Past

I will explain how to define a notion of stable-independence in NIP

theories, which is an attempt to capture the "stable part" of types.

The displacement of a liquid by an air finger is a generic two-phase flow that

underpins applications as diverse as microfluidics, thin-film coating, enhanced

oil recovery, and biomechanics of the lungs. I will present two intriguing

examples of such flows where, firstly, oscillations in the shape of propagating

bubbles are induced by a simple change in tube geometry, and secondly, flexible

vessel boundaries suppress viscous fingering instability.

1) A simple change in pore geometry can radically alter the behaviour of a

fluid displacing air finger, indicating that models based on idealized pore

geometries fail to capture key features of complex practical flows. In

particular, partial occlusion of a rectangular cross-section can force a

transition from a steadily-propagating centred finger to a state that exhibits

spatial oscillations via periodic sideways motion of the interface at a fixed

location behind the finger tip. We characterize the dynamics of the

oscillations and show that they arise from a global homoclinic connection

between the stable and unstable manifolds of a steady, symmetry-broken

solution.

2) Growth of complex dendritic fingers at the interface of air and a viscous

fluid in the narrow gap between two parallel plates is an archetypical problem

of pattern formation. We find a surprisingly effective means of suppressing

this instability by replacing one of the plates with an elastic membrane. The

resulting fluid-structure interaction fundamentally alters the interfacial

patterns that develop and considerably delays the onset of fingering. We

analyse the dependence of the instability on the parameters of the system and

present scaling arguments to explain the experimentally observed behaviour.

The Arnoldi method for standard eigenvalue problems possesses several

attractive properties making it robust, reliable and efficient for

many problems. We will present here a new algorithm equivalent to the

Arnoldi method, but designed for nonlinear eigenvalue problems

corresponding to the problem associated with a matrix depending on a

parameter in a nonlinear but analytic way. As a first result we show

that the reciprocal eigenvalues of an infinite dimensional operator.

We consider the Arnoldi method for this and show that with a

particular choice of starting function and a particular choice of

scalar product, the structure of the operator can be exploited in a

very effective way. The structure of the operator is such that when

the Arnoldi method is started with a constant function, the iterates

will be polynomials. For a large class of NEPs, we show that we can

carry out the infinite dimensional Arnoldi algorithm for the operator

in arithmetic based on standard linear algebra operations on vectors

and matrices of finite size. This is achieved by representing the

polynomials by vector coefficients. The resulting algorithm is by

construction such that it is completely equivalent to the standard

Arnoldi method and also inherits many of its attractive properties,

which are illustrated with examples.

We discuss shock reflection problem for compressible gas dynamics, and von Neumann conjectures on transition between regular and Mach reflections. Then we will talk about some recent results on existence, regularity and geometric properties of regular reflection solutions for potential flow equation. In particular, we discuss optimal regularity of solutions near sonic curve, and stability of the normal reflection soluiton. Open problems will also

be discussed. The talk will be based on the joint work with Gui-Qiang Chen, and with Myoungjean Bae.

The mid 1980s saw a shift in the nature of the relationship between mathematics and physics. Differential equations and geometry applied in a classical setting were no longer the principal players; in the quantum world topology and algebra had come to the fore. In this talk we discuss a method of classifying 2-dim invertible Klein topological quantum field theories (KTQFTs). A key object of study will be the unoriented cobordism category $\mathscr{K}$, whose objects are closed 1-manifolds and whose morphisms are surfaces (a KTQFT is a functor $\mathscr{K}\rightarrow\operatorname{Vect}_{\mathbb{C}}$). Time permitting, the open-closed version of the category will be considered, yielding some surprising results.

Particle-based stochastic reaction-diffusion models have recently been used to study a number of problems in cell biology. These methods are of interest when both noise in the chemical reaction process and the explicit motion of molecules are important. Several different mathematical models have been used, some spatially-continuous and others lattice-based. In the former molecules usually move by Brownian Motion, and may react when approaching each other. For the latter molecules undergo continuous time random-walks, and usually react with fixed probabilities per unit time when located at the same lattice site.

As motivation, we will begin with a brief discussion of the types of biological problems we are studying and how we have used stochastic reaction-diffusion models to gain insight into these systems. We will then introduce several of the stochastic reaction-diffusion models, including the spatially continuous Smoluchowski diffusion limited reaction model and the lattice-based reaction-diffusion master equation. Our work studying the rigorous relationships between these models will be presented. Time permitting, we may also discuss some of our efforts to develop improved numerical methods for solving several of the models.

A perfect obstruction theory for a commutative ring is a morphism from a perfect complex to the cotangent complex of the ring

satisfying some further conditions. In this talk I will present work in progress on how to associate in a functorial manner commutative

differential graded algebras to such a perfect obstruction theory. The key property of the differential graded algebra is that its zeroth homology

is the ring equipped with the perfect obstruction theory. I will also indicate how the method introduced can be globalized to work on schemes

without encountering gluing issues.

We call a graph $H$ \emph{Ramsey-unsaturated} if there is an edge in the

complement of $H$ such that the Ramsey number $r(H)$ of $H$ does not

change upon adding it to $H$. This notion was introduced by Balister,

Lehel and Schelp who also showed that cycles (except for $C_4$) are

Ramsey-unsaturated, and conjectured that, moreover, one may add {\em

any} chord without changing the Ramsey number of the cycle $C_n$, unless

$n$ is even and adding the chord creates an odd cycle.

We prove this conjecture for large cycles by showing a stronger

statement: If a graph $H$ is obtained by adding a linear number of

chords to a cycle $C_n$, then $r(H)=r(C_n)$, as long as the maximum

degree of $H$ is bounded, $H$ is either bipartite (for even $n$) or

almost bipartite (for odd $n$), and $n$ is large.

This motivates us to call cycles \emph{strongly} Ramsey-unsaturated.

Our proof uses the regularity method.

Recent results (starting with Scheffer and Shnirelman and continuing with De Lellis and Szekelhyhidi ) underline the importance of considering solutions of the incompressible Euler equations as limits of solutions of more physical examples like Navier-Stokes or Boltzmann.
I intend to discuss several examples illustrating this issue.

In Riemannian geometry there are several notions of rank

defined for non-positively curved manifolds and with natural extensions

for groups acting on non-positively curved spaces.

The talk shall explain how various notions of rank behave for

mapping class groups of surfaces. This is joint work with J. Behrstock.

We will give a quick overview of the semigroup perspective on splitting schemes for S(P)DEs which present a robust, "easy to implement" numerical method for calculating the expected value of a certain payoff of a stochastic process driven by a S(P)DE. Having a high numerical order of convergence enables us to replace the Monte Carlo integration technique by alternative, faster techniques. The numerical order of splitting schemes for S(P)DEs is bounded by 2. The technique of combining several splittings using linear combinations which kills some additional terms in the error expansion and thus raises the order of the numerical method is called the extrapolation. In the presentation we will focus on a special extrapolation of the Lie-Trotter splitting: the symmetrically weighted sequential splitting, and its subsequent extrapolations. Using the semigroup technique their convergence will be investigated. At the end several applications to the S(P)DEs will be given.

Based on ideas from rough path analysis and operator splitting, the Kusuoka-Lyons-Victoir scheme provides a family of higher order methods for the weak approximation of stochastic differential equations. Out of this family, the Ninomiya-Victoir method is especially simple to implement and to adjust to various different models. We give some examples of models used in financial engineering and comment on the performance of the Ninomiya-Victoir scheme and some modifications when applied to these models.

String theory on a torus requires the introduction of dual coordinates

conjugate to string winding number. This leads to physics and novel geometry in a doubled space. This will be

compared to generalized geometry, which doubles the tangent space but not the manifold.

For a d-torus,   string theory can be formulated in terms of an infinite

tower of fields depending on both the d torus coordinates and the d dual

coordinates. This talk focuses on a finite subsector  consisting of a metric

and B-field (both d x d matrices) and a dilaton all depending on the 2d

doubled torus coordinates.

The double field theory is constructed and found to have a novel symmetry

that reduces to diffeomorphisms and anti-symmetric tensor gauge

transformations in certain circumstances. It also has manifest T-duality

symmetry which provides a generalisation of the usual Buscher rules to

backgrounds without isometries. The theory has a real dependence on the full

doubled geometry:  the dual dimensions are not auxiliary. It is concluded

that the doubled geometry is physical and dynamical.

Inverse methods are frequently used in geosciences to estimate model parameters from indirect measurements. A common inverse problem encountered when modelling the flow of large ice masses such as the Greenland and the Antarctic ice sheets is the determination of basal conditions from surface data. I will present an overview over some of the inverse methods currently used to tackle this problem and in particular discuss the use of Bayesian inverse methods in this context. Examples of the use of adjoint methods for large-scale optimisation problems that arise, for example, in flow modelling of West-Antarctica will be given.

Absence of arbitrage is a highly desirable feature in mathematical models of financial markets. In its pure form (whether as NFLVR or as the existence of a variant of an equivalent martingale measure R), it is qualitative and therefore robust towards equivalent changes of the underlying reference probability (the "real-world" measure P). But what happens if we look at more quantitative versions of absence of arbitrage, where we impose for instance some integrability on the density dR/dP? To which extent is such a property robust towards changes of P? We discuss these uestions and present some recent results.

The talk is based on joint work with Tahir Choulli (University of Alberta, Edmonton).

In this lecture I will exploit a model of asset prices where speculators overconfidence is a source of heterogeneous beliefs and arbitrage is limited. In the model, asset buyers are the most positive investors, but prices exceed their optimistic valuation because the owner of an asset has the option of reselling it in the future to an even more optimistic buyer. The value of this resale option can be identified as a bubble. I will focus on assets with a fixed terminal date, as is often the case with credit instruments. I will show that the size of a bubble satisfies a Partial Differential Equation that is similar to the equation satisfied by an American option and use the PDE to evaluate the impact of parameters such as interest rates or a “Tobin tax” on the size of the bubble and on trading volume.