Structures and dynamics

Panagiotopoulos, Aristotelis

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https://hdl.handle.net/2142/98378 Copy

Description

Title

Structures and dynamics

Author(s)

Panagiotopoulos, Aristotelis

Issue Date

2017-07-13

Director of Research (if dissertation) or Advisor (if thesis)

Solecki, Slawek

Doctoral Committee Chair(s)

Hieronymi, Philipp

Committee Member(s)

• Tserunyan, Anush
• van den Dries, Lou

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Polish groups
• Fraisse
• Turbulence
• Hjorth
• Left invariant
• Becker
• Projective Fraisse
• Infinite games
• Borel complexity

Abstract

"Our results are divided in three independent chapters. In Chapter 2, we show that if g is a generic isometry of a generic subspace X of the Urysohn metric space U then g does not extend to a full isometry of U. The same applies to the Urysohn sphere S. Let M be a Fraisse L-structure, where L is a relational countable language and M has no algebraicity. We provide necessary and sufficient conditions for the following to hold: ""For a generic substructure A of M, every automorphism f in Aut(A) extends to a full automorphism f' in Aut(M)."" From our analysis, a dichotomy arises and some structural results are derived that, in particular, apply to omega-stable Fraisse structures without algebraicity. Results in Chapter 2 are separately published in [Pan15]. In Chapter 3, we develop a game-theoretic approach to anti-classi cation results for orbit equivalence relations and use this development to reorganize conceptually the proof of Hjorth's turbulence theorem. We also introduce a new dynamical criterion providing an obstruction to classi cation by orbits of Polish groups which admit a complete left invariant metric (CLI groups). We apply this criterion to the relation of equality of countable sets of reals and we show that the relations of unitary conjugacy of unitary and selfadjoint operators on the separable in nite-dimensional Hilbert space are not classi able by CLI-group actions. Finally we show how one can adapt this approach to the context of Polish groupoids. Chapter 3 is joint work with Martino Lupini and can also be found in [LP16]. In Chapter 4, we develop a theory of projective Fraisse limits in the spirit of Irwin-Solecki. The structures here will additionally support dual semantics as in [Sol10, Sol12]. Let Y be a compact metrizable space and let G be a closed subgroup of Homeo(Y ). We show that there is always a projective Fraisse limit K and a closed equivalence relation r on its domain K that is de finable in K, so that the quotient of K under r is homeomorphic to Y and the projection of K to Y induces a continuous group embedding of Aut(K) in G with dense image. The main results of Chapter 4 can also be found in [Pan16]."

Graduation Semester

2017-08

Type of Resource

text

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http://hdl.handle.net/2142/98378

Copyright and License Information

Copyright 2017 Aristotelis Panagiotopoulos

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