# SCIENTIFIC PROGRAMS AND ACTIVITIES

November 27, 2014
THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
 Set Theory Seminar Series 2014-15 Fields Institute, Room 210 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
 2014 Fridays Upcoming Seminars at 1:30 p.m. in the Fields Institute, Room 210 November 28, 2014 12:30-15:00 Stewart Library Antonio Avil\'{e}s A combinatorial lemma about cardinals $\aleph_n$ and its applications on Banach spaces 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. Miodrag Sokic Lindel\"{o}f spaces of small extent are $\omega$-resolvable I intend to present the proof of the following result, joint with L. Soukup and Z. Szentmikl\'{o}ssy: 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 Lindel\"{o}f spaces of uncountable dispersion character are $\omega$-resolvable. This improves some results of Pavlov and of Filatova, respectively, concerning Malychin’s problem if regular Lindelof spaces of uncountable dispersion character are resolvable at all. The question if regular Lindelof spaces of uncountable dispersion character are maximally resolvable, i.e. $\Delta(X)$-resolvable, remains wide open. 2014-15 Past Seminars Speaker and Talk Title November 21, 2014 Istv\'{a}n Juh\'{a}sz 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