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Analysis, Geometry, and Dynamical Systems Seminar   RSS

Past sessions

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26/11/2009, 11:00 — 12:00 — Room P3.10, Mathematics Building
Messoud Efendiev, Technical University of Munich

On a criteria for veryfing mathematical models

24/11/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building
Lucia Scardia, Hausdorff Center for Mathematics, Bonn, Germany

Convergence of equilibria for thin elastic plates under physical growth conditions

The asymptotic behaviour of the equilibrium configurations of a thin elastic plate is studied, as the thickness h of the plate goes to zero. More precisely, it is shown that critical points of the functional E h describing the elastic energy of the thin plate converge to critical points of the Γ-limit of E h, in the von Kármán regime. This is proved under the physical assumption that the elastic energy blows up in the case of total compression.

03/11/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building
Henry van Roessel, University of Alberta, Edmonton, Canadá

Some recent work on coagulation equations

16/10/2009, 11:00 — 12:00 — Room P3.10, Mathematics Building
, CMAP, Ecole Polytechnique

Quadratic growth BSDE and applications to pricing and hedging of derivatives based on non-tradable underlyings.

We consider Forward-Backward stochastic differential equations with generators that grow quadratically in the control variable (qgFBSDE). This type of equations is used to solve a utility maximization problem in incomplete markets tackling the problem of pricing a financial derivative based on non-traded underlyings such as weather derivatives. We provide new differentiability results for qgFBSDE that in turn allow closed form formulas for the hedging strategy. We present numerical results.

28/07/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building
Hossein Tehrani, University of Nevada, Las Vegas

Fucik spectrum of wave operator and related jumping nonlinearity problems

07/07/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building
Josep Sardanyés, Complex Systems Lab-Universitat Pompeu Fabra, Barcelona

On the dynamics of RNA viruses through nonlinear differential equations

30/06/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building
Louis Kauffman, University of Illinois, Chicago

Virtual Knot Theory and Oriented Extensions of the Jones Polynomial

The original Jones polynomial for classical knots depends only weakly on the orientation of the knot or link to which it is applied. Careful examination of the orientations assigned to states in an oriented bracket polynomial model for the Jones polynomial reveals that there is much more topological information available from orientation when the knot or link is embedded in a thickened surface or regarded as a virtual link. This talk will discuss a large scale generalization that we call the extended bracket polynomial (taking polynomial and graphical values) and the arrow polynomial (joint work with Heather Dye) taking polynomial values with infinitely many new variables. The arrow polynomial is related to the Miyazawa polynomials for virtual knots. The talk will discuss applications of these new invariants to finding virtual crossing number and genus of virtual knots, and we shall discuss extensions of Khovanov homology related to these invariants.

30/06/2009, 11:00 — 12:00 — Room P4.35, Mathematics Building
Saber Elaydi, Trinity University - Texas

Towards a theory of periodic difference equations and population biology

23/06/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building
Massimiliano Morini, SISSA, Itália

Equilibrium configurations of epitaxially strained elastic films: existence, regularity, and qualitative properties of solutions

We consider a variational model used in the physical literature to describe the equilibrium configurations of an elastic film epitaxially deposited on a flat rigid substratum, when a lattice mismatch is present between the two materials. After specifying the functional set-up, in which the minimization problem can be properly formulated, we study the regularity and several qualitative properties of equilibrium configurations; that is, of local and global minimizers of the energy functional.

16/06/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building
Zdenek Kocan and Veronika Kurkova, Silesian University in Opava, Czech Republic

Entropy, horseshoes and homoclinic trajectories on trees, graphs and on dendrites

09/06/2009, 11:30 — 12:30 — Room P3.10, Mathematics Building
Pierre Cartier, Institut des Hautes Études Scientifiques, Paris

Arithmetics: motives, periods and motivic Galois group

08/06/2009, 11:30 — 12:30 — Room P3.10, Mathematics Building
Pierre Cartier, Institut des Hautes Études Scientifiques, Paris

Geometry: sheaves, topos and fields

05/06/2009, 11:30 — 12:30 — Room P3.10, Mathematics Building
Pierre Cartier, Institut des Hautes Études Scientifiques, Paris

Geometry: the notions of spectrum and scheme

08/05/2009, 14:00 — 15:00 — Room P4.35, Mathematics Building
Emma D'Aniello, Seconda Universitá degli Studi di Napoli

Topological dynamical systems and odometers

10/03/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building
Alex Vladimirsky, Cornell University

Homogenization and multiobjective optimization - computational challenges

I will present two recent projects related to front propagation and optimal control.

The first of these (joint with A. Oberman and R. Takei) deals with 2-scale and 3-scale computations in geometric optics. We propose a new and efficient method to homogenize first-order Hamilton-Jacobi PDEs. Unlike the prior cell-problem methods, our algorithm is based on homogenizing the related Finsler metric. We illustrate by computing the effective velocity profiles for a number of periodic and "random" composite materials.

The second project (joint with A. Kumar) deals with multiple criteria for optimality (e.g., fastest versus shortest trajectories) and optimality under integral constraints. We show that an augmented PDE on a higher-dimensional domain describes all Pareto-optimal trajectories. Our numerical method uses the causality of this PDE to approximate its discontinuous viscosity solution efficiently. The method is illustrated by problems in robotic navigation (e.g., minimizing the path length and exposure to an enemy observer simultaneously).

17/02/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building
Andreas Döring, Imperial College London

A topos approach to quantum theory

I will report on work with Chris Isham on the application of topos theory to physics. The Kochen-Specker theorem shows that a naive realist description of quantum systems is impossible. This can be understood as the inapplicability of Boolean logic to quantum systems. In order to arrive at a more realist description, one can use the internal logic of a certain topos of presheaves. The choice of this topos is directly motivated from the Kochen-Specker theorem. I will show which structures within this topos are of physical significance and how propositions about physical quantities are assigned truth-values. The relation with constructive Gel'fand duality is sketched.

10/02/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building
Alexandre Baraviera, Universidade Federal do Rio Grande do Sul

Iterated function systems, classical and quantum

An iterated function system (IFS) is a collection of maps that are chosen and iterated according to some probability. The geometric and measurable structure of the limit set is usually very interesting and studied with the use of the well known technique of transfer operator. In this talk we will review some results about this problem and its quantum version, where the maps are unitary operators acting on some suitable Hilbert space.

03/02/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building
Florin Radulescu, Università di Roma-Tor Vergata

Type II1 Von Neumann Representations for Hecke Operators on Maass Forms and Inequalities for their Eigenvalues

We prove that classical Hecke operators on Maass forms are a special case of completely positive maps on II1 factors, associated to a pair of isomorphic subfactors. This representation induces several matrix inequalities on the eigenvalues of the Hecke operators on Maass forms. These inequalities are the consequence of a "double" action of the Hecke algebra which can be seen only in the type II1 representation. The classical Hecke operators are then a "diagonal" of this double action.

27/01/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building
Josep Sardanyés, Complex Systems Lab, Universitat Pompeu Fabra, Barcelona

Theoretical models of cooperation with ODEs: from prebiotic evolution to complex ecosystems

The dynamics of cooperation can be studied using mean field models given by nonlinear ordinary differential equations (ODEs). Such a dynamics is modeled considering heterocatalytic feedback loops, with density-dependent enhancement of reproduction rates between cooperating replicators. These models are used to study the dynamics of catalytic networks named hypercycles which have a particular graph architecture and which has been suggested to be of importance in earlier stages of prebiotic evolution. In this talk we will introduce the mathematical formalism used for the dynamical study of such networks, focusing on the origin of life problem and on the dynamics of ecological systems. We will analyze several low-dimensional catalytic networks providing their stability and bifurcation scenarios. We will also discuss some interesting dynamical phenomenon associated to the bifurcations implying the transition from survival to extinction phases. We will then also compare the results obtained with the ODEs for these kind of networks with other computational tools typical of complex systems analysis.

13/01/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building
Artur Lopes, Universidade Federal do Rio Grande do Sul

Transport and the involution kernel in Ergodic Theory

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