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Previous talks this semester:
25.01.2022, Tuesday 13.30, room 403
Piotr Koszmider (IM PAN)
The diamond principle and its applications
Abstract: "We will show the standard proof of the consistency of the diamond principle by forcing and how this principle is motivated by forcing considerations. We will also present some standard applications of the diamond principle"
14.12.2021, Tuesday 13.30, room 403
Agnieszka Widz (Lodz University of Technology)
Magic Sets
Abstract: "Given a family of real functions F we say that a set M ⊆ ℝ is magic for F if
for all f, g ∈ F we have f [M ] ⊆ g[M ] ⇒ f = g. This notion was introduced by
Diamond, Pomerance and Rubel in 1981 [1]. Recently some results about magic
sets were proved by Halbeisen, Lischka and Schumacher [2]. Inspired by their work
I constructed two families of magic sets one of them being almost disjoint and the
other one being independent. During my talk I will sketch the background and
present the proof for the independent family, which uses a Kurepa tree."
References:
- H. G. Diamond, C. Pomerance, L. Rubel, Sets on which an entire function is determined by
its range, Mathematische Zeitschrift, 176 (1981), 383-398.
- L. Halbeisen, M. Lischka, S. Schumacher, Magic Sets, Real Anal. Exchange, 43 (2018), 187-204.
7.12.2021, Tuesday 13.30, room 403
Kacper Kucharski (UW)
Overcomplete sets in selected nonseparable Banach spaces
Abstract: "A subset Y of a Banach space X is called overcomplete if
|Y|=dens(X) and for any set Z⊆Y, such that |Z|=|Y|, Z is
linearly dense in X. A classical result says that every separable
Banach space admits an overcomplete set. The main goal of the talk is to
show how, using a certain Aronszajn tree, one can step-up this property
for selected nonseparable Banach spaces. If there is enough time, one
consistency result will also be stated and proved."
30.11.2021, Tuesday 13.30, room 403
Piotr Koszmider (IMPAN)
Bidiscrete system in compact spaces
Abstract: "A set X of the square of a compact Hausdorff space K is called bidiscrete if for every (x, y) in X there is
a continuous real valued function f on K such that f(x)=1, f(y)=0 and f(x')=f(y') for any (x', y') in X-{(x, y)}. Bidiscrete sets play role in investigations related to biorthogonal systems in Banach spaces and irredundant sets in many algebraic structures induced by the compact K, but the question if there is in ZFC a nonmetrizable compact K with no uncountable bidiscrete set remains open. There are such examples under special set-theoretic assumptions (Kunen) and there are no such totally disconnected examples under other assumptions (Todorcevic). We will discuss these and other know results and open problems."
23.11.2021, Tuesday 13.30, room 403
Kamil Ryduchowski (UW)
A Banach space admitting few operators
Abstract: "Using the colouring discussed in our previous talk we will construct a Banach space admitting few operators in the following sense: it will be a non-separable Banach space X such that every operator on X is of the form sI + S, where s is a scalar, S is an operator with a separable range and I stands for the identity on X, i.e. every operator on X is a homothety modulo the ideal of operators with separable range. The construction is due to Shelah and Steprans."
16.11.2021, Tuesday 13.30, room 403
Kamil Ryduchowski (UW)
An antiramsey coloring of pairs
Abstract: "We present a fundamental theorem by Todorcevic, stating that there exists a coloring of the complete graph of the first uncountable cardinality in uncountably many colors without an uncountable monochromatic clique. We also discuss other results of Todorcevic of similar nature, which are in some sense multidimensional variants of that coloring."
Literature:
- Todorcevic, Stevo, Partitioning pairs of countable ordinals. Acta Math. 159 (1987), no. 3-4, 261--294.
- Velleman, Dan, Partitioning pairs of countable sets of ordinals. J. Symbolic Logic 55 (1990), no. 3, 1019--1021.
2.11.2021, Tuesday 13.30, room 105
Piotr Koszmider (IMPAN)
An application of Schreier's family
Abstract:
We will present the original solution due to J. Schreier of a problem of S. Banach which uses a subset of
the Boolean algebra of clopen subsets of [0, ωω] induced by
a family of finite subsets of ℕ known as Schreier's family.
26.10.2021, Tuesday 13.30, room 105
Damian Głodkowski (UW/IMPAN)
The poset of projections in the Calkin algebra, continuation
Abstract: We will discuss the set-theoretic properties of the poset of projections in the Calkin algebra of a separable Hilbert space, taking into account possible types of maximal well-ordered sequences and maximal antichains. We will show that it is consistent that among the mentioned subsets there are some with cardinality less than continuum and that Martin's axiom implies that all of them have cardinality continuum. We will also discuss relations between the poset of projections and the Boolean algebra P(ω)/Fin.
Based on: Wofsey, Eric; P(ω)/fin and projections in the Calkin algebra. Proc. Amer. Math. Soc. 136 (2008), no. 2, 719-726.
19.10.2021, Tuesday 13.30, room 105
Damian Głodkowski (UW/IMPAN)
The poset of projections in the Calkin algebra
Abstract: We will discuss the set-theoretic properties of the poset of projections in the Calkin algebra of a separable Hilbert space, taking into account possible types of maximal well-ordered sequences and maximal antichains. We will show that it is consistent that among the mentioned subsets there are some with cardinality less than continuum and that Martin's axiom implies that all of them have cardinality continuum. We will also discuss relations between the poset of projections and the Boolean algebra P(ω)/Fin.
Based on: Wofsey, Eric; P(ω)/fin and projections in the Calkin algebra. Proc. Amer. Math. Soc. 136 (2008), no. 2, 719-726.
Talks in the second semester of 2021-22.
Talks in the second semester of 2020-21.
Talks in the first semester of 2020-21.
Talks in the second semester of 2019-20.
Talks in the first semester of 2019-20.
Talks in the second semester of 2018-19.
Talks in the first semester of 2018-19.
Talks in the second semester of 2017-18.
Talks in the first semester of 2017-18.
Talks in the second semester of 2016-17.
Talks in the first semester of 2016-17.
Talks in the second semester of 2015-16.
Talks in the first semester of 2015-16.
Talks in the second semester of 2014-15.
Talks in the first semester of 2014-15.
Talks in the second semester of 2013-14.
Talks in the first semester of 2013-14.
Talks in the second semester of 2012-13.
Talks in the first semester of 2012-13.
Talks in the second semester of 2011-12.
Talks in the first semester of 2011-12.
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