Literature and a list of topics for presentations

Some of the proposed topics below are from this book: Katok-Nitica book (KNbook) Download Katok-Nitica book (KNbook)

Here is the list of some topics suggestions for presentations (in parentheses I wrote some commentaries on the kind of mathematics learning the topic would entail. Also I might be updating this page more before Feb. 3rd): 

 

  1. Time changes of homogeneous flows and first cohomology:

    Starkov: Dynamical Systems on Homogeneous Spaces   Download Starkov: Dynamical Systems on Homogeneous Spaces  

    (Selected parts of Sections 1- 5 in Ch 24. The book is about flows on homogeneous spaces, but this particular part can be done just for flows on the torus, or the flows on a negatively curved surface, analysis is not very deep, but is nice)

  2. First cohomology and group extensions over dynamical systems 

    (Selected parts of J-P Conze paper Links to an external site. , this is low key analysis, nicely illustrates the connection between extensions of systems and cohomology)

  3. Zimmer cocycle superrigidity for higher rank semisimple Lie groups and lattices, selected parts from Feres book Links to an external site.   (this is for those of you who like Lie groups and more algebraic setting, it requires also measure theory, and more other stuff, but it is very rewording. This is one of the cornerstone results in a big theory concerning actions of very large groups).
  4.  Connection between SL(2, R) valued cocycles and the discrete Schrödinger equation (Introduction and some parts of K. Bjerklöv's paper Links to an external site.).
  5. First cohomology over Anosov systems: Livsic theorem(s) (Ch 4.1 and 4.2 in KNbook) (down to earth analysis, some basic dynamics)
  6. First cohomology over an ergodic toral automorphism: harmonic analysis approach (Ch 4.2.4 KN book) and vanishing of first cohomology for higher rank abelian actions (Ch 4.4 in KNbook up to, and excluding, Ch 4.4.3) (this uses elementary harmonic analysis, but the result is striking).
  7. First cohomology for accessible partially hyperbolic diffeomorphism: periodic cycle functionals, Ch 4.3 in KNbook. (This uses foliations on manifolds, and the notion of accessibility which originally comes from control theory and was then adopted in dynamics. It does not require any special pre-knowledge about foliations though).
  8. Lie group valued cocycles over Anosov systems: Sec 5.7 NKbook. (This is extension of topic 4 to cocycles with values in Lie groups rather than just real vector space).
  9. Higher order cohomology: Ch 6 in NKbook (this uses again only elementary harmonic analysis on the torus and some dynamics, no advance technology. The good thing is that it opens several conjectures and questions).
  10.  Connection between trivial first cohomology and local stability of dynamical systems: Hamilton's theorem (this is pure analysis reading, inverse function theorem type of thing, ask Danijela for a reference if you are interested).
  11. Vanishing cohomology and stability of fixed point and criteria for topological rigidity of actions (On and around proposition 6.1.2 in N. Qian's work Download N. Qian's work)
  12.  Cohomology for geodesic flow on hyperbolic surface: Gullemin-Kazhdan. Links to an external site.  (this is a bit old paper, it has been overcome by many other newer papers, but it isa nice starting point for those who prefer more geometric methods and situations).

  13. Central limit theorem and cohomological equation on homogeneous spaces: paper by Ronggang Shi Links to an external site.  (nice newer result connecting the things on CLT which Minsung will talk about to cohomology for certain type of homogeneous dynamics)

  14. Cohomological equations and invariant distributions for minimal circle diffeomorphisms: Paper by Avila-Kocsard Links to an external site.. (Just how special is dynamics on circle diffeos. Also, Avila is a Fields medallist, its good to read some of his work :)

14. In most of the above topics it is possible to pick an open ended thread and do something novel; here is a suggestion that leads into completely unexplored territory: read about connection between higher (than first) cohomology groups in the classical group cohomology setting (e.g. book "Lie groups, Lie algebras, cohomology and some applications in physics" by de Azcarraga and Izquierdo, section 7.5, or other text book on group cohomology) and equivalence classes of exact sequences. Then come up with analogous interpretation of higher cohomology for some concrete example in dynamical systems for which we already have some information about second (or higher) cohomology. What kind of classification can higher cohomology give? Talk to Danijela if you are interested in this.