One of the biggest open problems in mathematical physics has been the problem of formulating a complete and consistent theory of quantum gravity. Some of the core technical and epistemological difficulties come from the fact that General Relativity (GR) is fundamentally a geometric theory and, as such, it oughts to be "generally covariant", i.e., invariant under change of coordinates by any element of the diffeomorphism group \(\text{Diff}(M)\) of the ambient manifold \(M\). The Problem of Observables is a famous instance of the difficulties associated with general covariance, and one directly related to ineffectiveness of classical quantization recipes when it comes to GR. In a nutshell, the problem of observables asks whether GR admits a complete set of smooth observables. That is, whether the family of all diffeomorphism-invariant, real-valued, smooth maps on the space \(\text{Ein}(M)\) of all Einstein metrics on \(M\) is rich enough to separate all physical spacetimes. So far the only smooth observables known (when \(M={\mathbb R}^4)\) are the constant maps. In this talk, we will employ methods from descriptive set theory in order to answer the problem of observables in the negative. These results are inspired by old discussions with Marios Christodoulou and are based on recent work with George Sparling.
We use the projective Fraissé approach and Ramsey's theorem to show that the universal minimal flow of the homeomorphism group of the universal Knaster continuum is homeomorphic to the universal minimal flow of the free abelian group on countably many generators.We will define a projective Fraissé class whose limit approximates the universal Knaster continuum in such a way that the group \(\textrm{Aut}(\mathbb{K})\) of automorphisms of the Fraissé limit is a dense subgroup of the group, \(\textrm{Homeo}(K)\), of homeomorphisms of the universal Knaster continuum. The computation of the universal minimal flow involves modifying the Fraissé class in a natural way so that it approximates an open, normal, extremely amenable subgroup of \(\textrm{Homeo}(K)\).
We give a method of producing a Polish module over an arbitrary subring of \(\mathbb Q\) from an ideal of subsets of \(\mathbb N\) and a sequence in \(\mathbb N\). The method allows us to construct two Polish \(\mathbb Q\)-vector spaces, \(U\) and \(V\), such that– both \(U\) and \(V\) embed into \(\mathbb R\), but– \(U\) does not embed into \(V\) and \(V\) does not embed into \(U\),where by an embedding we understand a continuous \(\mathbb Q\)-linear injection. This construction answers a question of Frisch and Shinko. In fact, our method produces a large number of incomparable with respect to embeddings Polish \(\mathbb Q\)-vector spaces.This is joint work with Slawomir Solecki.
Let \(G\) be a countably infinite group and let \(\text{Sub}_{G}\) be the compact space of subgroups \(H \leqslant G\). Then an invariant random subgroup (IRS) of \(G\) is a probability measure \(\nu\) on \(\text{Sub}_{G}\) which is invariant under the conjugation action of \(G\) on \(\text{Sub}_{G}\).In this talk, after a brief introduction to the theory of invariant random subgroups, I will discuss some of the many basic questions in this relatively new area. For example, if \(\nu\) is an ergodic IRS of a countable group \(G\), then we obtain a corresponding zero-one law on \(\text{Sub}_{G}\) for the class of group-theoretic properties \(\Phi\) such that the set \(\{\, H \in \text{Sub}_{G} \mid H \text{ has property } \Phi \,\}\) is \(\nu\)-measurable; and thus \(\nu\) concentrates on a collection of subgroups which are quite difficult to distinguish between. Consequently, it is natural to ask whether there exists an ergodic IRS of a countable group \(G\) which does not concentrate on the subgroups \(H \leqslant G\) of a single isomorphism type.
In this talk, we will show that elementary bi-embeddability is an analytic complete equivalence relation under Borel reducibility by giving a reduction from the bi-embeddability relation on graphs. We will then discuss the degree spectra realized by these relations. The degree spectrum of a countable structure with respect to an equivalence relation \(E\), a central notion in computable structure theory, is the set of Turing degrees of structures \(E\)-equivalent to it. By analyzing the computability theoretic properties of our reduction from bi-embeddability to elementary bi-embeddability we show that the degree spectra of these two relations are related. Suppose a set of Turing degrees \(X\) is the bi-embeddability spectrum of a graph. Then the set of degrees whose Turing jump is in \(X\) is the elementary bi-embeddability spectrum of a graph. Combining results of Harrison-Trainor and the coauthor one sees that this is sharp: There is a bi-embedddability spectrum that is not an elementary bi-embeddability spectrum.
Given a finite relational language \(\mathcal L\) and a (possibly infinite) set \(\mathcal F\) of finite irreducible \(\mathcal L\)-structures, the class \(\operatorname{Forb}(\mathcal F)\) describes those finite \(\mathcal L\)-structures which do not embed any member of \(\mathcal F\). Classes of the form \(\operatorname{Forb}(\mathcal F)\) exactly describe those classes of finite \(\mathcal L\)-structures with free amalgamation. In recent joint work with Balko, Chodounsky, Dobrinen, Hubicka, Konecny, and Vena, we exactly characterize big Ramsey degrees for those classes \(\operatorname{Forb}(\mathcal F)\) where the forbidden set \(\mathcal F\) is finite. This characterization proceeds by defining tree-like objects called diagonal diaries, then showing that the big Ramsey degree of any \(A\) in \(\operatorname{Forb}(\mathcal F)\) is exactly the number of diagonal diaries which code the structure \(A\). After giving a brief description of these objects, the talk will then consider those infinite diagonal diaries which code the Fraisse limit of \(\operatorname{Forb}(\mathcal F)\). In upcoming joint work with Dobrinen, we prove a Galvin-Prikry theorem for any such infinite diagonal diary, giving new examples of objects satisfying the Galvin-Prikry theorem which dramatically fail to satisfy Todorcevic's Ramsey space axioms A1 through A4.
Kaleidoscopic groups are infinite permutation groups recently introduced by Duchesne, Monod, and Wesolek by analogy with a classical construction of Burger and Mozes of subgroups of automorphism groups of regular trees. In contrast with the Burger-Mozes groups, kaleidoscopic groups are never locally compact and they are realized as groups of homeomorphisms of Wazewski dendrites (tree-like, compact spaces whose branch points are dense). The input for the construction is a finite or infinite permutation group \(\Gamma\) and the output is the kaleidoscopic group \(K(\Gamma)\).In this talk, I will discuss recent joint work with Gianluca Basso, in which we explain how these groups can be viewed as automorphism groups of homogeneous structures and characterize the universal minimal flow of \(K(\Gamma)\) in terms of the original group \(\Gamma\).
In the decade 1994 - 2004 I wrote five papers applying techniques from descriptive set theory to a question posed by the dynamics group of Barcelona concerning the possible lengths of iterations. In August 2021 I gave two talks in the CUNY set theory Zoominar of Vika Gitman which were largely devoted to expounding the last and hardest of my constructions in this area. The present talk will be devoted to my earlier and more basic results, some recent work, and various open problems which I hope might attract logicians working in areas such as the descriptive set theory of group actions.