Past

We study the asymptotic behavior of deterministic dynamics of many interacting particles with random initial data in the limit where the number of particles tends to infinity. A famous example is hard sphere flow, we restrict our attention to the simpler case where particles are removed after the first collision. A fixed number of particles is drawn randomly according to an initial density $f_0(u,v)$ depending on $d$-dimensional position $u$ and velocity $v$. In the Boltzmann Grad scaling, we derive the validity of a Boltzmann equation without gain term for arbitrary long times, when we assume finiteness of moments up to order two and initial data that are $L^\infty$ in space. We characterize the many particle flow by collision trees which encode possible collisions. The convergence of the many-particle dynamics to the Boltzmann dynamics is achieved via the convergence of associated probability measures on collision trees. These probability measures satisfy nonlinear Kolmogorov equations, which are shown to be well-posed by semigroup methods.

 Many geophysical flows over topography can be modeled by two-dimensional
depth-averaged fluid dynamics equations.  The shallow water equations
are the simplest example of this type, and are often sufficiently
accurate for simulating tsunamis and other large-scale flows such
as storm surge.  These hyperbolic partial differential equations
can be modeled using high-resolution finite volume methods.  However,
several features of these flows lead to new algorithmic challenges,
e.g. the need for well-balanced methods to capture small perturbations
to the ocean at rest, the desire to model inundation and flooding,
and that vastly differing spatial scales that must often be modeled,
making adaptive mesh refinement essential. I will discuss some of
the algorithms implemented in the open source software GeoClaw that
is aimed at solving real-world geophysical flow problems over
topography.  I'll also show results of some recent studies of the
11 March 2011 Tohoku Tsunami and discuss the use of tsunami modeling
in probabilistic hazard assessment.

Robust pricing of an exotic derivative with payoff $\Phi$ can be viewed as the task of estimating its expectation $E_Q \Phi$ with respect to a martingale measure $Q$ satisfying marginal constraints. It has proven fruitful to relate this to the theory of Monge-Kantorovich optimal transport. For instance, the duality theorem from optimal transport leads to new super-replication results. Optimality criteria from the theory of mass transport can be translated to the martingale setup and allow to characterize minimizing/maximizing models in the robust pricing problem. Moreover, the dual viewpoint provides new insights to the classical inequalities of Doob and Burkholder-Davis-Gundy.

• Joseph Parker - Numerical algorithms for the gyrokinetic equations and applications to magnetic confinement fusion
• Rita Schlackow - Global and functional analyses of 3' untranslated regions in fission yeast
• Peter Stewart - Creasing and folding of fibre-reinforced materials

PLEASE NOTE EARLY START TIME TO AVOID CLASH WITH OCCAM GROUP MEETING

The human influenza A virus causes three to five million cases of severe illness and about 250 000 to 500 000 deaths each year. The 1918 Spanish Flu may have killed more than 40 million people. Yet, the underlying cause of the seasonality of the human influenza virus, its preferential transmission in winter in temperate climates, remains controversial. One of the major forms of the human influenza virus is a sphere made up of lipids selectively derived from the host cell along with specialized viral proteins. I have employed molecular dynamics simulations to study the biophysical properties of a single transmissible unit--an approximately spherical influenza A virion in water (i.e., to mimic the water droplets present in normal transmission of the virus). The surface area per lipid can't be calculated as a ratio of the surface area of the sphere to the number of lipids present as there are many different species of lipid for which different surface area values should be calculated. The 'mosaic' of lipid surface areas may be regarded quantitatively as a Voronoi diagram, but construction of a true spherical Voronoi tessellation is more challenging than the well-established methods for planar Voronoi diagrams. I describe my attempt to implement an approach to the spherical Voronoi problem (based on: Hyeon-Suk Na, Chung-Nim Lee, Otfried Cheong. Computational Geometry 23 (2002) 183–194) and the challenges that remain in the implementation of this algorithm.

Newelski suggested that topological dynamics could be used to extend "stable group theory" results outside the stable context. Given a group G, it acts on the left on its type space S_G(M), i.e. (G,S_G(M)) is a G-flow. If every type is definable, S_G(M) can be equipped with a semigroup structure *, and it is isomorphic to the enveloping Ellis semigroup of the flow. The topological dynamics of (G,S_G(M)) is coded in the Ellis semigroup and in its minimal G-invariant subflows, which coincide with the left ideals I of S_G(M). Such ideals contain at least an idempotent r, and r*I forms a group, called "ideal group". Newelski proved that in stable theories and in o-minimal theories r*I is abstractly isomorphic to G/G^{00} as a group. He then asked if this happens for any NIP theory. Pillay recently extended the result to fsg groups; we found instead a counterexample to Newelski`s conjecture in SL(2,R), for which G/G^{00} is trivial but we show r*I has two elements. This is joint work with Jakub Gismatullin and Anand Pillay.

In this talk I shall describe two rather different, but not entirely unrelated,

problems involving thin-film flow of a viscous fluid which I have found of interest

and which may have some application to a number of practical situations,

including condensation in heat exchangers and microfluidics.

The first problem,

which is joint work with Adam Leslie and Brian Duffy at the University of Strathclyde,

concerns the steady three-dimensional flow of a thin, slowly varying ring of fluid

on either the outside or the inside of a uniformly rotating large horizontal cylinder.

Specifically, we study ``full-ring'' solutions, corresponding to a ring of continuous,

finite and non-zero thickness that extends all the way around the cylinder.

These full-ring solutions may be thought of as a three-dimensional generalisation of

the ``full-film'' solutions described by Moffatt (1977) for the corresponding two-dimensional problem.

We describe the behaviour of both the critical and non-critical full-ring solutions.

In particular,

we show that, while for most values of the rotation speed and the load the azimuthal velocity is

in the same direction as the rotation of the cylinder, there is a region of parameter space close

to the critical solution for sufficiently small rotation speed in which backflow occurs in a

small region on the upward-moving side of the cylinder.

The second problem,

which is joint work with Phil Trinh and Howard Stone at Princeton University,

concerns a rigid plate moving steadily on the free surface of a thin film of fluid.

Specifically, we study two problems

involving a rigid flat (but not, in general, horizontal) plate:

the pinned problem, in which the upstream end of plate is pinned at a fixed position,

the fluid pressure at the upstream end of the plate takes a prescribed value and there is a free surface downstream of the plate, and

the free problem, in which the plate is freely floating and there are free surfaces both upstream and downstream of the plate.

For both problems, the motion of the fluid and the position of the plate

(and, in particular, its angle of tilt to the horizontal) depend in a non-trivial manner on the

competing effects of the relative motion of the plate and the substrate,

the surface tension of the free surface, and of the viscosity of the fluid,

together with the value of the prescribed pressure in the pinned case.

Specifically, for the pinned problem we show that,

depending on the value of an appropriately defined capillary number and on the value of the

prescribed fluid pressure, there can be either none, one, two or three equilibrium solutions

with non-zero tilt angle.

Furthermore, for the free problem we show that the solutions

with a horizontal plate (i.e.\ zero tilt angle) conjectured by Moriarty and Terrill (1996)

do not, in general, exist, and in fact there is a unique equilibrium solution with,

in general, a non-zero tilt angle for all values of the capillary number.

Finally, if time permits some preliminary results for an elastic plate will be presented.

Part of this work was undertaken while I was a

Visiting Fellow in the Department of Mechanical and Aerospace Engineering

in the School of Engineering and Applied Science at Princeton University, Princeton, USA.

Another part of this work was undertaken while I was a

Visiting Fellow in the Oxford Centre for Collaborative Applied Mathematics (OCCAM),

University of Oxford, United Kingdom.

This publication was based on work supported in part by Award No KUK-C1-013-04,

made by King Abdullah University of Science and Technology (KAUST).

We will discuss the Littlewood conjecture from Diophantine approximation, and recent variants of the conjecture in which one of the real components is replaced by a p-adic absolute value (or more generally a "pseudo-absolute value''). The Littlewood conjecture has a dynamical formulation in terms of orbits of the action of the diagonal subgroup on SL_3(R)/SL_3(Z). It turns out that the mixed version of the conjecture has a similar formulation in terms of homogeneous dynamics, as well as meaningful connections to several other dynamical systems. This allows us to apply tools from combinatorics and ergodic theory, as well as estimates for linear forms in logarithms, to obtain new results.

Fix a connected manifold-with-boundary M and a closed, connected submanifold P of its boundary. The set of all possible submanifolds of M whose components are pairwise unlinked and each isotopic to P can be given a natural topology, and splits into a disjoint union depending on the number of components of the submanifold. When P is a point this is just the usual (unordered) configuration space on M. It is a classical result, going back to Segal and McDuff, that for these spaces their homology in any fixed degree is eventually independent of the number of points of the configuration (as the number of points goes to infinity). I will talk about some very recent work on extending this result to higher-dimensional submanifolds: in the above setup, as long as P is of sufficiently large codimension in M, the homology in any fixed degree is eventually independent of the number of components. In particular I will try to give an idea of how the codimension restriction arises, and how it can be improved in some special cases.

ALE formulations are useful when approximating solutions of problems in deformable domains, such as fluid-structure interactions. For realistic simulations involving fluids in 3d, it is important that the ALE method is at least of second order of accuracy. Second order ALE methods in time, without any constraint on the time step, do not exist in the literature and the role of the so-called geometric conservation law (GCL) for stability and accuracy is not clear. We propose discontinuous Galerkin (dG) methods of any order in time for a time dependent advection-diffusion model problem in moving domains. We prove that our proposed schemes are unconditionally stable and that the conservative and non conservative formulations are equivalent. The same results remain true when appropriate quadrature is used for the approximation of the involved integrals in time. The analysis hinges on the validity of a discrete Reynolds' identity and generalises the GCL to higher order methods. We also prove that the computationally less intensive Runge-Kutta-Radau (RKR) methods of any order are stable, subject to a mild ALE constraint. A priori and a posteriori error analysis is provided. The final estimates are of optimal order of accuracy. Numerical experiments confirm and complement our theoretical results.

This is joint work with Andrea Bonito and Ricardo H. Nochetto.

Market impact, leverage, systemic risk, and the perils of mark-to-market accounting

Market impact is the price change associated with new buy or sell orders entering the market. It provides a useful alternative to thinking in terms of supply and demand for several reasons, the most important being that there is theoretical and empirical evidence that it follows a universal law. Understanding market impact is essential for adjusting investment size, for optimizing execution tactics, and provides a useful tool for understanding market ecology and systemic risk. I will present a new method for impact-adjusted accounting, and show how it can avoid the serious problems of marking-to-market when leverage is used. Then I will discuss how market impact can be combined with network theory to understand the problem of overlapping portfolios and market crowding. Since I am a new faculty member, at the beginning of the talk I will say a bit about my interests and current projects.

Recent advances in engineered muscle tissue attached to a synthetic substrate motivates the development of appropriate constitutive and numerical models. Applications of active materials can be expanded by using robust, non-mammalian muscle cells, such as those of Manduca sexta. In this talk we present a   continuum model that accounts for the stimulation of muscle fibers by introducing multiple stress-free reference configurations and for the hysteretic response by specifying a pseudo-elastic energy function. A simple example representing uniaxial loading-unloading is used to validate and verify the characteristics of the model. Then, based on experimental data of muscular thin films, a more complex case shows the qualitative potential of Manduca muscle tissue in active biohybrid constructs.

 We present a new, more conceptual proof of our result that, when a finite group acts on a polynomial ring, the regularity of the ring of invariants is at most zero, and hence one can write down bounds on the degrees of the generators and relations. This new proof considers the action of the group on the Cech complex and looks at when it splits over the group algebra. It also applies to a more general class of rings than just polynomial ones.

How many nodal degree d plane curves are tangent to a given line? The celebrated Caporaso-Harris recursion formula gives a complete answer for any number of nodes, degrees, and all possible tangency conditions. In this talk, I will report my recent work on the generalization of the above problem to count singular curves with given tangency condition to a fixed smooth divisor on general surfaces. I will relate the enumeration to tautological integrals on Hilbert schemes of points and show the numbers of curves in question are given by universal polynomials. As a result, we can obtain infinitely many new formulas for nodal curves and understand the asymptotic behavior for all singular curves with any tangency conditions.

We will discuss the origin of the conjectured colour-kinematics

duality in perturbative gauge theory, according to which there is a

symmetry between the colour dependence and the kinematic dependence of the

S-matrix. Based on this duality, there is a prescription to obtain gravity

amplitudes as the "double copy" of gauge theory amplitudes. We will first

consider tree-level amplitudes, where a diffeomorphism algebra underlies

the structure of MHV amplitudes, mirroring the colour algebra. We will

then draw on the progress at tree-level to consider one-loop amplitudes.

Topological field theories give a connection between

topology and algebra. This connection can be exploited in both

directions: using algebra to construct topological invariants, or

using topology to prove algebraic theorems. In this talk, I will

explain an interesting example of the latter phenomena. Radford's

theorem, as generalized by Etingof-Nikshych-Ostrik, says that in a

finite tensor category the quadruple dual functor is easy to

understand. It's somewhat mysterious that the double dual is hard to

understand but the quadruple dual is easy. Using topological field

theory, we show that Radford's theorem is exactly the consequence of

the Dirac belt trick in topology. That is, the double dual

corresponds to the generator of $\pi_1(\mathrm{SO}(3))$ and so the

quadruple dual is trivial in an appropriate sense exactly because

$\pi_1(\mathrm{SO}(3)) \cong \mathbb{Z}/2$. This is part of a large

project, joint with Chris Douglas and Chris Schommer-Pries, to

understand local field theories with values in the 3-category of

tensor categories via the cobordism hypothesis.

Given a flat torus, we consider certain discrete graph approximations of

it and determine the asymptotics of the number of spanning trees

("complexity") of these graphs as the mesh gets finer. The constants in the

asymptotics involve various notions of determinants such as the

determinant of the Laplacian ("height") of the torus. The analogy between

the complexity of graphs and the height of manifolds was previously

commented on by Sarnak and Kenyon. In dimension two, similar asymptotics

were established earlier by Barber and Duplantier-David in the context of

statistical physics.

Our proofs rely on heat kernel analysis involving Bessel functions, which

in the torus case leads into modular forms and Epstein zeta functions. In

view of a folklore conjecture it also suggests that tori corresponding to

densest regular sphere packings should have approximating graphs with the

largest number of spanning trees, a desirable property in network theory.

Joint work with G. Chinta and J. Jorgenson.

We construct explicitly a bridge process whose distribution, in its own filtration, is the same as the difference of two independent Poisson processes with the same intensity and its time 1 value satisfies a specific constraint. This construction allows us to show the existence of Glosten-Milgrom equilibrium and its associated optimal trading strategy for the insider. In the equilibrium the insider employs a mixed strategy to randomly submit two types of orders: one type trades in the same direction as noise trades while the other cancels some of the noise trades by submitting opposite orders when noise trades arrive. The construction also allows us to prove that Glosten-Milgrom equilibria converge weakly to Kyle-Back equilibrium, without the additional assumptions imposed in \textit{K. Back and S. Baruch, Econometrica, 72 (2004), pp. 433-465}, when the common intensity of the Poisson processes tends to infinity. This is a joint work with Umut Cetin.

Ice-dammed lakes form next to, on the surface of, and beneath glaciers

and ice sheets. Some lakes are known to drain catastrophically,

creating hazards, wasting water resources and modulating the flow of

the adjacent ice. My work aims to increase our understanding of such

drainage. Here I will focus on lakes that form next to glaciers and

drain subglacially (between ice and bedrock) through a channel. I will

describe how such a system can be modelled and present results from

model simulations of a lake that fills due to an input of meltwater

and drains through a channel that receives a supply of meltwater along

its length. Simulations yield repeating cycles of lake filling and

drainage and reveal how increasing meltwater input to the system

affects these cycles: enlarging or attenuating them depending on how

the meltwater is apportioned between the lake and the channel. When

inputs are varied with time, simulating seasonal meteorological

cycles, the model simulates either regularly repeating cycles or

irregular cycles that never repeat. Irregular cycles demonstrate

sensitivity to initial conditions, a high density of periodic orbits

and topological mixing. I will discuss how these results enhance our

understanding of the mechanisms behind observed variability in these

systems.

This is the session for our industrial sponsors to propose project ideas. Academic staff are requested to attend to help shape the problem statements and to suggest suitable internal supervisors for the projects. 

Recent years have seen increasing attention to the subtle effects on

intracellular transport caused when multiple molecular motors bind to

a common cargo. We develop and examine a coarse-grained model which

resolves the spatial configuration as well as the thermal fluctuations

of the molecular motors and the cargo. This intermediate model can

accept as inputs either common experimental quantities or the

effective single-motor transport characterizations obtained through

systematic analysis of detailed molecular motor models. Through

stochastic asymptotic reductions, we derive the effective transport

properties of the multiple-motor-cargo complex, and provide analytical

explanations for why a cargo bound to two molecular motors moves more

slowly at low applied forces but more rapidly at high applied forces

than a cargo bound to a single molecular motor. We also discuss how

our theoretical framework can help connect in vitro data with in vivo

behavior.

The Discontinuous Galerkin (DG) method has been used to solve a wide range of partial differential equations. Especially for advection dominated problems it has proven very reliable and accurate. But even for elliptic problems it has advantages over continuous finite element methods, especially when parallelization and local adaptivity are considered.

In this talk we will first present a variation of the compact DG method for elliptic problems with varying coefficients. For this method we can prove stability on general grids providing a computable bound for all free parameters. We developed this method to solve the compressible Navier-Stokes equations and demonstrated its efficiency in the case of meteorological problems using our implementation within the DUNE software framework, comparing it to the operational code COSMO used by the German weather service.

After introducing the notation and analysis for DG methods in Euclidean spaces, we will present a-priori error estimates for the DG method on surfaces. The surface finite-element method with continuous ansatz functions was analysed a few years ago by Dzuik/Elliot; we extend their results to the interior penalty DG method where the non-smooth approximation of the surface introduces some additional challenges.

In 1977, Cline Parshall, Scott and van der Kallen wrote a seminal paper `Rational and generic cohomology' which exhibited a connection between the cohomology for algebraic groups and the cohomology for finite groups of Lie type, showing that in many cases one can conclude that there is an isomorphism of cohomology through restriction from the algebraic to the finite group.

One unfortunate problem with their result is that there remain infinitely many modules for which their theory---for good reason---tells us nothing. The main result of this talk (recent work with Parshall and Scott) is to show that almost all the time, one can manipulate the simple modules for finite groups of Lie type in such a way as to recover an isomorphism of its cohomology with that of the algebraic group.

H. Johnston and J.G. Liu proposed in 2004 a numerical scheme for approximating numerically solutions of the incompressible Navier-Stokes system. The scheme worked very well in practice but its analytic properties remained elusive.\newline

In order to understand these analytical aspects they considered together with R. Pego a continuous version of it that appears as an extension of the incompressible Navier-Stokes to vector-fields that are not necessarily divergence-free. For divergence-free initial data one has precisely the incompressible Navier-Stokes, while for non-divergence free initial data, the divergence is damped exponentially.\newline

We present analytical results concerning this extended system and discuss numerical implications. This is joint work with R. Pego, G. Iyer (Carnegie Mellon) and J. Kelliher, M. Ignatova (UC Riverside).

Saturated fusion systems are a next generation approach to the theory of finite groups- one major motivation being the opportunity to borrow techniques from homotopy theory. Extending work of Broto, Levi and Oliver, we introduce a new object - a 'tree of fusion systems' and give conditions (in terms of the orbit graph) for the completion to be saturated. We also demonstrate that these conditions are 'best possible' by producing appropriate counterexamples. Finally, we explain why these constructions provide a powerful way of building infinite families of fusion systems which are exotic (i.e. not realisable as the fusion system of a finite group) and give some concrete examples.

Although the importance of hydrologic uncertainty is widely recognized it is rarely considered in control problems, especially real-time control. One of the reasons is that stochastic control is computationally expensive, especially when control decisions are derived from spatially distributed models. This talk reviews relevant control concepts and describes how reduced order models can make stochastic control feasible for computationally demanding applications. The ideas are illustrated with a classic problem -- hydraulic control of a moving contaminant plume.

Let $X$ be a variety and $Z$ be a sub-variety. Can one glue vector bundles on $X-Z$ with vector bundles on some small neighborhood of $Z$? We survey two recent results on the process of gluing a vector bundle on the complement of a sub-variety with a vector bundle on some 'small' neighborhood of the sub-variety. This is joint work. The first with M. Temkin and is about gluing categories of coherent sheaves over the category of coherent sheaves on a Berkovich analytic space. The second with J. Block and is about gluing dg enhancements of the derived category of coherent sheaves.

In an Erdős--R\'enyi random graph above the phase transition, i.e.,

where there is a giant component, the size of (number of vertices in)

this giant component is asymptotically normally distributed, in that

its centred and scaled size converges to a normal distribution. This

statement does not tell us much about the probability of the giant

component having exactly a certain size. In joint work with B\'ela

Bollob\'as we prove a `local limit theorem' answering this question

for hypergraphs; the graph case was settled by Luczak and Łuczak.

The proof is based on a `smoothing' technique, deducing the local

limit result from the (much easier) `global' central limit theorem.

High-pressure freezing processes are a novel emerging technology in food processing,
offering significant improvements to the quality of frozen foods. To be able to simulate
plateau times and thermal history under different conditions, a generalized enthalpy
model of the high-pressure shift freezing process is presented. The model includes
the effects of pressure on conservation of enthalpy and incorporates the freezing point
depression of non-dilute food samples. In addition, the significant heat-transfer effects of
convection in the pressurizing medium are accounted for by solving the two-dimensional
Navier–Stokes equations.
The next question is: is high-pressure shift freezing good also in the long run?
A growth and coarsening model for ice crystals in a very simple food system will be discussed.

Topological quantum error correcting codes, such as the Toric code, are
ideal candidates for protecting a logical quantum bit against local noise.
How are we to get the best performance from these codes when an unknown
local perturbation is applied? This talk will discuss how knowledge, or lack
thereof, about the error affects the error correcting threshold, and how
thresholds can be improved by introducing randomness to the system. These
studies are directed at trying to understand how quantum information can be
encoded and passively protected in order to maximise the span of time between successive rounds of error correction, and what properties are
required of a topological system to induce a survival time that grows
sufficiently rapidly with system size. The talk is based on the following
papers: arXiv:1208.4924 and Phys. Rev. Lett. 107, 270502 (2011).

The Preisach model of hysteresis has a long history, a convenient algorithmic form and "nice" mathematical properties (for a given value of nice) that make it suitable for use in differential equations and other dynamical systems. The difficulty lies in the fact that the "parameter" for the Preisach model is infinite dimensional—in full generality it is a measure on the half-plane. Applications of the Preisach model (two interesting examples are magnetostrictive materials and vadose zone hydrology) require methods to specify a measure based on experimental or observed data. Current approaches largely rely on direct measurements of experimental samples, however in some cases these might not be sufficient or direct measurements may not be practical. I will present the Preisach model in all its glory, along with some history and applications, and introduce an open inverse problem of fiendish difficulty.

This presentation concerns five topics in computational partial differential equations with the overall goals of reliable error control and efficient simulation.

The presentation is also an advertisement for nonstandard discretisations in linear and nonlinear Computational PDEs with surprising advantages over conforming

finite element schemes and the combination

of the two. The equivalence of various first-order methods is explained for the linear Poisson model problem with conforming

(CFEM), nonconforming (NC-FEM), and mixed finite element methods (MFEM) and others discontinuous Galerkin finite element (dGFEM). The Stokes

equations illustrate the NCFEM and the pseudo-stress MFEM and optimal convergence of adaptive mesh-refining as well as for guaranteed error bounds.

An optimal adaptive CFEM computation of elliptic eigenvalue

problems and the computation of guaranteed upper and lower eigenvalue bounds based on NCFEM. The obstacle problem and its guaranteed error

control follows another look due to D. Braess with guaranteed error bounds and their effectivity indices between 1 and 3. Some remarks on computational

microstructures with degenerate convex minimisation

problems conclude the presentation.