# SCIENTIFIC PROGRAMS AND ACTIVITIES

August 22, 2017
THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
 Set Theory Seminar Series 2014-15 Fields Institute Friday 1:30 pm Organizing Committee: Miguel Angel Mota, Ilijas Farah, Juris Steprans, Paul Szeptycki
 List of previous talks Seminars held during 2013-14 York U Set Theory Page
 Seminars from July 1, 2015 onwards can be found on the 2015-2016 Set Theory Seminar Page 2015 Fridays Seminars June 30, 2015 H. Jerome Keisler, Randomization of scattered theories Consider a sentence $\phi$ of the infinitary logic $L_{\omega_1, \omega}$. In 1970, Morley introduced the notion of a scattered sentence, and showed that if $\phi$ is scattered then the class $I(\phi)$ of isomorphism types of countable models of $\phi$ has cardinality at most $\aleph_1$, and if $\phi$ is not scattered then $I(\phi)$ has cardinality continuum. The absolute form of Vaught's conjecture for $\phi$ says that if $\phi$ is scattered then $I(\phi)$ is at most countable. Generalizing previous work of Ben Yaacov and the author, we introduce here the notion of a separable model of $\phi^R$, which is a separable continuous structure whose elements are random elements of a model of $\phi$. We say that $\phi^R$ has few separable models if every separable model of $\phi^R$ is uniquely characterized up to isomorphism by a function that assigns probabilities summing to one to countably many elements of $I(\phi)$. In a previous paper, Andrews and the author showed that if $\phi$ is a complete first order theory and $I(\phi)$ is at most countable then $\phi^R$ has few separable models. We show here that this result holds for all $\phi$, and that if $\phi^R$ has few separable models then $\phi$ is scattered. Hence if the absolute Vaught conjecture holds for $\phi$, then $\phi^R$ has few separable models if and only if $I(\phi)$ is countable, and also if and only if $\phi$ is scattered. Moreover, assuming Martin's axiom for $\aleph_1$, we show that if $\phi$ is scattered then $\phi^R$ has few separable models. June 26, 2015 Franklin Tall, PFA(S)[S] II This is a continuation of last week's lecture. Last week's lecture was largely motivation; this lecture will be mainly technical, developing the method. If you really want to attend and missed last week, contact me and I will give you something to read. June 19, 2015 Franklin Tall, PFA(S)[S] and locally countable subspaces of compact countably tight spaces. I have lectured many times in the seminar on Stevo’s method of forcing with a coherent Souslin tree S over a model of PFA restricted to posets that preserve S, since it has many interesting applications in set-theoretic topology. However I believe the current cohort of graduate students has not seen an actual proof of this sort. Since the seminar is suffering from a lack of speakers, I plan to give a sporadic series of lectures featuring such proofs. In particular, as soon as I understand it sufficiently well, I want to give Alan Dow’s proof that in such models, first countable perfect pre-images of omega_1 include copies of omega_1. This is the capstone of the proof of the consistency of every hereditarily normal manifold of dimension > 1 being metrizable. First of all, however, I want to prove a technical theorem that is necessary for the manifold result, and for many other results concerning under what conditions locally compact normal spaces are paracompact. This particular theorem – getting locally countable collections to be sigma-discrete - is perhaps not of wide interest, but the method of getting an uncountable set in such a model to be the union of countably many “nice” subsets (rather than just including an uncountable nice subset) should have more applications. The “proof” I gave of this result in the seminar five years ago turned out to have a gap. The gap is bridged by a clever idea of Stevo. The proof will appear in a joint paper. June 12, 2015 Asger Törnquist, Definable maximal orthogonal families in forcing extensions Two Borel probability measures nu and mu on Cantor space are orthogonal if there is a Borel set which has measure 1 for nu, but measure 0 for mu. An orthogonal family of measures is a family of pairwise orthogonal measures; it is maximal if it is maximal under inclusion. Maximal orthogonal families of measures can't be analytic; this is a theorem of Preiss and Rataj (1985). A few years ago, Vera Fischer and I showed that in L there is a Pi-1-1 (lightface) maximal orthogonal family (a "mof") of measures in L, but that adding a Cohen real to L destroys all Pi-1-1 mofs. Subsequently, it was shown that the same holds if we add a random real (Friedman-Fischer-T.). This motivated the question: Can a Pi-1-1 mof coexist with a non-constructible real? In this talk we answer this by showing there is a Pi-1-1 mof in the Sacks and Miller extensions of L. By contrast, we will see that in the Mathias extension of L there are no Pi-1-1 mofs, and in the process of doing so we will obtain a new proof of the Preiss-Rataj theorem. This is joint work with David Schrittesser. May 22, 2015 Francisco Kibedi, Maximal Saturated Linear Orders In his 1907 paper about pantachies (maximal linearly ordered subsets of the space of real-valued sequences partially ordered by eventual domination), Felix Hausdorff poses several questions that he was unable to answer, including a question he labels $(\alpha)$: Is there a pantachie with no $(\omega_1, \omega_1)$-gaps? Hausdorff knew that CH implies the answer is no; in other words, under CH, a pantachie must have $(\omega_1, \omega_1)$-gaps. However, Hausdorff's question turns out to be independent of ZFC. We answer question $(\alpha)$ by proving something a bit stronger, namely, Con(ZFC + $\lnot$CH + $\exists$ a maximal saturated linear order of size continuum in the space of real-valued sequences partially ordered by eventual domination). We then extend this result to include Martin's Axiom --- i.e., we prove Con(ZFC + MA + $\lnot$CH + $\exists$ a maximal saturated linear order of size continuum in the space of real-valued sequences partially ordered by eventual domination). Note: This seminar will be held in the Bahen Center, room BA 1220. May 1, 2015 Alessandro Vignati Forcing axioms and Operator algebras: a lifting theorem for reduced products of matrix algebras Inspired by the work of Farah and others in the application of forcing axioms to operator algebras, we prove a correspondent of a lifting theorem in a continuous setting. Analyzing different kinds of maps from the reduced product of matrix algebra into a corona of a nuclear C*-algebra, we provide different notions of well-behaved lifting, and we show how forcing axioms imply their existence, in contrast to the results obtained under the Continuum Hypothesis. Secondly, we show some consequences of such a behavior. All required definitions will be given. This is joint work with Paul McKenney. April 17, 2015 Robert Raphael On the countable lifting property for C(X) Suppose that Y is a subspace of a Tychonoff space X so that the induced ring homomorphism $C(X) \rightarrow C(Y)$ is onto. We show that a countable set of pairwise orthogonal functions in C(Y) can be lifted to a pairwise orthogonal preimage in C(X). The question originally arose in vector lattices. Topping published the result for vector lattices using an erroneous induction, but two years later Conrad gave a counterexample. This is joint work with A.W.Hager. April 10, 2015 Christopher Eagle Model Theory of Compacta Topological spaces do not fit well into the framework of first-order model theory; nevertheless, tools from model theory have had some success in applications to compacta. Model-theoretic ideas have been used in topology in two ways: First, by finding suitable first-order structures to use as stand-ins for topological spaces, and second, by directly "dualizing" notions from model theory. We will describe both of these methods, and compare them to a newer approach which applies real-valued logic to the rings of continuous complex-valued functions of compact spaces. Using continuous logic we show that the pseudo-arc is a co-existentially closed continuum, answering a question of P. Bankston. We also show that the only compact metrizable spaces $X$ where $C(X)$ has quantifier elimination in continuous logic are the one-point space, the two-point space, and the Cantor set. This is joint work with Isaac Goldbring and Alessandro Vignati. March 27, 2015 Dana Bartosova About the conjecture that oligomorphic groups have metrizableuniversal minimal flows We will discuss a conjecture of Lionel Nguyen van Th\'e as in the title. It was shown by Andy Zucker to be equivalent to whether every class of finitary approximations of a countable ultrahomogeneous structure with oligomorphic automorphism group has a finite Ramsey degree. We look at the problem from the Boolean algebra point of view. An interesting example in this context is the automorphism group of a topological structure whose natural quotient is the pseudo-arc, which is a work in progress with Aleksandra Kwiatkowksa (UCLA). March 20, 2015 Room 230 Mike Pawliuk Amenability and Directed Graphs Part 2 : Cherlin's List Last week Miodrag spoke in general about Amenability, Fraisse classes and consistent random expansions. This talk will be more specific and focus on checking the amenability and unique ergodicity of the automorphism groups of the directed graphs on Cherlin's list. In addition, we will present a type of product of Fraisse classes that behaves nicely with respect to amenability and unique ergodicity. March 13, 2015 Miodrag Sokic Amenability and directed graphs Amenability for locally compact and countable groups has been extensively studied. In this talk we will give some results in the case of non-archimedean groups. In particular, we consider groups of anthropomorphism of structures from the Cherlin list of ultrahomogeneous directed graphs. February 27, 2015 Frank Tall Some observations on the Baireness of C_k(X) for a locally compact space X The area in-between Empty not having a winning strategy and Nonempty having a winning strategy in the Banach-Mazur game has attracted interest for many decades. We answer some questions Marion Scheepers asked when he was here last year, and also prove results related to his recent paper with Galvin and to a paper of Gruenhage and Ma. Our tools include PFA(S)[S] and non-reflecting stationary sets. February 27, 2015 12:00pm Saeed Ghasemi Rigidity of corona algebras In my thesis I use techniques from set theory and model theory to study the isomorphisms between certain classes of C*-algebras. In particular we look at the isomorphisms between corona algebras of direct sums of sequences of full matrix algebras. We will see that the question "whether any isomorphism between these C*-algebras is trivial" is independent from the usual axioms of set theory (ZFC). I also extend the classical Feferman-Vaught theorem to reduced products of metric structures. This theorem has a number of interesting consequences. In particular it implies that the reduced powers of elementarily equivalent structures are elementarily equivalent. We also use this to find examples of corona algebras of direct sums of sequences of full matrix algebras which are non-trivially isomorphic under the Continuum Hypothesis. This gives the first example of genuinely non-commutative structures with this property. In the last chapter of my thesis I have shown that SAW*-algebras are not isomorphic to tensor products of two infinite dimensional C*-algebras, for any C*-tensor product. This answers a question of S. Wassermann who asked whether the Calkin algebra has this property. February 11, 2015 3:30pm Stewart Library Alessandro Vignati Set theory and amenable operator algebras I will present my past and present work on logic and operator algebras. First I will show the construction of a nonseparable amenable operator algebra A with the property that every nonseparable subalgebra of A is not isomorphic to a C*-algebra, yet A is an inductive limit of algebras isomorphic to C*-algebras. Secondly, I will sketch possible techniques, associated to Model Theory in a continuous setting, that can be applied to operator algebras. February 6, 2015 Room 230 Logan Hoehn A complete classification of homogeneous plane compacta In this topology talk, we will discuss homogeneous spaces in the plane $R^2$. A space X is homogeneous if for every pair of points in X, there is a homeomorphism of X to itself taking one point to the other. Kuratowski and Knaster asked in 1920 whether the circle is the only connected homogeneous compact space in the plane. Explorations of this problem fueled a significant amount of research in continuum theory, and among other things, led to the discovery of two new homogeneous spaces in the plane: the pseudo-arc and the circle of pseudo-arcs. I will describe our recent result which implies that there are no more undiscovered homogeneous compact spaces in the plane. This is joint work with Lex Oversteegen of the University of Alabama at Birmingham. January 23, 2015 David Fernandez A model of ZFC with strongly summable ultrafilters, small covering of meagre and large dominating number Strongly summable ultrafilters are a variety of ultrafilters that relate with Hindman's finite sums theorem in a way that is somewhat analogous to that in which Ramsey ultrafilters relate to Ramsey's theorem. It is known that the existence of these ultrafilters cannot be proved in ZFC, however such an existencial statement follows from having the covering of meagre to equal the continuum. Furthermore, using ultraLaver forcing in a short finite support iteration, it is possible to get models with strongly summable ultrafilters and a small covering of meagre, and these models will also have small dominating number. Using this ultraLaver forcing in a countable support iteration to get a model with small covering meagre and strongly summable ultrafilters is considerably harder, but it can be done and in this talk I will explain how (it involves a characterisation of a certain kind of strongly summable ultrafilter in terms of games). Interesingly, this way we also get the dominating number equal to the continuum, unlike the previously described model. January 16, 2015 Marcin Sabok Automatic continuity for isometry groups We present a general framework for automatic continuity results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the groups of isometries of the Urysohn space and the Urysohn sphere, i.e. that any homomorphism from either of these groups into a separable group is continuous. This answers a question of Melleray. As a consequence, we get that the group of isometries of the Urysohn space has unique Polish group topology and the group of isometries of the Urysohn sphere has unique separable group topology. Moreover, as an application of our framework we obtain new proofs of the automatic continuity property for the group $\mathrm{Aut}([0,1],\lambda)$, due to Ben Yaacov, Berenstein and Melleray and for the unitary group of the infinite-dimensional separable Hilbert space, due to Tsankov. The results and proofs are stated in the language of model theory for metric structures. December 16, Tuesday *CANCELLED* BAHEN CENTRE, Room BA6183, 13:30-15:00, talk by Neil Hindman. December 12, 2015 13:30-15:00 Dilip Raghavan. Embedding $P(\omega)/FIN$ into the $P$-points We show under $\mathfrak{p}=\mathfrak{c}$ that $P(\omega)/FIN$ can be embedded into the $P$-points under RK and Tukey reducibility. December 5, 2015 no seminar scheduled due to Dow Conference. November 28, 2014 12:30-3:00 Stewart Library Antonio Avilés A combinatorial lemma about cardinals $\aleph_n$ and its applications on Banach spa The lemma mentioned in the title was used by Enflo and Rosenthal to show that the Banach space $L_p[0,1]^\Gamma$ does not have an unconditional basis when $|\Gamma|\geq \aleph_\omega$. In a joint work with Witold Marciszewski, we used some variation of it to show that there are no extension operators between balls of different radii in nonseparable Hilbert spaces. Istvan Juhász Lindelof spaces of small extent are $\omega$-resolvable I intend to present the proof of the following result, joint with L. Soukup and Z. Szentmiklossy: Every regular space $X$ that satisfies $\Delta(X) > e(X)$ is $\omega$-resolvable, i.e. contains infinitely many pairwise disjoint dense subsets. Here $\Delta(X)$, the dispersion character of $X$, is the smallest size of a nonempty open set in $X$ and $e(X)$, the extent of $X$, is the supremum of the sizes of all closed-and-discrete subsets of $X$. In particular, regular Lindelof spaces of uncountable dispersion character are $\omega$-resolvable. November 21, 2014 Miodrag Sokic Functional classes We consider the class of finite structures with functional symbols with respect to the Ramsey property. November 14, 2014 Martino Lupini Fraisse limits of operator spaces and the noncommutative Gurarij space We realize the noncommutative Gurarij space introduced by Oikhberg as the Fraisse limit of the class of finite-dimensional 1-exact operator spaces. As a consequence we deduce that such a space is unique, homogeneous, universal among separable 1-exact operator spaces, and linearly isometric to the Gurarij Banach space. November 7, 2014 Stewart Library Juris Steprans The descriptive set theoretic complexity of the weakly almost periodic functions in the dual of the group algebra The almost periodic functions on a group G are those functions F from G to the complex number such that the uniform norm closure of all shifts of F is compact in the uniform norm. The weakly almost periodic functions are those for which the analogous statement holds for the weak topology. The family of sets whose characteristic functions are weakly almost periodic forms a Boolean algebra. The question of when this family is a complete $\Pi^1_1$ set will be examined. October 31, 2014 Speaker 1 (from 12:30 to 13:30): Vera Fischer Definable Maximal Cofinitary Groups and Large Continuum A cofinitary group is a subgroup of the group of all permutations of the natural numbers, all non-identity elements of which have only finitely many fixed points. A cofinitary group is maximal if it is not properly contained in any other cofinitary group. We will discuss the existence of nicely definable maximal cofinitary groups in the presence of large continuum and in particular, we will see the generic construction of a maximal cofinitary group with a $\Pi^1_2$ definable set of generators in the presence of $2^\omega=\aleph_2$. Speaker 2 (from 13:30 to 15:00): Menachem Magidor On compactness for being $\lambda$ collectionwise hausdorff A compactness property is the statement for a structure in a given class, if every smaller cardinality substructure has a certain property then the whole structure has this property. In this talk we shall deal with the compactness for the property of a topological space being collection wise Hausdorff. The space is X is said to be $\lambda$--collection wise Hausdorff ($\lambda$--cwH) if every closed discrete subset of X of cardinality less than $\lambda$ can be separated by a family of open sets. X is cwH if it is $\lambda$--cwH for every cardinal $\lambda$. We shall deal with the problem of when $\lambda$--cwH implies cwH, or just when does $\lambda$--cwH implies $\lambda^+$--cwH. A classical example of Bing provides for every cardinal $\lambda$ a space $X_\lambda$ which is $\lambda$-cwH but not $\lambda^+$--cwH. So if we hope to get any level of compactness for the the property of being cwH, we have to restrict the class of spaces we consider. A fruitful case is the case where we restrict the local cardinality of the space. A motivating result is the construction by Shelah (using supercompact cardinal) of a model of Set Theory in which a space which is locally countable and which is $\omega_2$--cwH is cwH. Can the Shelah result be generalized to larger cardinals , e.g. can you get a model in which for spaces which are locally of cardinality $\leq \omega_1$ and which are $\omega_3$-cwH are cwH? In general for which pair of cardinals $(\lambda, \mu)$ we can have models in which a space which is locally of cardinality $< \mu$ and which is $\lambda$--cwH are $\lambda^+$--cwH? In this lecture we shall give few examples where we get some ZFC theorems showing that for some pairs $(\lambda, \mu)$ compactness necessarily fails, and cases of pairs for which one can consistently have compactness for the property of being cwH. October 24, 2014 Jordi Lopez Abad Ramsey properties of embeddings between finite dimensional normed spaces Given d=m , let E m,n be the set of all m×d matrices (a i,j ) such that (a) ? d j=1 |a i,j |=1 for every 1=i=m . (b) max m i=1 |a i,j |=1 for every 1=j=d . These matrices correspond to the linear isometric embeddings from the normed space l d 8 :=(R d ,?·? 8 ) into l d 8 , in their unit bases. We will discuss and give (hints of) a proof of the following new approximate Ramsey result: For every integers d , m and r and every e>0 there exists n such that for every coloring of E d,n into r -many colors there is A?E m,n and a color i1 and e>0 , and every integer r , there is some n such that for every coloring of Emb ? 2 (F,l n 8 ) into r -many colors there is T?Emb ? (G,l n 8 ) and some color i