Institute of Mathematics

Czech Academy of Sciences

Žitná 25, 115 67 Praha 1

Czech Republic

Abstract: In this recent joint work with Alessandro Carderi we systematically study (commensurably) maximal amenable subgroups of arithmetic groups (in characteristic 0). We prove a classification result for those and, using the notion of singularity introduced in earlier work by Remi Boutonnet and Alessandro Carderi, we prove that they are singluar and thus give rise to maximal amenable von Neumann subalgebras of the corresponding group von Neumann algebras. This provides a rich source of new examples of maximal amenable von Neumann subalgebras. A particular feature of this approach is that the groups can be constructed very explicitly using some algebraic number theory which will also be briefly discussed in the talk.

Abstract: (Joint work with Wiesław Kubiś)
Fraïssé theory may be viewed as a study of *universal homogeneous objects*. These notions can be formulated abstractly in the language of category theory. In a suitable category it is possible to construct a universal homogeneous object as a (co)limit of a *Fraïssé sequence*. This framework includes and generalizes the classical model-theoretic Fraïssé theory as well as quite recent projective Fraïssé theory by Irwin and Solecki. In the talk I will summarize the framework and briefly sketch several ways how to extend it – beyond the countable case; by weakening the amalgamation property, which is closely connected with the abstract Banach–Mazur game and generic objects; and beyond the discrete case, i.e. when the strict commutativity of diagrams is replaced by ε-commutativity with better and better ε.

Abstract: (Joint work with Bertalan Bodor) We study structures where the number of orbits of the componentwise action of the automorphism group on n-tuples with pairwise distinct entries is bounded by some function in n. We prove that if the bound is cn^{dn} for some constants c,d with c<1, then the resulting class of structures is precisely the class K of finite covers of first-order reducts of unary structures. It follows that K is closed under taking model companions and model-complete cores, which is an important property when studying the computational complexity of the constraint satisfaction problem for structures from K. We also show that Thomas’ conjecture holds for every structure in K.

Abstract:
With a simple generic approach, we develop a classification that encodes and measures the strength of
completeness (or compactness) properties in various types of spaces and ordered structures. The approach also allows us to encode notions of functions being contractive in these spaces and structures. Possible applications
include metric spaces, ultrametric spaces, ordered groups and fields, topological spaces, partially ordered
sets, and lattices. We describe several notions of completeness in these spaces and structures and
determine their respective strengths. Consequences of the levels of strength can be illustrated
by generic fixed point theorems, which then can be specialized to theorems in various applications
which work with contracting functions and some completeness property of the underlying space.

Ball spaces are nonempty sets of nonempty subsets of a given set. They are called spherically complete if every
chain of balls has a nonempty intersection. This is all that is needed for the encoding of completeness notions.
Operations on the sets of balls are investigated to determine when they lead to larger sets of balls; if so, then the properties of the so obtained new ball spaces are described. The operations can lead to increased level of
strength, or to ball spaces of newly constructed structures, such as products. Further, the general framework makes it possible to transfer concepts and approaches from one application to the other; as examples we discuss, if time permits, theorems analogous to the Knaster-Tarski Fixed Point Theorem for lattices and theorems
analogous to the Tychonoff Theorem for topological spaces.

Link:
https://math.usask.ca/fvk/Balls.pdf

Abstract:
Let $k$ be an algebraically closed non-trivially valued field of rank
1. After recalling the definition of Berkovich's analytification of
the affine line $\mathbb{A}_k^{1,\mathrm{an}}$, we will consider its
relation to the set of closed balls of $k$, which we denote by
$\mathbb{B}_k$. We will provide a characterization of definable
subsets of $\mathbb{B}_k$ in a natural first-order language. If time
permits, we will discuss the more general case of the analytification
of a curve over $k$. No prior knowledge on Berkovich spaces will be
required. This is a joint work with Jérôme Poineau.

[PDF version]

Abstract:
The category of finite n-regular graphs, with locally bijective graph homomorphisms, is a Fraïssé class. I will present a description of its Fraïssé limit, as the Cayley graph of an action of a free product of 2-elements groups.

Joint work with Jan Hubička.

Abstract: The famous Gottschalk surjunctivity conjecture asserts that for any group G and any finite alphabet A, every injective cellular automaton over G and A is surjective. Groups satisfying the conjecture are called surjunctive, and so far there are no examples of non-surjunctive groups, while it is known that every sofic group is surjunctive. Recently, Capobianco, Kari and Taati introduced, in a sense, a dual version of the Gottschalk conjecture and proved that every sofic group is `dual surjunctive'. We investigate the properties of dual surjunctive groups further. For instance, we prove that this class of groups is closed under taking ultraproducts, these groups satisfy the Kaplansky direct finiteness conjecture for finite unital rings, etc. We also consider dual surjunctivity of more general topological dynamical systems. The talk will be a report on current ongoing joint work with Jakub Gismatullin.

Abstract: A unital separable C*-algebra (other than the C*-algebra of all complex numbers) is strongly self-absorbing if it is isomorphic to its (minimal) tensor product with itself in a "strong" sense. Strongly self-absorbing C*-algebras play crucial roles in Elliott's classification program of separable nuclear C*-algebras by K-theoretic data, and among them the Jiang-Su algebra is the most notable. In fact, the classification of separable, simple, unital, nuclear C*-algebras that tensorially absorb the Jiang-Su algebra and satisfy the UCT has been the pinnacle of the classification results. In their original paper from 1999, Jiang and Su already proved that the Jiang-Su algebra is strongly self-absorbing. However, their proof is quite complicated and uses heavy tools from classification, such as KK-theory. We give a rather elementary and direct proof for the fact that the Jiang-Su algebra is strongly self-absorbing. This is achieved by establishing a general connection between the strongly self-absorbing C*-algebras and the "Fraisse limits" of categories of C*-algebras that are sufficiently closed under tensor products.

Abstract: We consider the derived category of a valuation domain and study its definable subcategories, showing in particular that these subcategories are always fully determined on the cohomology. Then we turn to t-structures in the derived category such that their coaisle is definable. These t-structures are structurally important because they include both the smashing localisations and the t-structures induced by cosilting complexes. We give a complete classification of these in terms of certain topological invariants, providing a non-stable generalization of a result by Bazzoni-Šťovíček. We discuss some consequences of the classification for the silting theory of the ring. Joint work with Silvana Bazzoni.

Abstract: Definable topological dynamics shows that the classical Bohr compactification of a discrete group $G$ can be seen as a special case of "definable" Bohr compactifications. In turn, the definable Bohr compactification of a definable group can be described in terms of its so-called model-theoretic connected components. I will introduce the ring analogues to such connected components, and show how they can be used in the description of the connected components of some linear matrix groups. With this, I will obtain the description of (classical) Bohr compactifications of some classical groups, such as the discrete Heisenberg group.

Abstract:
A linear order L is countably saturated if for any countable subsets A,B of L,
such that any element of A is less than any element of B, we can find an element of
L between them. This obvious generalization of density corresponds to ”countable
saturation” from model theory. We’ll say, that a countably saturated linear order
L is prime, if every countably saturated linear order contains an isomorphic copy
of L.
I’d like to present a characterization of the prime countably saturated linear orders
and say something about related results concerning certain classes of uncountable
graphs.

Link:
[arXiv:1907.00432]

Abstract:
We show by an elementary argument that no functor from the category of Hilbert spaces and isometries to the category of sets preserves directed colimits (i.e. direct limits), meaning that this category cannot, in a very strong sense, be axiomatized as an elementary class---that is, by a first order theory---nor even as an abstract elementary class.
Simple arguments yield corresponding negative axiomatizability results for Banach spaces, complete metric spaces, commutative unital C*-algebras, and KH^op, the opposite of the category of compact Hausdorff spaces.

Joint work with J. Rosický and S. Vasey.

Abstract: Based on the theory of ball spaces as introduced in the talk of Hanna Cmiel, we define and study Caristi-Kirk and Oettli-Théra ball spaces on metric spaces. If the underlying metric space is complete, then these have a very strong property: every ball contains a singleton ball. This fact provides concise proofs for several results which are equivalent to the Caristi-Kirk Fixed Point Theorem, namely Ekeland's Variational Principles, the Oettli-Théra Theorem, Takahashi's Theorem and the Flower Petal Theorem.

Abstract: When we consider the topological dynamic objects related to a group definable in a model, a question comes to mind: Do they reflect some real model-theoretic properties of the group or just some accidental features of the particular model where the group lives? This is the absoluteness question. In the talk I will discuss this question and some answers to it in some special cases.

Abstract:
A twisted sum of Banach spaces $Y$ and $Z$ is a short exact sequences
$0\to Y\to X\to Z\to 0$, where $X$ is another Banach spaces and morphisms are bounded linear operators.
Such a twisted sum is {\em trivial} if it is equivalent to $0\to Y\to Y\oplus Z\to Z\to 0$ (that is, if $Y$ is isomorphically embedded onto a complemented subspace of $X$).

By the classical Sobczyk theorem, a twisted sum $0\to c_0\to X\to Z\to 0$ is always trivial for separable spaces $Z$. We discuss the class $\mathfrak Z$ of nonseparable
spaces $Z$ admitting a nontrivial twisted sum $0\to c_0\to X\to Z\to 0$. Problems concerning that class are often set-theoretic in nature.
In particular, we have proved that the question of Cabello Sanch\'ez, Castillo, Kalton and Yost, if $C(K)\in {\mathfrak Z}$ for every nonmetrizable compactum $K$
is undecidable within the usual set theory.

[PDF version]

Link: [arXiv:1910.07273]

Abstract:
Given a countable structure $M$, we say that $M$ has \emph{$n$-ample generics} if there is an $n$-tuple in $\Aut(M)$ whose diagonal conjugacy class is comeagre in $(\Aut(M))^n$ (a \emph{generic $n$-tuple}).

An interesting open problem asks whether having $2$-ample generics implies having ample generics (i.e.\ having $n$-ample generics for all $n$). Another interesting question asks whether it is always the case that for a single generic automorphism, its square is always generic.

It is well-known that any generic $n$-tuple freely generates a subgroup of $\Aut(M)$. I will sketch the proof of the fact that if $M$ is the limit of a Fraisse class with some quite strong properties (including commutative canonical amalgamation), then any finite subset of such subgroup freely generating a subgroup (of any rank) is actually a generic tuple. In particular, the square of a generic automorphism is generic.

This is joint with Itay Kaplan and Nicholas Ramsey.

Abstract: In the model theory of modules, pure submodules and pure-injective modules play a very important role. The first ones emerge as a natural common generalization of the concepts of elementary submodel and direct summand. The latter ones serve as the sufficiently saturated structures in the theory. In the talk, we will concentrate on the infinitary notions of lambda-purity and lambda-pure-injectivity (for a regular uncountable cardinal lambda). Although their interplay does not turn out to be that straightforward and fruitful as in the finitary case, we shall show that under some large cardinal hypotheses, they can come out as pretty handy tools. In particular, we prove that, modulo the existence of proper class of strongly compact cardinals, it is consistent that the category of projective modules is accessible over any ring R.

Abstract:
Let $w\in\mathbb{F}_2$ be a non-trivial word.
We show that the image of the word map associated to $w$
on classical groups of Lie type is arbitrarily dense in
the normalized rank metric once the degree is large enough,
using a cohomological argument.

Link: [arXiv:1802.09289]

Abstract:
Baer's Criterion for Injectivity is a basic tool of the theory of modules and complexes of modules. Its dual version (DBC) is known to hold for all right perfect rings, but its validity for non-right perfect rings is a complex problem (first formulated by Faith in 1976). Recently, it has turned out that there are two classes of non-right perfect rings: (1) those for which DBC fails in ZFC, and (2) those for which DBC is independent of ZFC. We show that the latter class contains all small semiartinian von Neumann regular rings with primitive factors artinian.

Link: [arXiv:1901.01442]

Last modified: 29 October 2019

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