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
The talk will focus on results of two related strands of research undertaken by the speaker. The first is a model of quantum mechanics based on the idea of 'structural approximation'. The earlier paper 'The semantics of the canonical commutation relations' established a method of calculation, essentially integration, for quantum mechanics with quadratic Hamiltonians. Currently, we worked out a (model-theoretic) formalism for the method, which allows us to
perform more subtle calculations, in particular, we prove that our path integral calculation produce correct formula for quadratic Hamiltonians avoiding non-conventional limits used by physicists. Then we focus on the model-theoretic analysis of the notion of structural approximation and show that it can be seen as a positive model theory version of the theory of measurable structures, compact domination and integration (p-adic and adelic).
Using tropical geometry and new methods in the theory of Fukaya categories, we explain a mirror symmetry equivalence relating the Fukaya category of a hypersurface and the category of coherent sheaves on the boundary of a toric variety.
Given a graph $G$, we can form a hypergraph $H$ whose edges correspond to the triangles in $G$. If $G$ is the standard Erdős-Rényi random graph with independent edges, then $H$ is random, but its edges are not independent, because of overlapping triangles. This is (presumably!) a major complication when proving results about triangles in random graphs. However, it turns out that, for many purposes, we can treat the triangles as independent, in a one-sided sense (and losing something in the density): we can find an independent random hypergraph within the set of triangles. I will present two proofs, one of which generalizes to larger complete (and some non-complete) subgraphs.
In this talk, a novel discontinuous Galerkin (DG) method is introduced by utilising the principle of discrete least squares. The key idea is to build polynomial approximations by the method of (weighted) discrete least squares instead of usual interpolation or (discrete) $L^2$ projections. The resulting method hence uses more information of the underlying function and provides a more robust alternative to common DG methods. As a result, we are able to construct high-order schemes which are conservative as well as linear stable on any set of collocation points. Several numerical tests highlight the new discontinuous Galerkin discrete least squares (DG-DLS) method to significantly outperform present-day DG methods.
High-order methods for conservation laws can be highly efficient if their stability is ensured. A suitable means mimicking estimates of the continuous level is provided by summation-by-parts (SBP) operators and the weak enforcement of boundary conditions. Recently, there has been an increasing interest in generalised SBP operators both in the finite difference and the discontinuous Galerkin spectral element framework.
However, if generalised SBP operators are used, the treatment of boundaries becomes more difficult since some properties of the continuous level are no longer mimicked discretely —interpolating the product of two functions will in general result in a value different from the product of the interpolations. Thus, desired properties such as conservation and stability are more difficult to obtain.
In this talk, the concept of generalised SBP operators and their application to entropy stable semidiscretisations will be presented. Several recent ideas extending the range of possible methods are discussed, presenting both advantages and several shortcomings.
In this talk I will first briefly introduce 1-parameter persistent homology, and discuss some applications and the theoretical challenges in the multiparameter case. If time remains I will explain how tools from commutative algebra give invariants suitable for the study of data. This last part is based on the preprint https://arxiv.org/abs/1708.07390.
Shortly after Mason & Skinner introduced the so-called ambitwistor strings, Berkovits came up with a pure-spinor analogue of the theory, which was later shown to provide the supersymmetric version of the Cachazo-He-Yuan amplitudes. In the heterotic version, however, both models give somewhat unsatisfactory descriptions of the supergravity sector.
In this talk, I will show how the original pure-spinor version of the heterotic ambitwistor string can be modified in a consistent manner that renders the supergravity sector treatable. In addition to the massless states, the spectrum of the new model --- which we call sectorized heterotic string --- contains a single massive level. In the limit in which a dimensionful parameter is taken to infinity, these massive states become the unexpected massless states (e.g. a 3-form potential) first encountered by Mason & Skinner."
Klarrich showed that the Gromov boundary of the curve complex of a hyperbolic surface is homeomorphic to the space of ending laminations on that surface. Independent results of Bestvina-Reynolds and Hamenstädt give an analogous statement for the free factor graph of a free group, where the space of ending laminations is replaced with a space of equivalence classes of arational trees. I will give an introduction to these objects and describe some joint work with Bestvina and Horbez, where we show that the Gromov boundary of the free factor graph for a free group of rank N has topological dimension at most 2N-2.
A Morse function (and more generally a Morse-Bott function) on a compact manifold M has associated Morse inequalities. The aim of this
talk is to explain how we can associate Morse inequalities to any smooth function on M (reporting on work of/with G Penington).
Inverting the signature of a path with ideas from linear algebra with implementations.
A K3 surface is called attractive if and only if its Picard number is 20: The maximal possible. Attractive K3 surfaces possess complex multiplication. This property endows attractive K3 surfaces with rich and well understood arithmetic. For example, the associated Galois representation turns out to be a product of well known two dimensional representations and the Hasse-Weil L-function turns out to be a product of well known L-functions. On the other hand, attractive K3 surfaces show up as solutions of the attractor equations in type IIB string theory compactified on the product of a K3 surface with an elliptic curve. As such, these surfaces dictate the near horizon geometry of a charged black hole in this theory. We will try to see which arithmetic properties of the attractive K3 surfaces lend a stringy interpretation and use them to shed light on physical properties of the charged black hole.
After recalling some backgrounds and motivations, we'll report some recent results on the optimal L^2 extensions and multiplier ideal sheaves, with emphasizing the close relations between SCV and PDE.
We consider a generalization of low-rank matrix completion to the case where the data belongs to an algebraic variety, i.e., each data point is a solution to a system of polynomial equations. In this case, the original matrix is possibly high-rank, but it becomes low-rank after mapping each column to a higher dimensional space of monomial features. Many well-studied extensions of linear models, including affine subspaces and their union, can be described by a variety model. We study the sampling requirements for matrix completion under a variety model with a focus on a union of subspaces. We also propose an efficient matrix completion algorithm that minimizes a surrogate of the rank of the matrix of monomial features, which is able to recover synthetically generated data up to the predicted sampling complexity bounds. The proposed algorithm also outperforms standard low-rank matrix completion and subspace clustering techniques in experiments with real data.