Past

I will present two recent results concerning the stability of boundary layer asymptotic expansions of solutions of Navier-Stokes with small viscosity. First, we show that the linearization around an arbitrary stationary shear flow admits an unstable eigenfunction with small wave number, when viscosity is sufficiently small. In boundary-layer variables, this yields an exponentially growing sublayer near the boundary and hence instability of the asymptotic expansions, within an arbitrarily small time, in the inviscid limit. On the other hand, we show that the Prandtl asymptotic expansions hold for certain steady flows. Our proof involves delicate construction of approximate solutions (linearized Euler and Prandtl layers) and an introduction of a new positivity estimate for steady Navier-Stokes. This in particular establishes the inviscid limit of steady flows with prescribed boundary data up to order of square root of small viscosity. This is a joint work with Emmanuel Grenier and Yan Guo.

In this talk I will try to show how certain asymptotic properties of a random walk on a graph are related to geometric properties of the graph itself. A special focus will be put on spectral properties and isoperimetric inequalities, proving Kesten's criterion for amenability.

Given a vector space V over a field K, let End(V) denote the algebra of linear endomorphisms of V. If V is finite-dimensional, then it is well-known that the diagonalizable subalgebras of End(V) are characterized by their internal algebraic structure: they are the subalgebras isomorphic to K^n for some natural number n. 

In case V is infinite dimensional, the diagonalizable subalgebras of End(V) cannot be characterized purely by their internal algebraic structure: one can find diagonalizable and non-diagonalizable subalgebras that are isomorphic.  I will explain how to characterize the diagonalizable subalgebras of End(V) as topological algebras, using a natural topology inherited from End(V).  I will also illustrate how this characterization relates to an infinite-dimensional Wedderburn-Artin theorem that characterizes "topologically semisimple" algebras.

In recent years, several preconditioning strategies have been proposed for control problems in fluid dynamics. These are a special case of the general saddle point problem in optimisation. Here, we will extend a preconditionier for distributed Stokes control problems, developed by Rees and Wathen, to the case of boundary control. We will show the usefulness of low-rank structures in constructing a good approximation for the Schur complement of the saddle point matrix. In the end, we will discuss the applicability of this strategy for Navier-Stokes control.

We develop a spectral method for solving univariate singular integral equations over unions of intervals and circles, by utilizing Chebyshev, ultraspherical and Laurent polynomials to reformulate the equations as banded infinite-dimensional systems. Low rank approximations are used to obtain compressed representations of the bivariate kernels. The resulting system can be solved in linear time using an adaptive QR factorization, determining an optimal number of unknowns needed to resolve the solution to any pre-determined accuracy. Applications considered include fracture mechanics, the Faraday cage, and acoustic scattering. The Julia software package https://github.com/ApproxFun/SIE.jl implements our method with a convenient, user-friendly interface.

In the talk we present a survey of recent results (see [4]-[6]) on the existence theorems for the steady-state Navier-Stokes boundary value problems in the plane and axially symmetric 3D cases for bounded and exterior domains (the so called Leray problem, inspired by the classical paper [8]). One of the main tools is the Morse-Sard Theorem for the Sobolev functions $f\in W^2_1(\mathbb R^2)$ [1] (see also [2]-[3] for the multidimensional case). This theorem guaranties that almost all level lines of such functions are $C^1$-curves besides the function $f$ itself could be not $C^1$-regular.

Also we discuss the recent Liouville type theorem for the steady-state Navier-Stokes equations for  axially symmetric 3D solutions in the absence of swirl (see [1]).

References

1.  Bourgain J., Korobkov M. V., Kristensen J., On the Morse-Sard property and level sets of Sobolev and BV functions, Rev. Mat. Iberoam.,  29 , No. 1, 1-23  (2013).
2. Bourgain J., Korobkov M. V., Kristensen J., On the Morse-Sard property and level sets of $W^{n,1}$ Sobolev functions on $\mathbb R^n$, Journal fur die reine und angewandte Mathematik (Crelles Journal) (Online first 2013).
3. Korobkov M. V., Kristensen J., On the Morse-Sard Theorem for the sharp case of Sobolev mappings, Indiana Univ. Math. J., 63, No. 6, 1703-1724  (2014).
4. Korobkov M. V., Pileckas K., Russo R., The existence theorem for steady Navier-Stokes equations in the axially symmetric case, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 14, No. 1, 233-262  (2015).
5. Korobkov M. V., Pileckas K., Russo R., Solution of Leray's problem for stationary Navier-Stokes equations in plane and axially symmetric spatial domains,  Ann. of Math., 181, No. 2, 769-807  (2015).
6. Korobkov M. V., Pileckas K., Russo R., The existence theorem for the steady Navier-Stokes problem in exterior axially symmetric 3D domains, 2014, 75 pp., http://arXiv.org/abs/1403.6921.
7. Korobkov M. V., Pileckas K., Russo R., The Liouville Theorem for the Steady-State Navier-Stokes Problem for Axially Symmetric 3D Solutions in Absence of Swirl, J. Math. Fluid Mech. (Online first 2015).
8. Leray J., Étude de diverses équations intégrals nonlinéaires et de quelques problèmes que pose l'hydrodynamique, J. Math. Pures Appl., 9, No. 12, 1- 82 (1933).

After some generalities on étale fundamental groups and anabelian geometry, I will explore some of the current results on the section conjecture, including those of Koenigsmann and Pop on the birational section conjecture, and a recent unpublished result of Mohamed Saidi which reduces the section conjecture for finitely generated fields over the rationals to the case of number fields.

Erdos and Renyi introduced a model for studying random graphs of a given "density" and proved that there is a sharp threshold at which lower density random graphs are disconnected and higher density ones are connected.  Motivated by ideas in geometric group theory we will explain some new threshold theorems we have discovered for random graphs.  We will then, explain applications of these results to the geometry of Coxeter groups.  Some of this talk will be on joint work with Hagen and Sisto; other parts are joint work with Hagen, Susse, and Falgas-Ravry.

This talk investigates the discovery of an intriguing and fundamental connection between the famous but apparently unrelated work of two mathematicians of late antiquity, Pappus and Diophantus. This link went unnoticed for well over 1500 years until the publication of two groundbreaking but again ostensibly unrelated works by two German mathematicians at the close of the 19th century. In the interim, mathematics changed out of all recognition, with the creation of numerous new mathematical subjects and disciplines, without which the connection might never have been noticed in the first place. This talk examines the chain of mathematical events that led to the discovery of this remarkable link between two seemingly distinct areas of mathematics, encompassing number theory, finite-dimensional real normed algebras, combinatorial design theory, and projective geometry, and including contributions from mathematicians of all kinds, from the most distinguished to the relatively unknown.

Adrian Rice is Professor of Mathematics at Randolph-Macon College in Ashland, Virginia, where his research focuses on the history of 19th- and early 20th-century mathematics. He is a three-time recipient of the Mathematical Association of America's awards for outstanding expository writing.

There is a pleasing correspondence between splittings of a free group over finitely generated subgroups and systems of surfaces in a doubled handlebody. One can use this to describe a family of hyperbolic complexes on which Out(F_n) acts. This is joint work with Camille Horbez.

Two subsets A and B of R^n are equidecomposable if it is possible to partition A into pieces and rearrange them via isometries to form a partition of B. Motivated by what is nowadays known as Banach-Tarski paradox, Tarski asked if the unit square and the disc of unit area in R^2 are equidecomposable. 65 years later Laczkovich showed that they are, at least when the pieces are allowed to be non-measurable sets. I will talk about a joint work with A. Mathe and O. Pikhurko which implies in particular the existence of a measurable equidecomposition of circle and square in R^2.

I this talk I will try to give an overview of recent progress in
spatial stochastic modeling within the reaction-diffusion
framework. While much of the initial motivation for this work came
from problems in cell biology, I will also highlight some examples
from epidemics and neuroscience.

As a motivating introduction, some newly discovered properties of
optimal controls in stochastic enzymatic reaction networks will be
presented. I will next detail how diffusive and subdiffusive reactive
processes in realistic geometries at the cellular scale may be modeled
mesoscopically. Along the way, some different means by which these
models may be analyzed with respect to consistency and convergence
will also be discussed. These analytical techniques hint at how
effective (i.e. parallel) numerical implementations can be
designed. Large-scale simulations will serve as illustrations.

Divergence, thickness, and relative hyperbolicity are three geometric properties which determine aspects of the quasi-isometric geometry of a finitely generated group. We will discuss the basic properties of these notions and some of the relations between them. We will then then survey how these properties manifest in right-angled Coxeter groups and detail various ways to classify Coxeter groups using them.

This is joint work with Hagen and Sisto.

Standard commutative algebra is based on the notions of commutative monoid, Abelian group and commutative ring. In recent years, motivations from category theory, algebraic geometry, and mathematical logic led to the development of an area that may be called commutative 2-algebra, in which the notions used in commutative algebra are replaced by their category-theoretic counterparts (e.g. commutative monoids are replaced by  symmetric monoidal categories). The aim of this talk is to explain the analogy between standard commutative algebra and commutative 2-algebra, and to outline how this suggests counterparts of basic aspects of algebraic geometry. In particular, I will describe some joint work with Andre’ Joyal on operads and analytic functors in this context.

Stochastic optimization for sequential decision problems under uncertainty arises in many settings, and as a result as evolved under several canonical frameworks with names such as dynamic programming, stochastic programming, optimal control, robust optimization, and simulation optimization (to name a few).  This is in sharp contrast with the universally accepted canonical frameworks for deterministic math programming (or deterministic optimal control).  We have found that these competing frameworks are actually hiding different classes of policies to solve a single problem which encompasses all of these fields.  In this talk, I provide a canonical framework which, while familiar to some, is not universally used, but should be.  The framework involves solving an objective function which requires searching over a class of policies, a step that can seem like mathematical hand waving.  We then identify four fundamental classes of policies, called policy function approximations (PFAs), cost function approximations (CFAs), policies based on value function approximations (VFAs), and lookahead policies (which themselves come in different flavors).  With the exception of CFAs, these policies have been widely studied under names that make it seem as if they are fundamentally different approaches (policy search, approximate dynamic programming or reinforcement learning, model predictive control, stochastic programming and robust optimization).  We use a simple energy storage problem to demonstrate that minor changes in the nature of the data can produce problems where each of the four classes might work best, or a hybrid.  This exercise supports our claim that any formulation of a sequential decision problem should start with a recognition that we need to search over a space of policies.

I will discuss some recent results on the distribution of the real-analytic Eisenstein series on thin sets, such as a geodesic segment. These investigations are related to mean values of the Riemann zeta function, and have connections to quantum chaos.

Despite many years of intensive research, the modeling of contact lines moving by spreading and/or evaporation still remains a subject of debate nowadays, even for the simplest case of a pure liquid on a smooth and homogeneous horizontal substrate. In addition to the inherent complexity of the topic (singularities, micro-macro matching, intricate coupling of many physical effects, …), this also stems from the relatively limited number of studies directly comparing theoretical and experimental results, with as few fitting parameters as possible. In this presentation, I will address various related questions, focusing on the physics invoked to regularize singularities at the microscale, and discussing the impact this has at the macroscale. Two opposite “minimalist” theories will be detailed: i) a classical paradigm, based on the disjoining pressure in combination with the spreading coefficient; ii) a new approach, invoking evaporation/condensation in combination with the Kelvin effect (dependence of saturation conditions upon interfacial curvature). Most notably, the latter effect enables resolving both viscous and thermal singularities altogether, without needing any other regularizing effects such as disjoining pressure, precursor films or slip length. Experimental results are also presented about evaporation-induced contact angles, to partly validate the first approach, although it is argued that reality might often lie in between these two extreme cases.

We present a trust region algorithm for solving nonconvex optimization problems that, in the worst-case, is able to drive the norm of the gradient of the objective below a prescribed threshold $\epsilon > 0$ after at most ${\cal O}(\epsilon^{-3/2})$ function evaluations, gradient evaluations, or iterations.  Our work has been inspired by the recently proposed Adaptive Regularisation framework using Cubics (i.e., the ARC algorithm), which attains the same worst-case complexity bound.  Our algorithm is modeled after a traditional trust region algorithm, but employs modified step acceptance criteria and a novel trust region updating mechanism that allows it to achieve this desirable property.  Importantly, our method also maintains standard global and fast local convergence guarantees.

Inspired by a question posed by Lax, in recent years it has received  

an increasing attention the study of quantitative compactness  

estimates for the solution operator $S_t$, $t>0$ that associates to  

every given initial data $u_0$ the corresponding solution $S_t u_0$ of  

a conservation law or of a first order Hamilton-Jacobi equation.



Estimates of this type play a central roles in various areas of  

information theory and statistics as well as of ergodic and learning  

theory. In the present setting, this concept could provide a measure  

of the order of ``resolution'' of a numerical method for the  

corresponding equation.



In this talk we shall first review the results obtained in  

collaboration with O. Glass and K.T. Nguyen, concerning the  

compactness estimates for solutions to conservation laws. Next, we  

shall turn to the  analysis of the Hamilton-Jacobi equation pursued in  

collaboration with P. Cannarsa and K.T.~Nguyen.