Wednesday seminar

Dear all,

The seminar meets on Wednesday June 27th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Jindra Zapletal will present a talk.

Best,
David

Ilya Shapirovsky: Locally finite varieties of modal algebras

Place: Fields Institute (Room 210)
Date: June 15, 2018 (13:30-15:00)
Speaker: Ilya Shapirovsky
Title: Locally finite varieties of modal algebras
Abstract: A modal algebra is a Boolean algebra enriched with an additive operation. Equational theories of modal algebras are called modal logics. A logic L is said to be n-tabular if, up to the equivalence in L, there exist only finitely many n-variable formulas.  L is locally tabular if it is n-tabular for all finite n. Algebraically, n-tabularity of a logic means that its n-generated free algebra is finite (thus,  local tabularity of a logic is equivalent to local finiteness of its variety).

It is known that a variety of closure algebras  is locally finite iff its one-generated free algebra is finite (Larisa Maksimova, 1975). The following question has been open since 1970s: does this equivalence hold for every variety of modal algebras? (The analogous problem is open for varieties of Heyting algebras: does 2-tabularity of an intermediate logic imply local tabularity?)

Recently, in our joint work with Valentin Shehtman, it was shown how local tabularity of modal logics can be characterized in terms of partitions of relational structures. I will discuss this criterion and then use it to construct the first example  of a 1-tabular but not locally tabular modal logic.

Mirna Džamonja: Higher order versions of the logic of chains close

Dear all,

The seminar meets on Wednesday June 20th at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Mirna Džamonja — Higher order versions of the logic of chains
close

First order logic of chains was discovered by Carol Karp and revisited
in recent work of Dz. with Jouko Vaananen. The results have shown that
the logic, defined through a singular cardinal of countable cofinality,
behaves very much like the first order logic. In our new joint work, we
study higher order versions of the logic of chains and their fragments
to defend the thesis that in this context we can also recover
similarities with the ordinary logic. We also discuss the idea of
infinite computation.

Best,
David

Joshua Brot, Mengyang Cao, David J. Fernández-Bretón: Finiteness classes arising from Ramsey-theoretic statements in set theory without choice

Thursday, June 14, 2018, from 4 to 5:30pm
East Hall, room 4096

Speaker: Joshua Brot, Mengyang Cao, David J. Fernández-Bretón (University of Michigan)

Title: Finiteness classes arising from Ramsey-theoretic statements in set theory without choice

Abstract:

We investigate infinite sets that witness the failure of certain Ramsey-theoretic statements, such as Ramsey’s or (appropriately phrased) Hindman’s theorem; such sets may exist if one does not assume the Axiom of Choice. We will show very precise information as to where such sets are located within the hierarchy of infinite Dedekind-finite sets. The proofs involve both very pleasant combinatorial arguments (to establish certain implications) and the Fränkel-Mostowski technique to obtain permutation models of ZFA (to prove that certain other implications are not provable).

Zoltán Vidnyánszky: Borel chromatic numbers: basis and antibasis results

Talk held by Zoltán Vidnyánszky (KGRC) at the KGRC seminar on 2018-06-14.

Abstract: We give a full description of the existence of a homomorphism basis for Borel graphs of given Borel chromatic number. In particular, we show that there is a Borel graph with Borel chromatic number 3 that admits a homomorphism to any Borel graph of Borel chromatic number at least 3. We also discuss the relation of these results to Hedetniemi’s conjecture.

Andrea Vaccaro: Embedding C*-algebras into the Calkin algebra

Place: Bahen Centre Information (Room BA 2165)
Date: June 8, 2018 (13:30-15:00)
Speaker: Andrea Vaccaro
Title: Embedding C*-algebras into the Calkin algebra
Abstract:

Given a separable Hilbert space H, the Calkin algebra is the quotient C(H) = B(H)/K(H), B(H) being the algebra of all linear continuous operators from H into itself, and K(H) the closed ideal of compact operators. The Calkin algebra can be considered a noncommutative analogue of P(omega)/Fin, and it is known that these two objects share many structural properties. We show that yet another property of P(omega)/Fin has a noncommutative analogue for C(H). In particular, it is known that for every poset P there is a ccc poset H_P which forces the existence of an embedding of P into P(omega)/Fin. We prove that for any C*-algebra A there exists a ccc poset which forces the existence of an embedding of A into C(H).

Andrea Medini: Homogeneous spaces and Wadge theory

Talk held by Andrea Medini (KGRC) at the KGRC seminar on 2018-06-07.

Abstract: All spaces are assumed to be separable and metrizable. A space $X$ is homogeneous if for all $x,y\in X$ there exists a homeomorphism $h:X\longrightarrow X$ such that $h(x)=y$. A space $X$ is strongly homogeneous if all non-empty clopen subspaces of $X$ are homeomorphic to each other. We will show that, under the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (with the trivial exception of locally compact spaces). This extends results of van Engelen and complements a result of van Douwen. Our main tool will be Wadge theory, which provides an exhaustive analysis of the topological complexity of the subsets of $2^\omega$.

This is joint work with Raphaël Carroy and Sandra Müller.

Vera Trnkova

Dear all,

I regret to inform you that Věra Trnková passed away on Sunday May 27th. The memorial service will be held on Wednesday June 6th at 9:20 in Strasnice.

David

Viera Šottová: Ideal version of selection principle S1(P,R)

The seminar meets on Wednesday June 6t at 11:00 in the Institute of Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Speaker: Viera Šottová

Title: Ideal version of selection principle $S_1(\mathcal P,\mathcal R)$.

Abstract: attached.

Very Informal European Gathering, Bristol (England), 8-9 Jun 2018

The VIEG – 2018 will be held on Friday-Saturday June 8-9th 2018 at the School of Mathematics, University of Bristol. Invited participants include:

• David Aspero (UEA)
• Raffaela Cutolo (Naples)
• Mirna Dzamonja (UEA)
• Martin Goldstern (TU Vienna)
• Asaf Karagila (UEA)
• Benedikt Löwe (Hamburg, ILLC Amsterdam)
• Charles Morgan (Bristol)