Schedule


Week 1, October 14-18

 Mo
14.10.
Tu
15.10.
Wed
16.10.
Thu
17.10.
Fri
18.10.
9:00-9:30Welcome
9:30-10:00D. 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
10:00-11:00D. 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
11:00-11:30coffeecoffeecoffeecoffeecoffee
11:30-1:00T. 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.
1:00-2:00lunchlunchlunchlunchlunch
2:00-3:00S. Richard
Operator algebras: what are they good for ?
Talk 1Talk 2Talk 3B. Sirakov
Variational approach to boundary value problems.
3:00-3:30coffeecoffeecoffee
3:30-4:00Presentation
graduate program
tutorialtutorialtutorialcoffee
4:00-5:00coffeeClub de
Matematica

Week 2, October 21-22

 Mo
21.10.
Tu
22.10.
09:30-10:00Tutorial
10:00-11:00TutorialTalk 5
11:00-11:30coffeecoffee
11:30-12:30B. Sirakov
Variational approach to boundary value problems.
Talk 6
12:30-1:00B. Sirakov
Variational approach to boundary value problems.
interviews
1:00-2:00lunchBBQ
2:00-3:00interviewsBBQ
3:00-3:30coffeeBBQ
3:30-6:00interviewsBBQ

Courses

  • Geometrical aspects of spectral theory,por David Krejcirik (Czech Technical University).
  • 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.

  • Operator algebras: what are they good for ?, por Serge Richard (Nagoya University).

  • Minmax Methods in the Calculus of Variations of Curves and Surfaces, por Tristan Riviere (ETH Zürich).
  • 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 mini-course 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 mini-course 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 3-dimensional Euclidian space.

  • Variational approach to boundary value problems, por Boyan Sirakov (Pontificia Universidade Católica do Rio de Janeiro).
  • 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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