03815, submitted to J. Math. Phys.

Many problems in engineering can be understood as controlling the bifurcation structure of a given device. For example, one may wish to delay the onset of instability, or bring forward a bifurcation to enable rapid switching between states. In this talk, we will describe a numerical technique for controlling the bifurcation diagram of a nonlinear partial differential equation by varying the shape of the domain or a parameter in the equation. Our aim is to delay or advance a given branch point to a target parameter value. The algorithm consists of solving an optimization problem constrained by an augmented system of equations that characterize the location of the branch points. The flexibility and robustness of the method also allow us to advance or delay a Hopf bifurcation to a target value of the bifurcation parameter, as well as controlling the oscillation frequency. We will apply this technique on systems arising from biology, fluid dynamics, and engineering, such as the FitzHugh-Nagumo model, Navier-Stokes, and hyperelasticity equations.

In the past few years, the study of the geometric properties of random maps has been extended to a new regime, the "high genus regime", where we are interested in maps whose size and genus tend to infinity at the same time, at the same rate.
We consider here a slightly different case, where the genus also tends to infinity, but less rapidly than the size, and we study the law of simple cycles (with a well-chosen rescaling of the graph distance) in unicellular maps (maps with one face), thanks to a powerful bijection of Chapuy, Féray and Fusy.
The interest of this work is that we obtain exactly the same law as Mirzakhani and Petri who counted closed geodesics on a model of random hyperbolic surfaces in large genus (the Weil-Petersson measure). This leads us to conjecture that these two models are somehow "the same" in the limit. This is joint work with Svante Janson.

Detectors in collider experiments are modeled by light-ray operators in Quantum Field Theory. For example, energy detectors are certain null integrals of the stress-energy tensor, localized at an angle on the celestial sphere, where they collect quanta that escape in their direction. In this talk, I will discuss a series of work developing a nonperturbative, convergent operator product expansion (OPE) for light-ray operators in Conformal Field Theories (CFTs). Objects appearing in the expansion are more general light-ray operators, whose matrix elements can be computed by the generalized Lorentzian inversion formula. An important application is to event shapes in collider physics, which correspond to correlation functions of light-ray operators within the state created by the incoming particles. I will discuss some applications of the light-ray OPE in CFT, and mention some extensions to QCD which make contact with measurements at the LHC. Talk based primarily on [1905.01311] and [2010.04726].

We show that any irreducible representation $\rho$ of a finite group $G$ of exponent $n$, realisable over $\mathbb R$, is realisable over the field $E$ of real cyclotomic numbers of order $n$, and describe an algorithmic procedure transforming a realisation of $\rho$ over $\mathbb Q(\zeta_n)$ to one over $E$.

Synchronization, epidemic processes and information spreading are natural processes that emerge from the interaction between people. Mathematically, all of them can be described by models that, despite their simplicity, they can predict collective behaviors. In addition, they have in common a very interesting feature: a phase transition from an active state to an absorbing state. For example, the spread of an epidemic is characterized by the infection rate, the control parameter, which basically determines whether the epidemic will spread in the network or, if this rate is very low, few people become infected and the system falls into an absorbing state where there are no more infected people. In this presentation we will present the analytical and computational approaches used to investigate these classical models of statistical physics that present phase transitions and we will also show how the network topology influences such dynamical processes. The behavior of such dynamics can be much richer than we imagine.

We review old and new properties of systems of linear partial differential equations with constant coefficients. We discuss solvability in different function classes, to observe very different solution spaces. We examine the existence of vector potentials in the different spaces, by which we mean systems Av=0 with the property v=Bu, where A and B are linear PDE operators with constant coefficients. Properties of the systems and their solutions are examined both from linear algebra and algebraic geometry angles. A special class of operators that are examined is that of constant rank operators, which are prevalent in the nonlinear analysis of compensated compactness theory. We will discuss some of the challenges of extending this theory to non-constant rank operators.

This talk is about the mathematics behind an artistic project focusing on the vibrations of soap films.

The Onsager framework for linear irreversible thermodynamics provides a thermodynamically consistent model of mass transport in a phase consisting of multiple species, via the Stefan–Maxwell equations, but a complete description of the overall transport problem necessitates also solving the momentum equations for the flow velocity of the medium. We derive a novel nonlinear variational formulation of this coupling, called the (Navier–)Stokes–Onsager–Stefan–Maxwell system, which governs molecular diffusion and convection within a non-ideal, single-phase fluid composed of multiple species, in the regime of low Reynolds number in the steady state. We propose an appropriate Picard linearisation posed in a novel Sobolev space relating to the diffusional driving forces, and prove convergence of a structure-preserving finite element discretisation. The broad applicability of our theory is illustrated with simulations of the centrifugal separation of noble gases and the microfluidic mixing of hydrocarbons.

Black hole horizons are normally at finite spatial distance from the exterior region, but when they are degenerate (or extreme as they are usually referred to in this case) the spatial distance becomes infinite. One can still fall into the black hole in finite proper time but the crossing sphere is replaced by an "internal infinity". Near to the horizon of an extreme Kerr black hole, the scattering properties of test fields bear some similarities to what happens at an asymptotically flat infinity. This observation triggered a natural question concerning the peeling behaviour of test fields near such horizons. A geometrical tool known as the Couch-Torrence inversion is particularly well suited to studying this question. In this talk, I shall recall some essential notions on the peeling of fields at an asymptotically flat infinity and describe the Couch-Torrence inversion in the particular case of extreme Reissner-Nordström black holes, where it acts as a global conformal isometry of the spacetime. I will then show how to extend this inversion to more general spherically symmetric extreme horizons and describe what results can be obtained in terms of peeling. This is a joint ongoing project with Jack Borthwick (University of Besançon) and Eric Gourgoulhon (Paris Observatory).

The aim of this course is to give an introduction to the regularity theory of non-smooth spaces with lower bounds on the Ricci Curvature. This is a quickly developing field with motivations coming from classical questions in Riemannian and differential geometry and with connections to Probability, Geometric Measure Theory and Partial Differential Equations.

In the lectures we will focus on the non collapsed case, where much sharper results are available, mainly adopting the synthetic approach of the RCD theory, rather than following the original proofs.

The techniques used in this setting have been applied successfully in the study of Minimal surfaces, Elliptic PDEs, Mean curvature flow and Ricci flow and the course might be of interest also for people working in these subjects.

Recently, Terence Tao used a new quantitative approach to infer that certain ‘slightly supercritical’ quantities for the Navier–Stokes equations must become unbounded near a potential blow-up time. In this talk I’ll discuss a new strategy for proving quantitative bounds for the Navier–Stokes equations, as well as applications to behaviours of potentially singular solutions. This talk is based upon joint work with Christophe Prange (CNRS, Cergy Paris Université).

I will introduce a class of mean-field games under forward performance and for general risk preferences. Players interact through competition in fund management, driven by relative performance concerns in an asset diversification setting. This results in a common-noise mean field game. I will present the value and the optimal policies of such games, as well as some concrete examples. I will also discuss the partial information case, i.e.. when the risk premium is not directly observed. 

Via Poincaré duality, fundamental groups of aspherical manifolds have (appropriately shifted) isomorphisms between their homology and cohomology. In a 1973 Inventiones paper, Bieri and Eckmann defined a broader notion of a Duality Group, where the isomorphism between homology and cohomology can be twisted by what they called a Dualizing Module. Examples of these groups in geometric group theory (after passing to a finite-index subgroup) include $GL(n,\mathbb{Z})$, mapping class groups, and automorphism groups of free groups.

In work-in-progress with Thomas Wasserman we are looking into the following puzzle: the examples of duality groups that we know of that do not come from manifolds all have classifying spaces that satisfy a weaker local condition called the Cohen-Macaulay property. These spaces also satisfy weaker (twisted) versions of Poincaé duality via their local homology sheaves (or local cohomology cosheaves), and we are attempting to understand more about the links between these geometric versions of duality and the algebraic notion of a duality group. The goal of the talk is to explain more about the words used in the above paragraphs and say where we have got to so far.


 

A loop of Hamiltonian diffeomorphisms of a symplectic manifold $X$ defines, by clutching, a symplectic fibration over the two-sphere with fibre $X$.  We prove that the integral cohomology of the total space splits additively, answering a question of McDuff, and extending the rational cohomology analogue proved by Lalonde-McDuff-Polterovich in the late 1990’s. The proof uses a virtual fundamental class of moduli spaces of sections of the fibration in Morava K-theory. This talk reports on joint work with Mohammed Abouzaid and Mark McLean.

Non-smooth optimization is a core ingredient of many imaging or machine learning pipelines. Non-smoothness encodes structural constraints on the solutions, such as sparsity, group sparsity, low-rank and sharp edges. It is also the basis for the definition of robust loss functions such as the square-root lasso.  Standard approaches to deal with non-smoothness leverage either proximal splitting or coordinate descent. The effectiveness of their usage typically depend on proper parameter tuning, preconditioning or some sort of support pruning. In this work, we advocate and study a different route. By over-parameterization and marginalising on certain variables (Variable Projection), we show how many popular non-smooth structured problems can be written as smooth optimization problems. The result is that one can then take advantage of quasi-Newton solvers such as L-BFGS and this, in practice, can lead to substantial performance gains. Another interesting aspect of our proposed solver is its efficiency when handling imaging problems that arise from fine discretizations (unlike proximal methods such as ISTA whose convergence is known to have exponential dependency on dimension). On a theoretical level, one can connect gradient descent on our over-parameterized formulation with mirror descent with a varying Hessian metric. This observation can then be used to derive dimension free convergence bounds and explains the efficiency of our method in the fine-grids regime.

The line operators in a 2+1D TQFT form an algebraic structure called a modular tensor category (MTC). There is a natural action of a Galois group on MTCs which maps a given TQFT to other 'Galois conjugate' TQFTs. I will describe this Galois action and give several examples of Galois conjugate TQFTs. Galois action on a unitary TQFT can result in a non-unitary TQFT. I will derive a sufficient condition under which unitarity is preserved. Finally, I will describe the invariance of 0-form and 1-form symmetries of TQFTs under Galois action.