Mo 14.10.  Tu 15.10.  Wed 16.10.  Thu 17.10.  Fri 18.10. 


9:009:30  Welcome  
9:3010:00  D. Krejcirik Geometrical aspects of spectral theory  D. Krejcirik Geometrical aspects of spectral theory  T. Rivière Minimax methods in the calculus of variations of curves and surfaces  D. Krejcirik Geometrical aspects of spectral theory  
10:0011:00  D. Krejcirik Geometrical aspects of spectral theory  D. Krejcirik Geometrical aspects of spectral theory  T. Rivière Minimax methods in the calculus of variations of curves and surfaces  D. Krejcirik Geometrical aspects of spectral theory  Talk 4 
11:0011:30  coffee  coffee  coffee  coffee  coffee 
11:301:00  T. Rivière Minimax methods in the calculus of variations of curves and surfaces  T. Rivière Minimax methods in the calculus of variations of curves and surfaces  S. Richard Operator algebras: what are they good for ?  S. Richard Operator algebras: what are they good for ?  B. Sirakov Variational approach to boundary value problems. 
1:002:00  lunch  lunch  lunch  lunch  lunch 
2:003:00  S. Richard Operator algebras: what are they good for ?  Talk 1  Talk 2  Talk 3  B. Sirakov Variational approach to boundary value problems. 
3:003:30  coffee  coffee  coffee  
3:304:00  Presentation graduate program  tutorial  tutorial  tutorial  coffee 
4:005:00  coffee  Club de Matematica 
Mo 21.10.  Tu 22.10. 


09:3010:00  Tutorial  
10:0011:00  Tutorial  Talk 5 
11:0011:30  coffee  coffee 
11:3012:30  B. Sirakov Variational approach to boundary value problems.  Talk 6 
12:301:00  B. Sirakov Variational approach to boundary value problems.  interviews 
1:002:00  lunch  BBQ 
2:003:00  interviews  BBQ 
3:003:30  coffee  BBQ 
3:306:00  interviews  BBQ 
Abstract: Spectral theory is an extremely rich field which has found its application in many areas of physics and mathematics. One of the reason which makes it so attractive on the formal level is that it provides a unifying framework for problems in various branches of mathematics, for example partial differential equations, calculus of variations, geometry, stochastic analysis, etc.
The goal of the lecture is to acquaint the students with spectral methods in the theory of linear differential operators coming both from modern as well as classical physics, with a special emphasis put on geometrically induced spectral properties. We give an overview of both classical results and recent developments in the field, and we wish to always do it by providing a physical interpretation of the mathematical theorems.
Abstract: The study of the variations of curvature functionals takes its origins in the works of Euler and Bernouilli from the eighteenth century on the Elastica. Since these very early times, special curves and surfaces such as geodesics, minimal sur faces, elastica, Willmore surfaces, etc. have become central objects in mathematics much beyond the field of geometry stricto sensu with applications in analysis, in applied mathematics, in theoretical physics and natural sciences in general. Despite its venerable age the calculus of variations of length, area or curvature function als for curves and surfaces is still a very active field of research with important developments that took place in the last decades. In this minicourse we shall con centrate on the various minmax constructions of these critical curves and surfaces in euclidian space or closed manifolds. We will start by recalling the origins of minmax methods for the length functional and present in particular the “curve shortening process” of Birkhoff. We will explain the generalization of Birkhoff’s approach to surfaces and the ”harmonic map replacement” method by Colding and Minicozzi. We will then recall some fundamental notions of Palais Smale deformation theory in infinite dimensional spaces and apply it to the construction of closed geodesics and Elastica. In the second part of the minicourse we will present a new method based on smoothing arguments combined with Palais Smale deformation theory for performing successful minmax procedures for surfaces. We will present various applications of this so called “viscosity method” such as the problem of computing the cost of the sphere eversion in 3dimensional Euclidian space.
Abstract: Assuming some knowledge of functional analysis and Lebesgue spaces, we start by making a brief introduction to Sobolev spaces. We develop the variational formulation of boundary value problems for ordinary and partial differential equations, and use basic results from functional analsis to show their solvability in an appropriate weak sense. If time permits, we will also discuss regularity of weak solutions.
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