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Logic
Seminar |
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Tuesday,
October 4
Tuesday,
October 11
Abstract: Mycielski showed that for every comeager subset of the plane, there is a perfect subset of reals so that the collection of all pairs of non-identical reals from this perfect set is a subset of the original comeager set. A Mycielski type property can be formulated for other equivalence relations besides equality. Holshouser and Jackson showed that E_0 has the Mycielski property. In this talk I will show that E_1, E_2, and E_3 do not have the Mycielski property. This is joint work with Connor Meehan. Tuesday,
October 18
Abstract: It is an old question whether Thompson's group F is amenable or not. To prove that F is not amenable, we can show that F is not Liouville for any generating measure on F. By looking at its action on the set of dyadic rationals, one can show that for finitely supported generating measures, F is not Liouville, which could be a first step in proving that F is not amenable. There is a subclass of amenable groups, called strongly amenable groups. These are the groups for which every proximal action on a compact Hausdorff space has a fixed point. We can modify the usual action of F on dyadic rationals to prove that F is not strongly amenable. Tuesday,
October 25
We study profinite groups from a logician's point of view. Lubotzky and Jarden showed that topologically f.g. groups are given by their first-order theory. We raise the question whether a single sentence can suffice. Next, in the setting of computable profinite groups, the Haar measure is computable, so the usual notions of algorithmic randomness can be defined. We consider the strength of randomness necessary for effective versions of ``almost everywhere" type theorems to hold (with Fouche). Finally, we consider the complexity of isomorphism using the theory of Borel reducibility in descriptive set theory. For topologically finitely generated profinite groups this complexity is the same as the one of identity for reals. In general, it is the same as the complexity of isomorphism for countable graphs. The latter result can be adapted to locally compact closed subgroups of the group of permutations on N (joint with Kechris and Tent). This result has been obtained independently and by different means in work of Rosendal and Zielinski (arXiv 1610.00370, Oct 2016). Thursday,
November 17
Tuesday,
November 29
Abstract: I will present an overview of my work on universal objects in functional analysis, including the Gurarij space, the Poulsen simplex, and their noncommutative analogues. I will then explain how one can compute their universal minimal flows using methods from Ramsey theory and model theory, which is joint work with Bartosova, Lopez-Abad, and Mbombo. Thursday,
January 12
Abstract: We discuss a recent paper of Pequignot in which he answers a question of Kechris, Solecki, and Todorcevic concerning graphs generated by a single Borel function. It is shown that the shift graph on the space of infinite subsets of natural numbers does not admit a Borel homomorphism into every such graph with infinite Borel chromatic number. Notably, only definability arguments are used and no counterexample is produced. Tuesday,
January 17
Abstract: I will present a short conceptual proof of Hjorth's turbulence theorem in terms of a suitable open game. I will then explain how these ideas can be modified to prove a new dynamical criterion that provides an obstruction to classification by orbits of a CLI-group action. This is joint work with Aristotelis Panagiotopoulos. Tuesday,
January 24
Abstract: One of the most interesting results of Borel graph combinatorics is the $G_0$ dichotomy, i. e., the fact that an analytic graph has uncountable Borel chromatic number if and only if it contains a Borel homomorphic image of a graph called $G_0$. It was conjectured that an analogous statement could be true for graphs of infinite Borel chromatic number, or at least for some well-behaving subclass of the class of infinitely chromatic graphs. Using descriptive set theoretic methods we produce examples showing that some versions of these statements are false. Thursday,
January 26
Abstract: This is a general talk on locally compact quantum (lcq) groups aimed primarily for non-experts. I will motivate the axiomatic definition of lcq groups due to Kustermans and Vaes, and present some examples. I will then introduce the concepts of actions of lcq groups, noncommutative Poisson boundaries, closed and open quantum subgroups, unitary representations and their induction. Thursday,
January 26
Abstract: I will discuss recent joint work with C. Rosendal on the large-scale, or coarse, structure of the groups of self-homeomorphisms of manifolds. Classical geometric group theory is the study of the large-scale geometry of finitely generated discrete groups, or compactly generated topological groups. Rosendal recently developed a means of extending this to a more general class of Polish groups. I will explain some of this framework, which Polish groups are known to admit a well-defined coarse structure, and why groups of homeomorphisms are an important class of examples. Tuesday,
January 31
Abstract: This work is part of a project to understand the complexity of the equivalence relation of isomorphism of minimal flows of countable groups (and, more specifically, minimal subshifts) in the framework of Borel reducibility. We have two results concerning Toeplitz subshifts: one on the complexity of isomorphism of Z-Toeplitz subshifts with separated holes and one where the acting group is non-amenable (and residually finite). Those results only scratch the surface of the general problem and many interesting open questions remain. This is joint work with Marcin Sabok. Thursday,
February 2
Abstract: There exist examples of compact Hausdorff spaces with no non-trivial homeomorphisms, and in particular with no non-trivial group actions on them. In the opposite extreme, there are many spaces that have a huge number of homeomorphisms. Since C*-algebras are noncommutative generalizations of spaces, we may ask whether a given (separable) C*-algebra has `many' automorphisms, or, more generally, `many' actions of a given group. In this context, we must of course identify actions up to a reasonable notion of equivalence, since otherwise most C*-algebras will have uncountably many actions. We are not only interested in knowing ``how many" actions there are, but also determining the Borel complexity of the relation of equivalence between them. Questions of this sort have been studied in the context of von Neumann algebras, and particularly the hyperfinite II$_1$-factor $\mathcal{R}$. In this setting, the combination of a famous theorem of Ocneanu with a recent result by Brothier-Vaes shows that there is a dichotomy for the cardinality of the set of outer actions of a given group on $\mathcal{R}$: for an amenable group, there is a unique one, while for nonamenable groups there are uncountably many. In the context of C*-algebras, one may want to replace the hyperfinite II$_1$-factor with a UHF-algebra of infinite type. In this setting, far less is known, and all the uniqueness results that are available so far work only for abelian groups (but not even all of them). In this talk, I will report on some recent joint work with Martino Lupini, where we establish the existence of uncountably many, non equivalent ``free" actions of a given group with property (T) on a UHF-algebra. In fact, we show that the relation of equivalence for these actions is a complete analytic set. Thursday,
February 23
Abstract: The question of constructing hyperdegrees with some given properties appears naturally not only in recursion theory but also in effective descriptive set theory. We are particularly concerned with the hyperdegrees, which appear in the body of some certain categories of recursive trees (Kleene trees, Spector-Gandy trees). It turns out that a compactness-type result by Kreisel is an important tool in this study. In this talk we give some applications of the latter and we conclude with an open problem. Thursday, March 9
Abstract: We give a completely constructive solution to Tarski's circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If k > 0 and A,B \subset R^k are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than k, then A and B are equidecomposable by translations using Borel pieces. This answers a question of Wagon. Our proof uses ideas from the study of flows in graphs, and a recent result of Gao, Jackson, Krohne, and Seward on special types of witnesses to the hyperfiniteness of free Borel actions of Z^d. This is joint work with Spencer Unger. Tuesday,
March 14
Abstract: In the talk, we will discuss dynamical properties of the action of Thompson’s group F on the unit interval, using which we will show that the group F admits no non-trivial characters. Implications for the structure of invariant random subgroups will be also discussed. Wednesday, April 12
Abstract: In 2012, Palamourdas claimed that the graph G generated by n commuting Borel functions on a standard Borel space had Borel chromatic number (BCN) at most 2n + 1. In fact, his proof assumed that the functions have no fixed points. In joint work with Palamourdas, we revisit the problem and prove that if every vertex is connected to a fixed point, then there is an increasing filtration {X_i} of the Polish space for which G|X_i has BCN at most 2n for every i. Wednesday, April 19
Abstract: We will describe a new approach to convergence theorems in ergodic theory, which treats amenable groups and non-amenable groups on an equal footing. In particular, this approach gives a natural generalization of the von-Neumann, Birkhoff, and Shannon-McMillan-Breiman theorems to the case of negatively curved groups. We will focus in the talk on the case of free non-Abelian groups where the arguments are direct and easily accessible. Based on joint work with Lewis Bowen, and on joint work with Felix Pogorzelski. Wednesday, April 26
Abstract: I will describe some work in progress regarding the interplay between representations and topological actions. Wednesday, May 17
Abstract: Duality theory provides a general framework for strong completeness theorems yielding an exact correspondence between syntax and semantics for various fragments of logic. This talk will survey some known results in this framework, and then present a result relating the Cartesian fragment of ultrametric continuous logic to continuous locally presentable categories. The result treats structures on partial ultrametric spaces, on which predicates may be treated equivalently either as subspaces or as their [0, 1]-valued indicator functions. Wednesday, May 24
Abstract: We provide a direct proof of a recent theorem of Ben-Yaacov, Melleray and Tsankov. If $G$ is a Polish group and $X$ is a minimal metrizable $G$-flow with all orbits meager, we use $X$ to produce a new $G$-flow $S_G (X)$ which is minimal and non-metrizable. Tuesday, May 30
Abstract: Using the work of Hutchcroft and Nachmias on indistinguishability of the Wired Uniform Spanning Forest, R. Tucker-Drob recently proved a powerful theorem: any probability measure preserving locally countable ergodic Borel graph admits an ergodic hyperfinite subgraph. We give a completely different and self-contained proof of this theorem, which also goes through for a quasi-invariant probability measure, yielding a generalization of Tucker-Drob's theorem: any locally countable ergodic Borel graph on a standard probability space admits an ergodic hyperfinite subgraph. Our proof uses new tools whose combination yields an alternative way of exploiting nonamenability. In the first talk, we will state the main result and discuss one or two gadgets involved in the proof. In the second talk, we will finish the discussion of the new tools and give a sketch of our proof. Wednesday, May 31
Abstract: Using the work of Hutchcroft and Nachmias on indistinguishability of the Wired Uniform Spanning Forest, R. Tucker-Drob recently proved a powerful theorem: any probability measure preserving locally countable ergodic Borel graph admits an ergodic hyperfinite subgraph. We give a completely different and self-contained proof of this theorem, which also goes through for a quasi-invariant probability measure, yielding a generalization of Tucker-Drob's theorem: any locally countable ergodic Borel graph on a standard probability space admits an ergodic hyperfinite subgraph. Our proof uses new tools whose combination yields an alternative way of exploiting nonamenability. In the first talk, we will state the main result and discuss one or two gadgets involved in the proof. In the second talk, we will finish the discussion of the new tools and give a sketch of our proof.
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