Past

The ``k-commodity flow problem'' is: we are given k pairs of vertices of a graph, and we ask whether there are k flows in the graph, where the ith flow is between the ith pair of vertices, and has total value one, and for each edge, the sum of the absolute values of the flows along it is at most one. We may also require the flows to be 1/2-integral, or indeed 1/p-integral for some fixed p.

If the problem is feasible (that is, the desired flows exist) then it is still feasible after contracting any edge, so let us say a flow problem is ``critical'' if it is infeasible, but becomes feasible when we contract any edge. In many special cases, all critical instances have only two vertices, but if we ask for integral flows (that is, p = 1, essentially the edge-disjoint paths problem), then there arbitrarily large critical instances, even with k = 2. But it turns out that p = 1 is the only bad case; if p>1 then all critical instances have bounded size (depending on k, but independent of p), and the same is true if there is no integrality requirement at all.

The proof gives rise to a very simple algorithm for the k edge-disjoint paths problem in 4-edge-connected graphs.

Cutting-edge experiments in quantum communications are reaching regimes

where relativistic effects can no longer be neglected. For example, there

are advanced plans to use satellites to implement teleportation and quantum

cryptographic protocols. Relativistic effects can be expected at these

regimes: the Global Positioning System (GPS), which is a system of

satellites that is used for time dissemination and navigation, requires

relativistic corrections to determine time and positions accurately.

Therefore, it is timely to understand what are the effects of gravity and

motion on entanglement and other quantum properties exploited in quantum

information.

In this talk I will show that entanglement can be created or degraded by

gravity and non-uniform motion. While relativistic effects can degrade the

efficiency of teleportation between moving observers, the effects can also

be exploited in quantum information. I will show that the relativistic

motion of a quantum system can be used to perform quantum gates. Our

results, which will inform future space-based experiments, can be

demonstrated in table-top experiments using superconducting circuits.

This talk is regarding PDE systems from geophysical applications with multiple time scales, in which linear skew-self-adjoint operators of size 1/epsilon gives rise to highly oscillatory solutions. Analysis is performed in justifying the limiting dynamics as epsilon goes to zero; furthermore, the analysis yields estimates on the difference between the multiscale solution and the limiting solution. We will introduce a simple yet effective time-averaging technique which is especially useful in general domains where Fourier analysis is not applicable.

Random wavelet series were introduced in the mid 90s as simple and flexible models that allow to take into account observed statistics of wavelet coefficients in signal and image processing. One of their most interesting properties is that they supply random processes whose pointwise regularity jumps form point to point in a very erratic way, thus supplying examples of multifractal processes.

Interest in such models has been renewed recently under the spur of new applications coming from widely different fields; e.g.

-in functional analysis, they allow to derive the regularity properties of ``generic'' functions in a given function space (in the sense of

prevalence)

-they offer toy examples on which one can check the accuracy of numerical algorithms that allow to derive the multifractal parameters associated with signals and images.

We will give an overview of these properties, and we will focus on recent extensions whose sample paths are not locally bounded, and offer models for signals which share this property.

 In this talk I will introduce my joint work with Kreck on a classification of
certain 5-manifolds with fundamental group Z. This result can be interpreted as a
generalization of the classical Browder-Levine's fibering theorem to dimension 5.

The talk will survey some results about smooth and topological 4-manifolds obtained via surgery, and discuss some contrasting information provided by gauge theory about smooth finite group actions on 4-manifolds.

Free probability theory has been introduced by Voiculescu in the 80's for the study of the von Neumann algebras of the free groups. It consists in an algebraic setting of non commutative probability, where one encodes "non commutative random variables" in abstract (non commutative) algebras endowed with linear forms (which satisfies properties in order to play the role of the expectation). In this context, Voiculescu introduce the notion of freeness which is the analogue of the classical independence.

A decade later, he realized that a family of independent random matrices invariant in law by conjugation by unitary matrices are asymptotically free. This phenomenon is called asymptotic freeness. It had a deep impact in operator algebra and probability and has been generalized in many directions. A simple particular case of Voiculescu's theorem gives an estimate, for N large, of the spectrum of an N by N Hermitian matrix H_N = A_N + 1/\sqrt N X_N, where A_N is a given deterministic Hermitian matrix and X_N has independent gaussian standard sub-diagonal entries.

Nevertheless, it turns out that asymptotic freeness does not hold in certain situations, e.g. when the entries of X_N as above have heavy-tails. To infer the spectrum of a larger class of matrices, we go further into Voiculescu's approach and introduce the distributions of traffics and their free product. This notion of distribution is richer than Voiculescu's notion of distribution of non commutative random variables and it generalizes the notion of law of a random graph. The notion of freeness for traffics is an intriguing mixing between the classical independence and Voiculescu's notion of freeness. We prove an asymptotic freeness theorem in that context for independent random matrices invariant in law by conjugation by permutation matrices.

The purpose of this talk is to give an introductory presentation of these notions.

We discuss the superhedging problem under model uncertainty based on existence

and duality results for minimal supersolutions of backward stochastic differential equations.

The talk is based on joint works with Samuel Drapeau, Gregor Heyne and Reinhard Schmidt.

Nowadays there are a large number of satellite and airborne observations of the large ice sheet that covers Antarctica. These include maps of the surface elevation, ice thickness, surface velocity, the rate of snow accumulation, and the rate of change of surface elevation. Uncertainty in the possible rate of future sea level rise motivates using all of these observations and models of ice-sheet flow to project how the ice sheet will behave in future, but this is still a challenge. To make useful predictions, especially in the presence of potential dynamic instabilities, models will need accurate initial conditions, including flow velocity throughout the ice thickness. The ice sheet can be several kilometres thick, but most of the observations identify quantities at the upper surface of the ice sheet, not within its bulk. There is thus a question of how the subsurface flow can be inferred from surface observations. The key parameters that must be identified are the viscosity in the interior of the ice and the basal drag coefficient that relates the speed of sliding at the base of the ice sheet to the basal shear stress. Neither is characterised well by field or laboratory studies, but for incompressible flow governed by the Stokes equations they can be investigated by inverse methods analogous to those used in electric impedance tomography (which is governed by the Laplace equation). Similar methods can also be applied to recently developed 'hybrid' approximations to Stokes flow that are designed to model shallow ice sheets, fast-sliding ice streams, and floating ice shelves more efficiently. This talk will give a summary of progress towards model based projections of the size and shape of the Antarctic ice sheet that make use of the available satellite data. Some of the outstanding problems that will need to be tackled to improve the accuracy of these projections will also be discussed.

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.