Ideas for projects (for references send me an email or ask me after class):
- Groups without Polish topologies. There are many topological groups which do not admit any Polish
topology. Moreover the arguments involved in showing that such a group
is not Polish can be quite intricate. Collect examples of such groups and
conceptualize/unify existing proofs (see Shelah, Mann, Solecki, Rosendal, ...)
- Polish groups with stable metrics. We will see that there are Polish groups which have no non-trivial representation
on a reflexive Banach space. Conceptualize and present the arguments involved in showing that the family of continuous representations of G on reflexive
Banach spaces generates the topology on G if and only if G admits a left invariant metric d that is stable.
- Strong amalgamation a probabilistic limits. Let X be the Polish space of all countable L-structures and let [M] be the subset of X consisting of all structures
isomorphic to some fixed M. It is a theorem of Ackerman Freer Patel that there is an isomorphism invariant
measure concetrating on [M] if and only if M has trivial definable closure. Present the proof of this result, explain the connections with strong amalgamation property and discuss
bottom-up ways of building invariant measures on X. One open research question is whether one can build in a bottom-up fashion a measure concetrating on the triangle-free Henson graph.
- Amenability and convex Ramsey property. We will see the KPT correspondence between Ramsey property and extreme amenability. A correspondence between
amenability and a certain weakening of Ramsey property was later established by Justin Moore. We yet do not know of any natural convex Ramsey class that is not Ramsey.
- Ample Generics and automatic continuity. In Turbulence, amalgamation and generic automorphisms of homogeneous structures the authors establish a correspondece between
combiatorial properties of a Fraisse class and dynamical properties of the action of the automorphism group on itself by conjugation. One of these properties is ample genericity and it implies a very strong automatic continuity property for the group automorphism group.
- The homeomorphism group of the Cantor space has ample generics. In this paper Kwiatkowska used a projective analogue of Fraisse theory to establish that the homeomoephism group of the Cantor space has ample generics.
- Hrushovski property for finite metric spaces. In this paper Solecki established the Hrushovski property for finite metric space and derived various consequences for tthe isometry group of the Urysohn space.
About the course: this is a topics course in topological groups and dynamics. We will be focusing on techniques and phenomena that go beyond the realm of locally compact groups.
In the absence of classical techniques which rely on local compacteness, we will develop various frameworks for analyzing the structure of "large" topological groups, such as:
Fraisse theoretic methods; Baire category methods; structured completions and compactifications. As we develop these techniques we will provide a wide range of applications
and examples from topology, analysis, and logic. Here are three centerpieces that, among others, will be covered:
- Extreme amenability. Consider the automorphism group Aut(Q,<), of all order preserving bijections of the rationals. Endowed with the pointwise convergence topology it has the astonishing property
that whenever it acts continuously on a compact space K, K has a fixed point under the action. As we will see, this is intimately tied to a combinatorial principle known as the Ramsey theorem.
We will develop the theory of a more general phenomenon known as the KPT-correspondence (for Kechris, Pestov, Todorcevic).
- Anti-classification results. Dynamics play a very important role in logic because they allow us to prove negative results about classification projects in mathematics.
We will see for example that there is no "constructive way" of classifying countable graphs up to isomorphism by assigning to them enough meaningful invariants.
By developing more intricate tools (such as Hjorth's turbulence theory) we will see that similar classification projects from analysis and topology are even more hopeless
(e.g. classifying unitary operators up to unitary equivalence).
- Exotic groups. Consider the group Homeo+([0,1]), of all orientation preserving homeomorphisms of the compact space [0,1]. Endowed with the compact-open topology it has no non-trivial continuous
representations on a Hilbert space. Even more, by a recent result of Megrelishvili, it has no non-trivial representation on any reflexive Banach space. As we will see, this is intimately tied to the the fact that the group Homeo+([0,1]) has no compactification
on which the multiplication of Homeo+([0,1]) extends to a seperately continuous semigroup operation.
Background. This course is intended for graduate and advanced undergraduate students. We will assume that the students have some background in topology and the theory of metric spaces.
Some background in logic, functional analysis, and representation theory will be useful but not necessary. Through a broad syllabus I intend to motivate and raise interest about certain topics rather
than exclude the "non-specialists" so please do not hesitate to contact me if you have any questions!
Books and notes.
I will be posting notes on a weekly basis. That being said, some relevant books are: Su Gao's "Invariant descriptive set theory", Vladimir Pestov's " Dynamics of Infinite-Dimensional Groups: The Ramsey-Dvoretzky-Milman Phenomenon",
Greg Hjorth's "Classification and Orbit Equivalence Relations".
The grades will be based on participation and on a final project. For the project each of you will read some paper(s), present the main ideas in class, and write a short paper on the topic.
Math 191 Fall 2018
10:30 - 11:55 TR, 289 LINDE
Office. 102 Kellogg
panagio at caltech.edu
Any time (knock door or send an email)
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