Past

A mini-lecture series consisting of four 1 hour lectures.

It has long been a challenge to synthesize the complementary insights offered by model theory and category theory. A small fragment of that challenge is to understand ultraproducts categorically. I will show that, granted some general categorical machinery, the notions of ultrafilter and ultraproduct follow inexorably from the notion of finiteness of a set. The machine in question, known as the codensity monad, has existed in an underexploited state for nearly fifty years. To emphasize that it was not constructed specifically for this purpose, I will mention some of its other applications. This talk represents joint work with an anonymous referee. Little knowledge of category theory will be assumed.

Focused ion beam milling allows small scale single crystal cantilevers to be produced with cross-sectional dimensions on the order of microns which are then tested using a nanoindenter allowing both elastic and plastic materials properties to be measured. EBSD allows these cantilevers to be milled from any desired crystal orientation. Micro-cantilever bending experiments suggest that sufficiently smaller cantilevers are stronger, and the observation is believed to be related to the effect of the neutral axis on the evolution of the dislocation structure. A planar model of discrete dislocation plasticity was used to simulate end-loaded cantilevers to interpret the behaviour observed in the experiments. The model allowed correlation of the simulated dislocation structure to the experimental load displacement curve and GND density obtained from EBSD. The planar model is sufficient for identifying the roles of the neutral axis and source spacing in the observed size effect, and is particularly appropriate for comparisons to experiments conducted on crystals orientated for plane strain deformation. The effect of sample dimensions and dislocation source density are investigated and compared to small scale mechanical tests conducted on Titanium and Zirconium.

I will discuss conjectures relating cup products of cyclotomic units and modular symbols modulo an Eisenstein ideal. In particular, I wish to explain how these conjectures may be viewed as providing a refinement of the Iwasawa main conjecture. T. Fukaya and K. Kato have proven these conjectures under certain hypotheses, and I will mention a few key ingredients. I hope to briefly mention joint work with Fukaya and Kato on variants.

I will be taking us on a journey through low dimensional topology, starting in 2 dimensions motivating handles decompositions in a dimension that we can visualize, moving onto to a brief of note of what this means in 3 dimensions and then moving onto the wild world of 4 manifolds. I will be showing a way in which we can actually try and view a 4 manifold before moving onto a way of manipulating these diagrams to give diffeomorphic 4 manifolds. Hopefully, I will have time to go into some ways in which Kirby calculus has been used to show that certain potential exotic 4 spheres are not exotic and some results on stable diffeomorphims of 4 manifolds.

The behavior of lipid vesicles is due to a complex interplay between the mechanics of the vesicle membrane, the surrounding fluids, and any external fields which may be present. In this presentation two aspects of vesicle behavior are explored: vesicles in inertial flows and vesicles exposed to electric fields.

The first half of the talk will present work done investigating the behavior of two-dimensional vesicles in inertial flows. A level set representation of the interface is coupled to a Navier-Stokes projection solver. The standard projection method is modified to take into account not only the volume incompressibility of the fluids but also the surface incompressibility of the vesicle membrane. This surface incompressibility is enforced by using the closest point level set method to calculate the surface tension needed to enforce the surface incompressibility. Results indicate that as inertial effects increase vesicle change from a tumbling behavior to a stable tank-treading configuration. The numerical results bear a striking similarity to rigid particles in inertial flows. Using rigid particles as a guide scaling laws are determined for vesicle behavior in inertial flows.

The second half of the talk will move to immersed interface solvers for three-dimensional vesicles exposed to electric fields. The jump conditions for pressure and fluid velocity will be developed for the three-dimensional Stokes flow or constant density Navier-Stokes equations assuming a piecewise continuous viscosity and an inextensible interface. An immersed interface solver is implemented to determine the fluid and membrane state. Analytic test cases have been developed to examine the convergence of the fluids solver.

Time permitting an immersed interface solver developed to calculate the electric field for a vesicle exposed to an electric field will be discussed. Unlike other multiphase systems, vesicle membranes have a time-varying potential which must be taken into account. This potential is implicitly determined along with the overall electric potential field.

I will look at a toy model for an index in a large market. The aim is to

consider the pricing of volatility swaps on the index. This is very much

work in progress.

We consider the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the pre-Maxwell dynamics for the magnetic field. At the free-interface, driven by the plasma velocity, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasma density does not go to zero continuously at the interface, but has a jump, meaning that it is bounded away from zero in the plasma region and it is identically zero in the vacuum region. The plasma-vacuum system is not isolated from the outside world, because of a given surface current on the fixed boundary that forces oscillations.

Under a suitable stability condition satisfied at each point of the initial interface, stating that the magnetic fields on either side of the interface are not collinear, we show the existence and uniqueness of the solution to the nonlinear plasma-vacuum interface problem in suitable anisotropic Sobolev spaces.

The proof follows from the well-posedness of the homogeneous linearized problem and a basic a priori energy estimate, the analysis of the elliptic system for the vacuum magnetic field, a suitable tame estimate in Sobolev spaces for the full linearized equations, and a Nash-Moser iteration.

This is a joint work with Y. Trakhinin (Novosibirsk).

A generalized Baumslag-Solitar group is a group G acting co-compactly on a tree X, with all vertex- and edge stabilizers isomorphic to the free abelian group of rank n. We will discuss the $L^p$-metric and $L^p$-equivariant compression of G, and also the quasi-isometric embeddability of G in a finite product of binary trees. Complete results are obtained when either $n=1$, or the quotient graph $G\X$ is either a tree or homotopic to a circle. This is joint work with Yves Cornulier.

Group actions play an important role in both topological problems and coarse geometric conjectures. I will introduce the idea of a partial action of a group on a metric space and explain, in the case of certain classes of coarsely disconnected spaces, how partial actions can be used to give a geometric proof of a result of Willett and Yu concerning the coarse Baum-Connes conjecture.

Historically, decay rates have been used to provide quantitative and qualitative information on the solutions to hyperbolic conservation laws. Quantitative results include the establishment of convergence rates for approximating procedures and numerical schemes. Qualitative results include the establishment of results on uniqueness and regularity as well as the ability to visualize the waves and their evolution in time.

In this talk, I will present two decay estimates on the positive waves for systems of hyperbolic and genuinely nonlinear balance laws satisfying a dissipative mechanism. The result is obtained by employing the continuity of Glimm-type functionals and the method of generalized characteristics. Using this result on the spreading of rarefaction waves, the rate of convergence for vanishing viscosity approximations to hyperbolic balance laws will also be established. The proof relies on error estimates that measure the interaction of waves using suitable Lyapunov functionals. If time allows, a further application of the recent developments in the theory of balance laws to differential geometry will be addressed.

In this talk, I will discuss homotopy limits: The basics, and why you should care about them if you are a topologist, an algebraic geometer, or an algebraist (have I missed anyone?).

Since graph-coloring is an NP-complete problem in general, it is natural to ask how the complexity changes if the input graph is known not to contain a certain induced subgraph H. Due to results of Kaminski and Lozin, and Hoyler, the problem remains NP-complete, unless H is the disjoint union of paths. Recently the question of coloring graphs with a fixed-length induced path forbidden has received considerable attention, and only a few cases of that problem remain open for k-coloring when k>=4. However, little is known for 3-coloring. Recently we have settled the first open case for 3-coloring; namely we showed that 3-coloring graphs with no induced 6-edge paths can be done in polynomial time. In this talk we will discuss some of the ideas of the algorithm.

This is joint work with Peter Maceli and Mingxian Zhong.

The study of wave equations on black hole backgrounds provides important insights for the non-linear stability problem for black holes. I will illustrate this in the context of asymptotically anti de Sitter black holes and present both stability and instability results. In particular, I will outline the main ideas of recent work with J. Smulevici (Paris) establishing a logarithmic decay in time for solutions of the massive wave equation on Kerr-AdS black holes and proving that this slow decay rate is in fact sharp.

The mod p homology of a space is an unstable coalgebra over the Steenrod algebra at the prime p. This talk will be about the classical problem of realising an unstable coalgebra as the homology of a space. More generally, one can consider the moduli space of all such topological realisations and ask for a description of its homotopy type. I will discuss an obstruction theory which describes this moduli space in terms of the Andr\'{e}-Quillen cohomology of the unstable coalgebra. This is joint work with G. Biedermann and M. Stelzer.

In this talk we consider two questions about conformally invariant random curves known as Schramm-Loewner evolutions (SLE). The first question is about the "boundary zig-zags", i.e. the probabilities for a chordal SLE to pass through small neighborhoods of given boundary points in a given order. The second question is that of obtaining explicit descriptions of "multiple SLE pure geometries", i.e. those extremal multiple SLE probability measures which can not be expressed as non-trivial convex combinations of other multiple SLEs. For both problems one needs to find solutions of a system of partial differential equations with asymptotics conditions written recursively in terms of solution of the same problem with a smaller number of variables. We present a general correspondence, which translates these problems to linear systems of equations in finite dimensional representations of the quantum group U_q(sl_2), and we then explicitly solve these systems. The talk is based on joint works with Eveliina Peltola (Helsinki), and with Niko Jokela (Santiago de Compostela) and Matti Järvinen (Crete).

We will first review the return to equilibrium of the Ising model when a small external field is applied. The relaxation time is extremely long and can be estimated as the time needed to create critical droplets of the stable phase which will invade the whole system. We will then discuss the impact of disorder on this metastable behavior and show that for Ising model with random interactions (dilution of the couplings) the relaxation time is much faster as the disorder acts as a catalyst. In the last part of the talk, we will focus on the droplet growth and study a toy model describing interface motion in disordered media.

Suppose we are given a double continuum (in time and strike) of discounted

option prices, or equivalently a set of measures which is increasing in

convex order. Given sufficient regularity, Dupire showed how to construct

a time-inhomogeneous martingale diffusion which is consistent with those

prices. But are there other martingales with the same 1-marginals? (In the

case of Gaussian marginals this is the fake Brownian motion problem.)

In this talk we show that the answer to the question above is yes.

Amongst the class of martingales with a given set of marginals we

construct the process with smallest possible expected total variation.

Mechanics plays an important role during several biological phenomena such as morphogenesis,

wound healing, bone remodeling and tumorogenesis. Each one of these events is triggered by specific

elementary cell deformations or movements that may involve single cells or populations of cells. In

order to better understand how cell behave and interact, especially during degenerative processes (i.e.

tumorogenesis and metastasis), it has become necessary to combine both numerical and experimental

approaches. Particularly, numerical models allow determining those parameters that are still very

difficult to experimentally measure such as strains and stresses.

During the last few years, I have developed new finite element models to simulate morphogenetic

movements in Drosophila embryo, limb morphogenesis, bone remodeling as well as single and

collective cell migration. The common feature of these models is the multiplicative decomposition of

the deformation gradient which has been used to take into account both the active and the passive

deformations undergone by the cells. I will show how this mechanical approach, firstly used in the

seventies by Lee and Mandel to describe large viscoelastic deformations, can actually be very

powerful in modeling the biological phenomena mentioned above.

• Sean Lim - Full waveform inversion: a first look
• Alex Raisch - Bistable liquid crystal displays: modelling, simulation and applications
• Vladimir Zubkov - Mathematical model of kidney morphogenesis

A mini-lecture series consisting of four 1 hour lectures.

We will discuss arithmetic restriction phenomena and its relation to Waring's problem, focusing on how recent work of Wooley implies certain restriction bounds.

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.