Past

Globally Valued Fields, studied jointly with E. Hrushovski, are a formalism for fields in which the sum formula for valuations holds, such as number fields or function fields of curves. They form an elementary class (in continuous first order logic), and model-theoretic questions regarding this class give rise to difficult yet fascinating geometric questions.
I intend to present « Lyon school » approach to studying GVFs. This consists of reducing as much as possible to local considerations, among other things via the "fullness" axiom.
 

Given a Fano manifold we will consider two ways of attaching a (usually infinite) collection of polytopes, and a certain combinatorial transformation relating them, to it. The first is via Mirror Symmetry, following a proposal of  Coates--Corti--Kasprzyk--Galkin--Golyshev. The second is via symplectic topology, and comes from considering degenerating Lagrangian torus fibrations. We then relate these two collections using the Gross--Siebert program. I will also comment on the situation in higher dimensions, noting particularly that by 'inverting' the second method (degenerating Lagrangian fibrations) we can produce topological constructions of Fano threefolds.
 

Enumerating families of combinatorial objects with given properties and describing the typical structure of these objects are fundamental problems in extremal combinatorics. In this talk, we will investigate intersecting families of discrete structures in various settings, determining their typical structure as the size of the underlying ground set tends to infinity. Our new approach outlines a general framework for a number of similar problems; in particular, we prove analogous results for hypergraphs, permutations, and vector spaces using the same technique. This is joint work with József Balogh, Shagnik Das, Hong Liu, and Maryam Sharifzadeh.

In this talk we describe a new approach that enables the use of elliptic PDEs with white noise forcing to sample Matérn fields within the multilevel Monte Carlo (MLMC) framework.

When MLMC is used to quantify the uncertainty in the solution of PDEs with random coefficients, two key ingredients are needed: 1) a sampling technique for the coefficients that satisfies the MLMC telescopic sum and 2) a numerical solver for the forward PDE problem.

When the dimensionality of the uncertainty in the problem is infinite (i.e. coefficients are random fields), the sampling techniques commonly used in the literature are Karhunen–Loève expansions or circulant embeddings. In the specific case in which the coefficients are Gaussian fields of Mat ́ern covariance structure another sampling technique available relies on the solution of a linear elliptic PDE with white noise forcing.

When the finite element method (FEM) is used for the forward problem, the latter option can become advantageous as elliptic PDEs can be quickly and efficiently solved with the FEM, the sampling can be performed in parallel and the same FEM software can be used without the need for external packages. However, it is unclear how to enforce a good stochastic coupling of white noise between MLMC levels so as to respect the MLMC telescopic sum. In this talk we show how this coupling can be enforced in theory and in practice.

We give a short reminder about central results of classical tilting theory, 
including the Brenner-Butler tilting theorem, and
homological properties of tilted and quasi-tilted algebras. We then discuss 
2-term silting complexes and endomorphism algebras of such objects,
and in particular show that some of these classical results have very natural 
generalizations in this setting.
(joint work with Yu Zhou)

We propose and analyze a multilevel weighted least squares polynomial approximation method. Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that using an optimal distribution of sample locations, the number of samples required to achieve quasi-optimal approximation in a given polynomial subspace scales, up to a logarithmic factor, linearly in the dimension of this space. However, in many applications, the computation of samples includes a numerical discretization error. Thus, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose a multilevel method, which employs samples with different accuracies and is able to match the accuracy of single level approximations at reduced computational work. We prove complexity bounds under certain assumptions on polynomial approximability and sample work. Furthermore, we propose an adaptive
algorithm for situations where such assumptions cannot be verified a priori. Numerical experiments underline the practical applicability of our method.

Many metallurgical processes involve the heat treatment of granular material due to large alternating currents. To understand how the current propagates through the material, one must understand the bulk resistivity, that is, the resistivity of the granular material as a whole. The literature suggests that the resistance due to contacts between particles contributes significantly to the bulk resistivity, therefore one must pay particular attention to these contacts. 

My work is focused on understanding the precise impact of small contacts on the current propagation. The scale of the contacts is several order of magnitude smaller than that of the furnace itself, therefore we apply matched asymptotics methods to study how the current varies with the size of the contact.

How long does a uniformly accelerated observer need to interact with a
quantum field in order to record thermality in the Unruh temperature?
In the limit of large excitation energy, the answer turns out to be
sensitive to whether (i) the switch-on and switch-off periods are
stretched proportionally to the total interaction time T, or whether
(ii) T grows by stretching a plateau in which the interaction remains
at constant strength but keeping the switch-on and switch-off
intervals of fixed duration. For a pointlike Unruh-DeWitt detector,
coupled linearly to a massless scalar field in four spacetime
dimensions and treated within first order perturbation theory, we show
that letting T grow polynomially in the detector's energy gap E
suffices in case (i) but not in case (ii), under mild technical
conditions. These results limit the utility of the large E regime as a

probe of thermality in time-dependent versions of the Hawking and
Unruh effects, such as an observer falling into a radiating black
hole. They may also have implications on the design of prospective
experimental tests of the Unruh effect.

Based on arXiv:1605.01316 (published in CQG) with Christopher J
Fewster and Benito A Juarez-Aubry.

R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes showed in their celebrated paper from 1993 that numerous compensated compactness quantities such as Jacobians of mappings in $W^{1,n}(\mathbb{R}^n,\mathbb{R}^n)$ belong the real-variable Hardy space $\mathcal{H}^1(\mathbb{R}^n)$. They proceeded to ask what is the exact range of these nonlinear quantities and in particular whether the Jacobian operator $J$ maps $W^{1,2}(\mathbb{R}^2,\mathbb{R}^2)$ onto $\mathcal{H}^1(\mathbb{R}^2)$.

I present the proof of my recent result that $J \colon W^{1,n}(\mathbb{R}^n,\mathbb{R}^n) \to \mathcal{H}^1(\mathbb{R}^n)$ is non-surjective for every $n \ge 2$. The surjectivity question is still open when the domain of definition of $J$ is the inhomogeneous Sobolev space $\dot{W}^{1,n}(\mathbb{R}^n,\mathbb{R}^n)$. I also shortly discuss my work on T. Iwaniec's conjecture from 1997 which states that for every $n \ge 2$ and $p \in [1,\infty[$ the operator $J \colon W^{1,np}(\mathbb{R}^n,\mathbb{R}^n) \to \mathcal{H}^p(\mathbb{R}^n)$ has a continuous right inverse.

I will discuss ongoing work aimed at constructing higher categories of (enriched) higher categories. This should give the appropriate targets for many interesting examples of extended topological quantum field theories, including extended versions of the classical examples of TQFTs due to Turaev-Viro, Reshetikhin-Turaev, etc.

The signature of a path has many properties that make it an excellent feature to be used in machine learning. We exploit this properties to analyse a stream of data that arises from a psychiatric study whose objective is to analyse bipolar and borderline personality disorders. We build a machine learning model based on signatures that tries to answer two clinically relevant questions, based on observations of their reported state over a short period of time: is it possible to predict if a person is healthy, has bipolar disorder or has borderline personality disorder? And given a person or borderline personality disorder, it is possible to predict his or her future mood? Signatures proved to be very effective to tackle these two problems.

Abstract. As the first  step for approaching the uniqueness and blowup properties of the solutions of the stochastic wave equations with multi-plicative noise, we analyze the conditions for the uniqueness and blowup properties of the solution (Xt; Yt) of the equations dXt = Ytdt, dYt = jXtj_dBt, (X0; Y0) = (x0; y0). In particular, we prove that solutions arenonunique if 0 < _ < 1 and (x0; y0) = (0; 0) and unique if 1=2 < _ and (x0; y0) 6= (0; 0). We also show that blowup in _nite time holds if _ > 1 and (x0; y0) 6= (0; 0).

This is a joint work with A. Gomez, J.J. Lee, C. Mueller and M. Salins.

G2-manifolds are the Riemannian 7-manifolds with G2 holonomy and in many respects can be regarded as 7-dimensional analogues of Calabi-Yau 3-folds.
In joint work with Mark Haskins and Johannes Nordström we construct infinitely many families of new complete non-compact G2 manifolds (only four such manifolds were previously known). The underlying smooth 7-manifolds are all circle bundles over asymptotically conical Calabi-Yau 3-folds. The metrics are circle-invariant and have an asymptotic geometry that is the 7-dimensional analogue of the geometry of 4-dimensional ALF hyperkähler metrics. After describing the main features of our construction I will concentrate on some illustrative examples, describing how results in Calabi-Yau geometry about isolated singularities and their resolutions can be used to produce examples of complete G2-manifolds.

Nekrasov, Rosly and Shatashvili observed that the generating function of a certain space of SL(2) opers has a physical interpretation as the effective twisted superpotential for a four-dimensional N=2 quantum field theory. In this talk we describe the ingredients needed to generalise this observation to higher rank. Important ingredients are spectral networks generated by Strebel differentials and the abelianization method. As an example we find the twisted superpotential for the E6 Minahan-Nemeschansky theory. 
 

In this talk we consider a system of interacting Brownian particles. When diffusing particles interact with each other their motions are correlated, and the configuration space is of very high dimension. Often an equation for the one-particle density function (the concentration) is sought by integrating out the positions of all the others. This leads to the classic problem of closure, since the equation for the concentration so derived depends on the two-particle correlation function. We discuss two  common closures, the mean-field (MFA) and the Kirkwood-superposition approximations, as well as an alternative approach, which is entirely systematic, using matched asymptotic expansions (MAE). We compare the resulting (nonlinear) diffusion models with Monte Carlo simulations of the stochastic particle system, and discuss for which types of interactions (short- or long-range) each model works best. 

For a smooth variety over a number field, one defines various different homology groups (Betti, de Rham, etale, log-crystalline), which carry various kinds of enriching structure and are thought of as a system of realisations for a putative underlying (mixed) motivic homology group. Following Deligne, one can study fundamental groups in the same way, and the study of specific realisations of the motivic fundamental group has already found Diophantine applications, for instance in the anabelian proof of Siegel's theorem by Kim.

It is hoped that study of fundamental groups should give one access to ``higher'' arithmetic information not visible in the first cohomology, for instance classical and p-adic heights. In this talk, we will discuss recent work making this hope concrete, by demonstrating how local components of canonical heights on abelian varieties admit a natural description in terms of fundamental groups.

This talk explains how to formulate the now classical problem of optimal liquidation (or optimal trading) inside a Mean Field Game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader in front of a " background noise " (or " mean field "). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price. Here the trader faces the uncertainty of fair price changes too but not only. He has to deal with price changes generated by other similar market participants, impacting the prices permanently too, and acting strategically. Our MFG formulation of this problem belongs to the class of " extended MFG ", we hence provide generic results to address these " MFG of controls ", before solving the one generated by the cost function of optimal trading. We provide a closed form formula of its solution, and address the case of " heterogenous preferences " (when each participant has a different risk aversion). Last but not least we give conditions under which participants do not need to instantaneously know the state of the whole system, but can " learn " it day after day, observing others' behaviors.

In this talk, I will present a novel solution strategy to efficiently and accurately compute approximate solutions to semilinear optimal control problems, focusing on the optimal control of phase field formulations of geometric evolution laws.
The optimal control of geometric evolution laws arises in a number of applications in fields including material science, image processing, tumour growth and cell motility.
Despite this, many open problems remain in the analysis and approximation of such problems.
In the current work we focus on a phase field formulation of the optimal control problem, hence exploiting the well developed mathematical theory for the optimal control of semilinear parabolic partial differential equations.
Approximation of the resulting optimal control problem is computationally challenging, requiring massive amounts of computational time and memory storage.
The main focus of this work is to propose, derive, implement and test an efficient solution method for such problems. The solver for the discretised partial differential equations is based upon a geometric multigrid method incorporating advanced techniques to deal with the nonlinearities in the problem and utilising adaptive mesh refinement.
An in-house two-grid solution strategy for the forward and adjoint problems, that significantly reduces memory requirements and CPU time, is proposed and investigated computationally.
Furthermore, parallelisation as well as an adaptive-step gradient update for the control are employed to further improve efficiency.
Along with a detailed description of our proposed solution method together with its implementation we present a number of computational results that demonstrate and evaluate our algorithms with respect to accuracy and efficiency.
A highlight of the present work is simulation results on the optimal control of phase field formulations of geometric evolution laws in 3-D which would be computationally infeasible without the solution strategies proposed in the present work.

We present a new model for multi-agent dynamics where each agent is described by its position and body attitude: agents travel at a constant speed in a given direction and their body can rotate around it adopting different configurations. Agents try to coordinate their body attitudes with the ones of their neighbours. This model is inspired by the Vicsek model. The goal of this talk will be to present this new flocking model, its relevance and the derivation of the macroscopic equations from the particle dynamics.

Identity-based cryptography can be useful in situations where a full-scale public-key infrastructure is impractical.  Original identity-based proposals relied on elliptic curve pairings and so are vulnerable to quantum computers.  I will describe some on-going work to design a post-quantum identity-based encryption scheme using ideas from Ring Learning with Errors. Our scheme has the advantage that it can be extended to the hierarchical setting for more flexible key management.

In the game 'Set', players compete to pick out groups of three cards sharing common attributes. But how many cards must be dealt before such a group must appear? 
This is an example of a "cap set problem", a problem in Ramsey theory: how big can a set of objects get before some form of order appears? We will translate the cap set problem into a problem of geometry over finite fields, discussing the current best upper bounds and running through an elementary proof. We will also (very) briefly discuss one or two implications of the cap set problem over F_3 to other questions in Ramsey theory and computational complexity