# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

Newer session pages: Next 7 6 5 4 3 2 1 Newest

### Bundles, Operator Algebras and $K$-theory

The bundles that occur naturally in functional analysis are not necessarily locally trivial. Dixmier and Douady have shown that a continuous bundle with fiber an infinite dimensional separable Hilbert space $H$ over a compact contractible metric space does not need to be trivial, even though the unitary group of $H$ is contractible. Similar phenomena appear in the theory of operator algebras. The operator algebras with Hausdorff primitive spectrum are known to be isomorphic to algebras of sections in certain continuous bundles which are typically not locally trivial. We plan to give a gentle introduction illustrated by examples to the $K$-theory methods used in the study of these bundles.

### On Hausdorff Measures and KMS States

We explore an intriguing correspondence between Hausdorff measures arising from self-similar metrics and KMS states. This correspondence identifies the Hausdorff dimension of the space with the inverse temperature of the KMS state.

This talk is partially based on joint work with Jean Renault.

### A Global Scenario for Chaotic Systems from Poincaré to Present Time

In simple and reasonably nontechnical language, I shall discuss Poincaré's viewpoint that a key question in dynamics was to describe for most systems their global orbit structure. Poincaré suggested the question more than one hundred years ago and it is still a major question in the area. I will include in the discussion the important topic of homoclinic bifurcations, which curiously played a dramatic role in Poincaré's essay that led him to receive a prize from King Oscar II of Sweden.

### New geometric techniques in mechanics

We will develop a geometric description of Lagrangian Mechanics on Lie algebroids (generalizing Klein's formalism on the tangent bundle). The main motivation comes from reduction theory of Lagrangian systems. If time permits we will explain how Lie groupoids appear in discrete mechanics and the relation between both theories.

### A generic property of families of Lagrangian systems

We prove that a generic Lagrangian has finitely many minimizing measures in all the cohomology classes. In particular, for a generic Lagrangian, the quotient Aubry set is finite for all cohomology classes.

### Dissipative equations in locally uniform spaces

Semilinear damped wave equations, partly dissipative systems and reaction diffusion equations in ${ℝ}^{N}$ are considered in the locally uniform spaces under the assumptions on the nonlinear term similar to those in bounded domains.

### Analysis of a pressureless dynamical system and an open geometrical problem

We consider the pressureless Euler equation equipped with periodic boundary conditions. The criterion for the global smooth solvability can be given in terms of degeneracy of the initial state: the Jacoby Matrix must be everywhere nilpotent. A simple purely geometrical reformulation of the above condition is possible in the 2D case. The question is open for higher dimensions. An application to turbulence is also discussed.

### The set of periods for the Morse-Smale diffeomorphisms on ${T}^{2}$

We shall use the Lefschetz zeta function for studying the set of periods of the Morse-Smale diffeomorfisms defined on the 2-dimensional torus for every homotopy class.

### Phase transitions and multifractal analysis for horseshoes.

Older session pages: Previous 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 Oldest