March 2021

# Gonzalo Martinez Cervantes: L-orthogonal sequences versus L-orthogonal elements

25/03/21 02:47

Tuesday, March 30, 2021 17:00

This is a joint work (in progress) with Antonio Avilés and Abraham Rueda Zoca.

*Location:***Zoom.us**: if you want to participate please contact organizers*Gonzalo Martinez Cervantes (University of Murcia)*

Speaker:Speaker:

*Title*: L-orthogonal sequences versus L-orthogonal elements*Abstract*: Let \(X\) be a Banach space. We say that a sequence \(\{x_n\}_n\) in the sphere of a Banach space \(X\) is an L-orthogonal sequence if the norm of \(x+x_n\) converges to \(1+\|x\|\) for every \(x\) in \(X\). On the other hand, we say that an element \(x^{**}\) in the sphere of \(X^{**}\) is L-orthogonal to \(X\) if the norm of \(x^{**}+x\) is equal to \(1+\|x\|\) for every \(x\) in \(X\). In this talk we will recall some results due to G. Godefroy, N. J. Kalton, B. Maurey, V. Kadets, V. Shepelska and D.Werner relating these concepts to the containment of an isomorphic copy of \(\ell_1\). It is natural to conjecture that the weak*-closure of an L-orthogonal sequence always contains L-orthogonal elements in the bidual. Indeed, this is the case for separable Banach spaces. We will see that this conjecture is independent of ZFC. Namely, we provide an affirmative answer under the existence of selective ultrafilters, whereas a counterexample can be constructed if no Q-point exists.This is a joint work (in progress) with Antonio Avilés and Abraham Rueda Zoca.

# Władysław Wilczyński: Convergence with respect to measure and category

17/03/21 09:33

Tuesday, March 23, 2021 17:00

Some lemmas similar to that in the paper of Bucchioni were used earlier to prove the equivalence of the convergence in category and the Cauchy condition for this type of convergence.

*Location:***Zoom.us**: if you want to participate please contact organizers*Władysław Wilczyński (University of Łódź)*

Speaker:Speaker:

*Title*: Convergence with respect to measure and category*Abstract*: D. Fremlin in 1975 has proved that if \((X,S,m)\) is a probability space, then a sequence of measurable functions on \(X\) either has a subsequence convergent a.e., or there exists a subsequence without measurable pointwise cluster point. His proof is based upon the properties of weak convergent sequences in square integrable functions. The weaker form of the theorem was proved by Bucchioni and Goldman in1978. Their proof uses only some properties of the pair (family of measurable subsets of \([0,1]\), family of null sets). The pair (family of subsets of \([0,1]\) having the Baire property, family of sets of the first category) behaves similarly , so it was possible to obtain similar result for the convergence in category considered by E. Wagner in 1978.Some lemmas similar to that in the paper of Bucchioni were used earlier to prove the equivalence of the convergence in category and the Cauchy condition for this type of convergence.

# Arturo Antonio Martínez Celis Rodríguez: Rosenthal Families

10/03/21 15:24

Tuesday, March 16, 2021 17:00

*Location:***Zoom.us**: if you want to participate please contact organizers*Arturo Antonio Martínez Celis Rodríguez (University of Wroclaw)*

Speaker:Speaker:

*Title*: Rosenthal Families*Abstract*: A collection of infinite subsets of the natural numbers is a Rosenthal family if it can replace the family of all infinite subsets in a classical Lemma by Rosenthal concerning sequences of measures on pairwise disjoint sets. In this talk we will show that every ultrafilter is a Rosenthal family and that the minimal size of a Rosenthal family is the reaping number. We will also try to show some connections to functional analysis.# Grigor Sargsyan: The exact strength of Sealing

03/03/21 17:25

Tuesday, March 9, 2021 17:00

Sealing is a generic absoluteness statement which was introduced by Woodin. First given a generic object \(g\), let \(\Gamma^\infty_g\) be the set of universally Baire sets of \(V[g]\) and \(R_g\) be the set of reals of \(V[g]\).

Sealing (essentially) says that for all \(V\)-generic \(g\) and all \(V[g]\)-generic \(h\) there is an embedding

\(j: L(\Gamma^\infty_g, R_g)\to L(\Gamma^\infty_g*h, R_g*h).\)

Thus, in a way, Sealing says that there cannot be independence results about universally Baire sets, and as such it is a generalization of Shoenfield's absoluteness theorem.

It is an open problem if large cardinals imply Sealing. No canonical inner model can satisfy it, and so if some large cardinal implies it then its inner model theory must be significantly different than the current theory we have. Surprisingly, Woodin showed that if there are proper class of Woodin cardinals and delta is a supercompact then collapsing \(2^{2^\delta}\) to be countable forces Sealing. Because of its impact on the inner model problem and because of Woodin's result, it seemed that the set theoretic strength of Sealing must be at the level of supercompact cardinals. However, the speaker and Nam Trang showed that it is weaker than a Woodin cardinal that is a limit of Woodin cardinals (which are significantly smaller than supercompact cardinals). We will exposit this theorem and will also explain its consequences on the inner model problem.

*Location:***Zoom.us**: if you want to participate please contact organizers*Grigor Sargsyan (Rutgers & IMPAN)*

Speaker:Speaker:

*Title*: The exact strength of Sealing*Abstract*: Shoenfield's celebrated absoluteness theorem says that no \(\Sigma^1_2\) fact \(\phi\) can be shown to be independent of the axioms of ZFC via the method of forcing. A set of reals is universally Baire if its continuous preimages have the Baire property in all topological spaces. Can there be independence results about such sets?Sealing is a generic absoluteness statement which was introduced by Woodin. First given a generic object \(g\), let \(\Gamma^\infty_g\) be the set of universally Baire sets of \(V[g]\) and \(R_g\) be the set of reals of \(V[g]\).

Sealing (essentially) says that for all \(V\)-generic \(g\) and all \(V[g]\)-generic \(h\) there is an embedding

\(j: L(\Gamma^\infty_g, R_g)\to L(\Gamma^\infty_g*h, R_g*h).\)

Thus, in a way, Sealing says that there cannot be independence results about universally Baire sets, and as such it is a generalization of Shoenfield's absoluteness theorem.

It is an open problem if large cardinals imply Sealing. No canonical inner model can satisfy it, and so if some large cardinal implies it then its inner model theory must be significantly different than the current theory we have. Surprisingly, Woodin showed that if there are proper class of Woodin cardinals and delta is a supercompact then collapsing \(2^{2^\delta}\) to be countable forces Sealing. Because of its impact on the inner model problem and because of Woodin's result, it seemed that the set theoretic strength of Sealing must be at the level of supercompact cardinals. However, the speaker and Nam Trang showed that it is weaker than a Woodin cardinal that is a limit of Woodin cardinals (which are significantly smaller than supercompact cardinals). We will exposit this theorem and will also explain its consequences on the inner model problem.