Past

This is not intended to be a systematic History, but a selection of highlights, with some digressions, including:

The early days of the Computing Lab;

How the coming of the Computer changed some of the ways we do Computation;

A problem from the Study Groups;

Influence of the computing environment (hardware and software);

Convergence analysis for the heat equation, then and now.

A function $u: \mathbb{R}^{n} \to \mathbb{R}^{m}$ is one-homogeneous if $u(ax)=au(x)$ for any positive real number $a$ and all $x$ in $\R^{n}$. Phillips(2002) showed that in two dimensions such a function cannot solve an elliptic system in divergence form, in contrast to the situation in higher dimensions where various authors have constructed one-homogeneous minimizers of regular variational problems. This talk will discuss an extension of Phillips's 2002 result to $x-$dependent systems. Some specific one-homogeneous solutions will be constructed in order to show that certain of the hypotheses of the extension of the Phillips result can't be dropped. The method used in the construction is related to nonlinear elasticity in that it depends crucially on polyconvex functions $f$ with the property that $f(A) \to \infty$ as $\det A \to 0$.

We identify a natural way of ordering functions, which we call the interval dominance order, and show that this concept is useful in the theory of monotone comparative statics and also in statistical decision theory. This ordering on functions is weaker than the standard one based on the single crossing property (Milgrom and Shannon, 1994) and so our monotone comparative statics results apply in some settings where the single crossing property does not hold. For example, they are useful when examining the comparative statics of optimal stopping time problems. We also show that certain basic results in statistical decision theory which are important in economics - specifically, the complete class theorem of Karlin and Rubin (1956) and the results connected with Lehmann's (1988) concept of informativeness – generalize to payoff functions that obey the interval dominance order.

Geometrically, the problem of descent asks when giving some structure on a space is the same as giving some structure on a cover of the space, plus perhaps some extra data.
In algebraic geometry, faithfully flat descent says that if $X\rightarrow Y$ is a faithfully flat morphism of schemes, then giving a sheaf on $Y$ is the same as giving a collection of sheaves on a certain simplicial resolution constructed from $X$, satisfying certain compatibility conditions. Translated to algebra, it says that if $S\rightarrow R$ is a faithfully flat morphism of rings, then giving an $S$-module is the same as giving a certain simplical module over a simplicial ring constructed from $R$. In topology, given an etale cover $X\rightarrow Y$ one can recover $Y$ (at least up to homotopy equivalence) from a simplical space constructed from $X$.

How can one guarantee the presence of an oriented four-cycle in an oriented graph G? We shall see, that one way in which this can be done, is to demand that G contains no large `biased. subgraphs; where a `biased. subgraph simply means a subgraph whose orientation exhibits a strong bias in one direction.

Furthermore, we discuss the concept of biased subgraphs from another standpoint, asking: how can an oriented graph best avoid containing large biased subgraphs? Do random oriented graphs give the best examples? The talk is partially based on joint work with Omid Amini and Florian Huc.

There is a non-degenerate 2-form on S^6, which is compatible with the almost complex structure that S^6 inherits from its inclusion in the imaginary octonions. Even though this 2-form is not closed, we may still define Lagrangian submanifolds. Surprisingly, they are automatically minimal and are related to calibrated geometry. The focus of this talk will be on the Lagrangian submanifolds of S^6 which are fibered by geodesic circles over a surface. I will describe an explicit classification of these submanifolds using a family of Weierstrass formulae.

In the lecture, I am going to explain a connection between

local regularity theory for the Navier-Stokes equations

and Liouville type theorems for bounded ancient solutions to

these equations.

I talk about a recent article of mine that aims at giving an alternative proof to a formula by Carne on random walks. Consider a discrete, reversible random walk on a graph (not necessarily the simple walk); then one has a surprisingly simple formula bounding the probability of getting from a vertex x at time 0 to another vertex y at time t, where it appears a universal Gaussian factor essentially depending on the graph distance between x and y. While Carne proved that result in 1985, through‘miraculous’ (though very pretty!) spectral analysis reasoning, I will expose my own ‘natural' probabilistic proof of that fact. Its main interest is philosophical, but it also leads to a generalization of the original formula. The two main tools we shall use will be techniques of forward and backward martingales, and a tricky conditioning argument to prevent a random walk from being `’too transient'.

One of the outstanding successes of mathematical population genetics is Kingman's coalescent. This process provides a simple and elegant description of the genealogical trees relating individuals in a sample of neutral genes from a panmictic population, that is, one in which every individual is equally likely to mate with every other and all individuals experience the same conditions. But real populations are not like this. Spurred on by the recent flood of DNA sequence data, an enormous industry has developed that seeks to extend Kingman's coalescent to incorporate things like variable population size, natural selection and spatial and genetic structure. But a satisfactory approach to populations evolving in a spatial continuum has proved elusive. In this talk we describe the effects of some of these biologically important phenomena on the genealogical trees before describing a new approach (joint work with Nick Barton, IST Austria) to modelling the evolution of populations distributed in a spatial continuum.

Abstract: It is known that many Calabi-Yau manifolds form a connected web. The question of whether all CY manifolds form a single web depends on the degree of singularity that is permitted for the varieties that connect  the distinct families of smooth manifolds. If only conifolds are allowed then, since shrinking two-spheres and three-spheres to points cannot affect the fundamental group, manifolds with different fundamental groups will form disconnected webs. We examine these webs for the tip of the distribution of CY manifolds where the Hodge numbers $(h^{11},h^{21})$ are both small. In the tip of the distribution the quotient manifolds play an important role. We generate via conifold transitions from these quotients a number of new manifolds. These include a manifold with $\chi =-6$, that is an analogue of the $\chi=-6$ manifold found by Yau,  and manifolds with an attractive structure that may prove of interest for string phenomenology.

In 2006, a bad field was constructed (together with Baudisch, Hils and Wagner) collapsing Poizat's green fields. In this talk, we will not concentrate on the general methodology for collapsing specific structures, but more on a specific result in algebraic geometry, a weaker version of the Conjecture on Intersection with Tori (CIT). We will present a model theoretical proof of this result as well as discuss the possible generalizations to positive characteristic. We will try to make the talk  self-contained and aimed for an audience with a basic acquaintance with Model Theory.

Travelling waves are highly symmetric solutions to the Hamiltonian lattice equation and are determined by nonlinear advance-delay differential equations. They provide much insight into the microscopic dynamics and are moreover fundamental building blocks for macroscopic

lattice theories.

In this talk we concentrate on travelling waves in convex FPU chains and study both periodic waves (wave trains) and homoclinic waves (solitons). We present a new existence proof which combines variational and dynamical concepts.

In particular, we improve the known results by showing that the profile functions are unimodal and even.

Finally, we study the complete localization of wave trains and address additional complications that arise for heteroclinic waves (fronts).(joint work with Jens D.M. Rademacher, CWI Amsterdam)

A major issue in general relativity, from its earliest days to the

present, is how to extract physical information from any solution or

class of solutions to the Einstein equations. Though certain

information can be obtained for arbitrary solutions, e.g., via geodesic

deviation, in general, because of the coordinate freedom, it is often

hard or impossible to do. Most of the time information is found from

special conditions, e.g., degenerate principle null vectors, weak

fields close to Minkowski space (using coordinates close to Minkowski

coordinates) or from solutions that have symmetries or approximate

symmetries. In the present work we will be concerned with

asymptotically flat space times where the approximate symmetry is the

Bondi-Metzner-Sachs (BMS) group. For these spaces the Bondi

four-momentum vector and its evolution, found from the Weyl tensor at

infinity, describes the total energy-momentum of the interior source

and the energy-momentum radiated. By generalizing certain structures

from algebraically special metrics, by generalizing the Kerr and the

charged-Kerr metric and finally by defining (at null infinity) the

complex center of mass (the real center of mass plus 'i' times the

angular momentum) with its transformation properties, a large variety

of physical identifications can be made. These include an auxiliary

Minkowski space viewed from infinity, kinematic meaning to the Bondi

momentum, dynamical equations of motion for the center of mass, a

geometrically defined spin angular momentum and a conservation law with

flux for total angular momentum.

The Willmore Functional for surfaces has been introduced for the first time almost one century ago in the framework of conformal geometry (though it's one dimensional version already appears in thework of Daniel Bernouilli in the XVIII-th century). Maybe because of its simplicity and the depth of its mathematical relevance, it has since then played a significant role in various fields of sciences and technology such as cell biology, non-linear elasticity, general relativity...optical design...etc.

Critical points to the Willmore Functional are called Willmore Surfaces. They satisfy the so called Willmore Equations introduced originally by Gerhard Thomsen in 1923 . This equation, despite the elegance of it's formulation, is very inappropriate for dealing with analysis questions such as regularity, compactness...etc. We will present a new formulation of the Willmore Euler-Lagrange equation and explain how this formulation, together with the Integrability by compensation theory, permit to solve fundamental analysis questions regarding this functional, which were untill now totally open.

The proof of quasiconvexity based sufficient conditions for strong local minima in vectorial variational problems consists of three major parts: the Decomposition Theorem, the Orthogonality principle and the Localization principle. The first and the last are the most technical.
In these talks I will explain the technical difficulties and the ways in which they were overcome.

The workshop will address current issues related to the stability of solutions in nonlinear elasticity, including local energy minimizers, the stability of growing bodies, global existence for small data, bifurcation and continuation of solutions, and Saint-Venant’s principle.

In the past several years it has come to be appreciated that in low Reynolds number flow the nonlinearities provided by non-Newtonian stresses of a complex fluid can provide a richness of dynamical behaviors more commonly associated with high Reynolds number Newtonian flow. For example, experiments by V. Steinberg and collaborators have shown that dilute polymer suspensions being sheared in simple flow geometries can exhibit highly time dependent dynamics and show efficient mixing. The corresponding experiments using Newtonian fluids do not, and indeed cannot, show such nontrivial dynamics. To better understand these phenomena we study the Oldroyd-B viscoelastic model. We first explain the derivation of this system and its relation to more familiar systems of Newtonian fluids and solids and give some analytical results for small data perturbations. Next we study this and related models numerically for low-Reynolds number flows in two dimensions. For low Weissenberg number (an elasticity parameter), flows are "slaved" to the four-roll mill geometry of the fluid forcing. For sufficiently large Weissenberg number, such slaved solutions are unstable and under perturbation transit in time to a structurally dissimilar flow state dominated by a single large vortex, rather than four vortices of the four-roll mill state. The transition to this new state also leads to regions of well-mixed fluid and can show persistent oscillatory behavior with continued destruction and generation of smaller-scale vortices.

This talk first introduces generalized Young measures (or DiPerna/Majda measures) in an $L^1$-setting. This extension to classical Young measures is able to quantitatively account for both oscillation and concentration phenomena in generating sequences.

We establish several fundamental properties like compactness and representation of nonlinear integral functionals and present some examples. Then, generalized Young measures generated by $W^{1,1}$- and $BV$-gradients are more closely examined and several tools to manipulate them (including averaging and approximation) are presented.

Finally, we address the question of characterizing the set of generalized Young measures generated by gradients in the spirit of the Kinderlehrer-Pedregal Theorem.

This is joint work with Jan Kristensen.

We show that the leading terms of the number of absolutely indecomposable representations of a quiver over a finite field are related to counting graphs. This is joint work with Geir Helleloid.

This talk is devoted to adaptivity in optimal control of PDEs with special emphasis on barrier methods for pointwise state constraints. The talk is divided into to major parts, first we will discuss the case of additional pointwise inequality constraints on the state variable, then we will transfer the results to constraints on the gradient of the state. Each part will start with a discussion of necessary optimality conditions and a brief overview about what is known and what is not known concerning a priori analysis. Then a posteriori error estimates for the discretization error as well as for the error from the barrier method will be presented. Finally we show some simple examples to illustrate the behavior of the estimators. 
The talk will be followed by an informal tea in the Gibson Building seminar room giving an opportunity to chat with Winnifried Wollner and Amit Acharya (our other current OxMOS visitor)

Γ-convergence is a variational convergence on functionals. The explicit characterization of the integrand of the Γ-limit of sequences of integral functionals with periodic integrands is by now well known. Here we focus on the explicit characterization of the limit energy density of a sequence of functionals with non-periodic integrands. Such characterization is achieved in terms of the Young measure associated with relevant sequences of functions. Interesting examples are considered.

Multiaxial mechanical proportional loadings on shape memory alloys undergoing phase transformation permit to determine the yield curve of phase transformation initiation in the stress space. We show how to transport this yield surface in the set of effective transformation strains of producted phase M. Two numerical applications are done concerning a Cu Al Be and a Ni Ti polycrystallines shape memory alloys. A special attention is devoted to establish a convexity criterium of these surfaces.

Moreover an application to the determination of the phase transformation surface around the crack tip for SMA fracture is performed.

At last some datas are given concerning the SMA damping behavior

AUTHORS

Christian Lexcellent, Rachid Laydi, Emmanuel Foltete, Manuel collet and Frédéric Thiebaud

FEMTO-ST Département de Mécanique Appliquée Université de Franche Comte Besançon France

Cofinitary groups are subgroups of the symmetric group on the natural numbers

(elements are bijections from the natural numbers to the natural numbers, and

the operation is composition) in which all elements other than the identity

have at most finitely many fixed points. We will give a motivation for the

question of which isomorphism types are possible for maximal cofinitary

groups. And explain some of the results we achieved so far.

We present a selection of recent results pertaining to Hessian

and Monge-Ampere equations, where the Hessian matrix is augmented by a

matrix valued lower order operator. Equations of this type arise in

conformal geometry, geometric optics and optimal transportation.In

particular we will discuss structure conditions, due to Ma,Wang and

myself, which imply the regularity of solutions.These conditions are a

refinement of a condition used originally by Pogorelev for general

equations of Monge-Ampere type in two variables and called strong

ellipticity by him.

A 2-dimensional Topological Quantum Field Theory (TQFT) is a symmetric monoidal functor from the category of 2-dimensional cobordisms to the category of vector spaces. A classic result states that 2d TQFTs are classified by commutative Frobenius algebras.  I show how to extend this result to open-closed TQFTs using a class of 2-manifolds with corners, how to use the Moore-Segal relations in order to find a canonical form and a complete set of invariants for our cobordisms and how to classify open-closed TQFTs algebraically.  Open-closed TQFTs can be used to find algebraic counterparts of Bar-Natan's topological extension of Khovanov homology from links to tangles and in order to get hold of the braided monoidal 2-category that governs this aspect of Khovanov homology. I also sketch what open-closed TQFTs reveal about the categorical ladder of combinatorial manifold invariants according to Crane and Frenkel.

references:

1] A. D. Lauda, H. Pfeiffer:

Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras,

Topology Appl. 155, No. 7 (2008) 623-666, arXiv:math/0510664

2] A. D. Lauda, H. Pfeiffer: State sum construction of two-dimensional open-closed Topological Quantum Field Theories,

J. Knot Th. Ramif. 16, No. 9 (2007) 1121-1163,arXiv:math/0602047

3] A. D. Lauda, H. Pfeiffer: Open-closed TQFTs extend Khovanov homology from links to tangles, J. Knot Th. Ramif., in press, arXiv:math/0606331.

Mean curvature vector is the negative gradient of the area functional. Thus if we deform a submanifold along its mean curvature vector, which is called mean curvature flow (MCF), the area will decrease most rapidly. When the ambient manifold is Kahler-Einstein, being Lagrangian is preserved under MCF. It is thus very natural trying to construct special Lagrangian/ Lagrangian minimal through MCF. In this talk, I will make a brief introduction and report some of my recent works with my coauthors in this direction, which mainly concern the singularities of the flow.

There are two basic theories of curve counting on Calabi-Yau threefolds. Donaldson-Thomas theory arises by considering curves as subschemes; Gromov-Witten theory arises by considering curves as the image of maps. Both theories can also be formulated for orbifolds. Let X be a dimension three Calabi-Yau orbifold and let

Y --> X be a Calabi-Yau resolution. The Gromov-Witten theories of X and Y are related by the Crepant Resolution Conjecture. The Gromov-Witten and Donaldson-Thomas theories of Y are related by the famous MNOP conjecture. In this talk I will (with some provisos) formulate the remaining equivalences: the crepant resolution conjecture in Donaldson-Thomas theory and the MNOP conjecture for orbifolds. I will discuss examples to illustrate and provide evidence for the conjectures.