Past

We develop a physical understanding of how stress waves propagate in uniform, heterogeneous, ordered and disordered media composed of discrete granular particles. We exploit this understanding to create experimentally novel materials and devices at different scales, (for example, for application in energy absorption, acoustic imaging and energy harvesting). We control the constitutive behavior of the new materials selecting the particles’ geometry, their arrangement and materials properties. One-dimensional chains of particles exhibit a highly nonlinear dynamic response, allowing a completely new type of wave propagation that has opened the door to exciting fundamental physical observations (i.e., compact solitary waves, energy trapping phenomena, and acoustic rectification). This talk will focus on energy localization and redirection in one-, two- and three-dimensional systems. (For an extended abstract please contact Ruth @email).

Based on ideas from Eells and Sampson, the Ricci flow was introduced by R. Hamilton in 1982 to try to prove Thurston's Geometrization Conjecture (a path which turned out to be successful). In this talk we will introduce the Ricci flow equation and view it as a modified heat flow. Using this we will prove the basic results on existence and uniqueness, and gain some insight into the evolution of various geometric quantities under Ricci flow. With this results we will proceed to define Perelman's $\mathcal{F}$ and $\mathcal{W}$ entropy functionals to view the Ricci flow as a gradient flow. If time permits we will briefly sketch some results from Cheeger and Gromov's compactness theory, which, along with the entropy functionals, alow us to blow up singularities.This is meant to be an introductory talk so I will try to develop as much geometric intuition as possible and stay away from technical calculations.

Recently, there has been a great deal of interest in the theory of modules over algebraic quantizations of so-called symplectic
resolutions. In this talk I'll discuss some new work -joint, and very much in progress- that open the door to giving a geometric description to certain categories of such modules; generalizing classical theorems of Kashiwara and Bernstein in the case of D-modules on an algebraic variety.

The discontinuous Galerkin (DG) method has recently become one of the most widely researched and utilized discretization methodologies for solving problems in science and engineering. It is fundamentally based upon the mathematical framework of variational methods, and provides hope that computationally fast, efficient and robust methods can be constructed for solving real-world problems. By not requiring that the solution to be continuous across element boundaries, DG provides a flexibility that can be exploited both for geometric and solution adaptivity and for parallelization. This freedom comes at a cost. Lack of smoothness across elements can hamper simulation post-processing like feature extraction and visualization. However, these limitations can be overcome by taking advantage of an additional property of DG - that of superconvergence. Superconvergence can aid in addressing the lack of continuity through the development of Smoothness-Increasing Accuracy-Conserving (SIAC) filters. These filters respect the mathematical properties of the data while providing levels of smoothness so that commonly used visualization tools can be used appropriately, accurately, and efficiently. In this talk, the importance of superconvergence in applications such as visualization will be discussed as well as overcoming the mathematical barriers in making superconvergence useful for applications.

[1] E. Feireisl, O. Kreml, S. Nečasová, J. Neustupa, and J. Stebel. Weak solutions to the barotropic NavierStokes system with slip boundary conditions in time dependent domains. J. Differential Equations, 254:125–140, 2013.

[2] E. Feireisl, O. Kreml, S. Nečasová, J. Neustupa, and J. Stebel. Incompressible limits of fluids excited by moving boundaries. Submitted

The integers (while wonderful in many others respects) do not make for fascinating Geometric Group Theory. They are, however, essentially the only infinite finitely generated group which is both hyperbolic and amenable. In the class of locally compact topological groups, the intersection of these two notions is richer, and the major aim of this talk will be to give the structure of a classification of such groups due to Caprace-de Cornulier-Monod-Tessera, beginning with Milnor's proof that any connected Lie group admitting a left-invariant negatively curved Riemannian metric is necessarily soluble.

Categorification is a fancy word for a process that is pretty ubiquitous in mathematics, though it is usually not referred to as "a thing". With the advent of higher category theory it has, however, become "a thing". I will explain what people mean by this "thing" (sneak preview: it involves replacing sets by categories) and hopefully convince you it is not quite as alien as it may seem and maybe even tempt you to let it infect some of your maths. I'll then explain how this fits into the context of higher categories.

Given a Lie group $G$, one can construct a principal $G$-bundle on a manifold $M$ by taking a cover $U\to M$, specifying a transition cocycle on the cover, and then descending the trivialized bundle $U \times G$ along the cocycle. We demonstrate the existence of an analogous construction for local $n$-bundles for general $n$. We establish analogues for simplicial Lie groupoids of Moore's results on simplicial groups; these imply that bundles for strict Lie $n$-groupoids arise from local $n$-bundles. We conclude by constructing a simple finite dimensional model of the Lie 2-group String($n$) using cohomological data.

We consider two decision problems for linear recurrence sequences(LRS) over the integers, namely the Positivity Problem (are all terms of a given LRS positive?) and the Ultimate Positivity Problem (are all but finitely many terms of a given LRS positive?). We show decidability of both problems for LRS of order 5 or less, and for simple LRS (i.e. whose characteristic polynomial has no repeated roots) of order 9 or less. Moreover, we show by way of hardness that extending the decidability of either problem to LRS of order 6 would entail major breakthroughs in analytic number theory, more precisely in the field of Diophantine approximation of transcendental numbers.
This talk is based on a recent paper, available at
http://www.cs.ox.ac.uk/people/joel.ouaknine/publications/positivity13ab…
joint with James Worrell and Matt Daws.

I'm going to make the talk more of a general discussion about weather forecasts and how they are used for practical decision making in energy trading in the first half, then spend the second half focusing on how we think about assessing and using the notion of state dependent predictability in our decision making process.

I will present new results on local smooth embedding of Riemannian manifolds of dimension $n$ into Euclidean space of dimension $n(n+1)/2$.  This part of ac joint project with G-Q Chen ( OxPDE), Jeanne Clelland ( Colorado), Dehua Wang ( Pittsburgh), and Deane Yang ( Poly-NYU).

The spectral presheaf of a nonabelian von Neumann algebra or C*-algebra was introduced as a generalised phase space for a quantum system in the so-called topos approach to quantum theory. Here, it will be shown that the spectral presheaf has many features of a spectrum of a noncommutative operator algebra (and that it can be defined for other classes of algebras as well). The main idea is that the spectrum of a nonabelian algebra may not be a set, but a presheaf or sheaf over the base category of abelian subalgebras. In general, the spectral presheaf has no points, i.e., no global sections. I will show that there is a contravariant functor from unital C*-algebras to their spectral presheaves, and that a C*-algebra is determined up to Jordan *-isomorphisms by its spectral presheaf in many cases. Moreover, time evolution of a quantum system can be described in terms of flows on the spectral presheaf, and commutators show up in a natural way. I will indicate how combining the Jordan and Lie algebra structures may lead to a full reconstruction of nonabelian C*- or von Neumann algebra from its spectral presheaf.

We discuss several singular limits for a scaled system of equations

(barotropic Navier-Stokes system), where the characteristic numbers become

small or ``infinite''. In particular, we focus on the situations relevant

in certain geophysical models with low Mach, large Rossby and large

Reynolds numbers. The limit system is rigorously identified in the

framework of weak solutions. The relative entropy inequality and careful

analysis of certain oscillatory integrals play crucial role.

The talk is based on the joint papers [{\it Bourgain J., Korobkov

M.V. and Kristensen~J.}: Journal fur die reine und angewandte Mathematik

(Crelles

Journal).

DOI: 10.1515/crelle-2013-0002] \ and \

[{\it Korobkov~M.V., Pileckas~K. and Russo~R.}:

arXiv:1302.0731, 4 Feb 2013]

We establish Luzin $N$ and Morse--Sard

properties for functions from the Sobolev space

$W^{n,1}(\mathbb R^n)$. Using these results we prove

that almost all level sets are finite disjoint unions of

$C^1$-smooth compact manifolds of dimension

$n-1$. These results remain valid also within

the larger space of functions of bounded variation

$BV_n(\mathbb R^n)$.

As an application, we study the nonhomogeneous boundary value problem

for the Navier--Stokes equations of steady motion of a viscous

incompressible fluid in arbitrary bounded multiply connected

plane or axially-symmetric spatial domains. We prove that this

problem has a solution under the sole necessary condition of zero total

flux through the boundary.

The problem was formulated by Jean Leray 80 years ago.

The proof of the main result uses Bernoulli's law

for a weak solution to the Euler equations based on the above-mentioned

Morse-Sard property for Sobolev functions.

Convection in a porous medium plays an important role in many geophysical and industrial processes, and is of particular current interest due to its implications for the long-term security of geologically sequestered CO_2. I will discuss two different convective systems in porous media, with the aid of 2D direct numerical simulations: first, a Rayleigh-Benard cell at high Rayleigh number, which gives an accurate characterization both of the convective flux and of the remarkable dynamical structure of the flow; and second, the evolution and eventual `shut-down' of convection in a sealed porous domain with a source of buoyancy along only one boundary. The latter case is also studied using simple box models and laboratory experiments, and can be extended to consider convection across an interface that can move and deform, rather than across a rigid boundary. The relevance of these results in the context of CO_2 sequestration will be discussed.

We prove an analogue of the Tate isogeny conjecture and the
semi-simplicity conjecture for overconvergent crystalline Dieudonne modules
of abelian varieties defined over global function fields of characteristic
p, combining methods of de Jong and Faltings. As a corollary we deduce that
the monodromy groups of such overconvergent crystalline Dieudonne modules
are reductive, and after base change to the field of complex numbers they
are the same as the monodromy groups of Galois representations on the
corresponding l-adic Tate modules, for l different from p.

I will discuss some p-adic (and mod p) criteria ensuring that an elliptic curve over the rationals has algebraic and analytic rank one, as well as some applications.

In partially molten regions of Earth, rock and magma coexist as a two-phase aggregate in which the solid grains of rock form a viscously deformable matrix. Liquid magma resides within the permeable network of pores between grains. Deviatoric stress causes the distribution of contact area between solid grains to become anisotropic; this causes anisotropy of the matrix viscosity. The anisotropic viscosity tensor couples shear and volumetric components of stress/strain rate. This coupling, acting over a gradient in shear stress, causes segregation of liquid and solid. Liquid typically migrates toward higher shear stress, but under specific conditions, the opposite can occur. Furthermore, in a two-phase aggregate with a porosity-weakening viscosity, matrix shear causes porosity perturbations to grow into a banded structure. We show that viscous anisotropy reduces the angle between these emergent high-porosity features and the shear plane. This is consistent with lab experiments.