publications | Friedrich Martin Schneider

2024

Friedrich Martin Schneider, Sławomir Solecki

Groups without unitary representations, submeasures, and the escape property

Mathematische Annalen (2024), 46 pages.

Abs doi arXiv

We give new examples of topological groups that do not have non-trivial continuous unitary representations, the so-called exotic groups. We prove that all groups of the form L⁰(ϕ,G), where ϕ is a pathological submeasure and G is a topological group, are exotic. This result extends, with a different proof, a theorem of Herer and Christensen on exoticness of L⁰(ϕ,ℝ) for ϕ pathological. It follows that every topological group embeds into an exotic one. In our arguments, we introduce the escape property, a geometric condition on a topological group, inspired by the solution to Hilbert’s fifth problem and satisfied by all locally compact groups, all non-archimedean groups, and all Banach–Lie groups. Our key result involving the escape property asserts triviality of all continuous homomorphisms from L⁰(ϕ,G) to L⁰(μ,H), where ϕ is pathological, μ is a measure, G is a topological group, and H is a topological group with the escape property.
Vladimir Pestov, Friedrich Martin Schneider

On invariant means and pre-syndetic subgroups

Topology and its Applications (2024), article 109093, 17 pages.

Abs doi arXiv

Beyond the locally compact case, equivalent notions of amenability diverge, and some properties no longer hold, for instance amenability is not inherited by topological subgroups. This investigation is guided by some amenability-type properties of groups of paths and loops. It is shown that a version of amenability called skew-amenability is inherited by pre-syndetic subgroups in the sense of Basso and Zucker (in particular, by co-compact subgroups). It follows that co-compact subgroups of amenable topological groups whose left and right uniformities coincide are amenable. We discuss a version of amenability belonging to P. Malliavin and M.-P. Malliavin: the existence of a mean on bounded Borel functions that is invariant under the left action of a dense subgroup. We observe that this property is in general strictly stronger than amenability, and establish for it Reiter- and Følner-type criteria. Finally, there is a review of open problems.
Friedrich Martin Schneider

Group von Neumann algebras, inner amenability, and unit groups of continuous rings

International Mathematics Research Notices (2024), no. 8, 6422–6446.

Abs doi arXiv

We prove that, if a discrete group G is not inner amenable, then the unit group of the ring of operators affiliated with the group von Neumann algebra of G is non-amenable with respect to the topology generated by its rank metric. This provides examples of non-discrete irreducible, continuous rings (in von Neumann’s sense) whose unit groups are non-amenable with regard to the rank topology. Our argument establishes and uses connections with Eymard–Greenleaf amenability of the action of the unitary group of a II₁ factor on the associated space of projections of a fixed trace.

2023

Friedrich Martin Schneider

Concentration of invariant means and dynamics of chain stabilizers in continuous geometries

Geometric and Functional Analysis 33 (2023), no. 6, 1608–1681.

Abs doi arXiv

We prove a concentration inequality for invariant means on topological groups, namely for such adapted to a chain of amenable topological subgroups. The result is based on an application of Azuma’s martingale inequality and provides a method for establishing extreme amenability. Building on this technique, we exhibit new examples of extremely amenable groups arising from von Neumann’s continuous geometries. Along the way, we also answer a question by Pestov on dynamical concentration in direct products of amenable topological groups.
Maximilian Stubbemann, Tom Hanika, Friedrich Martin Schneider

Intrinsic Dimension for Large-Scale Geometric Learning

Transactions on Machine Learning Research (2023), 18 pages.

Abs URL arXiv

The concept of dimension is essential to grasp the complexity of data. A naive approach to determine the dimension of a dataset is based on the number of attributes. More sophisticated methods derive a notion of intrinsic dimension (ID) that employs more complex feature functions, e.g., distances between data points. Yet, many of these approaches are based on empirical observations, cannot cope with the geometric character of contemporary datasets, and do lack an axiomatic foundation. A different approach was proposed by V. Pestov, who links the intrinsic dimension axiomatically to the mathematical concentration of measure phenomenon. First methods to compute this and related notions for ID were computationally intractable for large-scale real-world datasets. In the present work, we derive a computationally feasible method for determining said axiomatic ID functions. Moreover, we demonstrate how the geometric properties of complex data are accounted for in our modeling. In particular, we propose a principle way to incorporate neighborhood information, as in graph data, into the ID. This allows for new insights into common graph learning procedures, which we illustrate by experiments on the Open Graph Benchmark.
Friedrich Martin Schneider, Caterina Viola

An application of Farkas’ lemma to finite-valued constraint satisfaction problems over infinite domains

Journal of Mathematical Analysis and Applications 517 (2023), no. 1, article 126591, 22 pages.

Abs doi arXiv

We show a universal algebraic local characterisation of the expressive power of finite-valued languages with domains of arbitrary cardinality and containing arbitrary many cost functions.

2022

Tom Hanika, Friedrich Martin Schneider, Gerd Stumme

Intrinsic dimension of geometric data sets

Tohoku Mathematical Journal 74 (2022), no. 1, 23–52.

Abs doi arXiv

The curse of dimensionality is a phenomenon frequently observed in machine learning (ML) and knowledge discovery (KD). There is a large body of literature investigating its origin and impact, using methods from mathematics as well as from computer science. Among the mathematical insights into data dimensionality, there is an intimate link between the dimension curse and the phenomenon of measure concentration, which makes the former accessible to methods of geometric analysis. The present work provides a comprehensive study of the intrinsic geometry of a data set, based on Gromov’s metric measure geometry and Pestov’s axiomatic approach to intrinsic dimension. In detail, we define a concept of geometric data set and introduce a metric as well as a partial order on the set of isomorphism classes of such data sets. Based on these objects, we propose and investigate an axiomatic approach to the intrinsic dimension of geometric data sets and establish a concrete dimension function with the desired properties. Our model for data sets and their intrinsic dimension is computationally feasible and, moreover, adaptable to specific ML/KD-algorithms, as illustrated by various experiments.

2021

Kate Juschenko, Friedrich Martin Schneider

Skew-amenability of topological groups

Commentarii Mathematici Helvetici 96 (2021), no. 4, 805–851.

Abs doi arXiv

We study skew-amenable topological groups, i.e., those admitting a left-invariant mean on the space of bounded real-valued functions left-uniformly continuous in the sense of Bourbaki. We prove characterizations of skew-amenability for topological groups of isometries and automorphisms, clarify the connection with extensive amenability of group actions, establish a Følner-type characterization, and discuss closure properties of the class of skew-amenable topological groups. Moreover, we isolate a dynamical sufficient condition for skew-amenability and provide several concrete variations of this criterion in the context of transformation groups. These results are then used to decide skew-amenability for a number of examples of topological groups built from or related to Thompson’s group F and Monod’s group of piecewise projective homeomorphisms of the real line.
Friedrich Martin Schneider, Sławomir Solecki

Concentration of measure, classification of submeasures, and dynamics of L₀

Journal of Functional Analysis 280 (2021), no. 5, article 108890, 51 pages.

Abs doi arXiv

Exhibiting a new type of measure concentration, we prove uniform concentration bounds for measurable Lipschitz functions on product spaces, where Lipschitz is taken with respect to the metric induced by a weighted covering of the index set of the product. Our proof combines the Herbst argument with an analogue of Shearer’s lemma for differential entropy. We give a quantitative “geometric” classification of diffuse submeasures into elliptic, parabolic, and hyperbolic. We prove that any non-elliptic submeasure (for example, any measure, or any pathological submeasure) has a property that we call covering concentration. Our results have strong consequences for the dynamics of the corresponding topological L₀-groups.

2020

Friedrich Martin Schneider, Andreas Thom

The Liouville property and random walks on topological groups

Commentarii Mathematici Helvetici 95 (2020), no. 3, 483–513.

Abs doi arXiv

We study harmonic functions and Poisson boundaries for Borel probability measures on general (i.e., not necessarily locally compact) topological groups, and we prove that a second-countable topological group is amenable if and only if it admits a fully supported, regular Borel probability measure with trivial Poisson boundary. This generalizes work of Kaimanovich–Vershik and Rosenblatt, confirms a general topological version of Furstenberg’s conjecture, and entails a characterization of the amenability of isometry groups in terms of the Liouville property for induced actions. Moreover, our result has non-trivial consequences concerning Liouville actions of discrete groups on countable sets.

2019

Friedrich Martin Schneider

Equivariant dissipation in non-archimedean groups

Israel Journal of Mathematics 234 (2019), no. 1, 281–307.

Abs doi arXiv

We prove that, if a topological group G has an open subgroup of infinite index, then every net of tight Borel probability measures on G UEB-converging to invariance dissipates in G in the sense of Gromov. In particular, this solves a 2006 problem by Pestov: for every left-invariant (or right-invariant) metric d on the infinite symmetric group Sym(ℕ), compatible with the topology of pointwise convergence, the sequence of the finite symmetric groups (Sym(n), d↾_Sym(n), μ_Sym(n))_n∈ℕ equipped with the restricted metrics and their normalized counting measures dissipates, thus fails to admit a subsequence being Cauchy with respect to Gromov’s observable distance.
Friedrich Martin Schneider

Equivariant concentration in topological groups

Geometry & Topology 23 (2019), no. 2, 925–956.

Abs doi arXiv

We prove that, if G is a second-countable topological group with a compatible right-invariant metric d and (μ_n)_n∈ℕ is a sequence of compactly supported Borel probability measures on G converging to invariance with respect to the mass transportation distance over d and such that (spt μ_n, d↾spt μ_n, μ_n↾spt μ_n)_n∈ℕ concentrates to a fully supported, compact mm–space (X,d_X,μ_X), then X is homeomorphic to a G–invariant subspace of the Samuel compactification of G. In particular, this confirms a conjecture by Pestov and generalizes a well-known result by Gromov and Milman on the extreme amenability of topological groups. Furthermore, we exhibit a connection between the average orbit diameter of a metrizable flow of an arbitrary amenable topological group and the limit of Gromov’s observable diameters along any net of Borel probability measures UEB–converging to invariance over the group.
Friedrich Martin Schneider, Jens Zumbrägel

MacWilliams’ extension theorem for infinite rings

Proceedings of the American Mathematical Society 147 (2019), no. 3, 947–961.

Abs doi arXiv

Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present paper we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudoinjective and its finitary left socle embeds into the semisimple quotient. Providing a topological perspective on the MacWilliams property, we also show that the finitary left socle of a left Artinian ring embeds into the semisimple quotient if and only if it admits a finitarily left torsion-free character, if and only if the Pontryagin dual of the regular left module is almost monothetic. In conclusion, an Artinian ring has the MacWilliams property if and only if it is finitarily Frobenius, i.e., it is quasi-Frobenius and its finitary socle embeds into the semisimple quotient.

2018

Friedrich Martin Schneider, Andreas Thom

On Følner sets in topological groups

Compositio Mathematica 154 (2018), no. 7, 1333–1361.

Abs doi arXiv

We extend Følner’s amenability criterion to the realm of general topological groups. Building on this, we show that a topological group G is amenable if and only if its left-translation action can be approximated in a uniform manner by amenable actions on the set G. As applications we obtain a topological version of Whyte’s geometric solution to the von Neumann problem and give an affirmative answer to a question posed by Rosendal.
Friedrich Martin Schneider

About von Neumann’s problem for locally compact groups

Journal of Noncommutative Geometry 12 (2018), no. 4, 1531–1549.

Abs doi arXiv

We note a generalization of Whyte’s geometric solution to the von Neumann problem for locally compact groups in terms of Borel and clopen piecewise translations. This strengthens a result of Paterson on the existence of Borel paradoxical decompositions for non-amenable locally compact groups. Along the way, we study the connection between some geometric properties of coarse spaces and certain algebraic characteristics of their wobbling groups.

2017

Friedrich Martin Schneider, Jens Zumbrägel

Profinite algebras and affine boundedness

Advances in Mathematics 305 (2017), 661–681.

Abs doi arXiv

We prove a characterization of profinite algebras, i.e., topological algebras that are isomorphic to a projective limit of finite discrete algebras. In general profiniteness concerns both the topological and algebraic characteristics of a topological algebra, whereas for topological groups, rings, semigroups, and distributive lattices, profiniteness turns out to be a purely topological property as it is equivalent to the underlying topological space being a Stone space. Condensing the core idea of those classical results, we introduce the concept of affine boundedness for an arbitrary universal algebra and show that for an affinely bounded topological algebra over a compact signature profiniteness is equivalent to the underlying topological space being a Stone space. Since groups, semigroups, rings, and distributive lattices are indeed affinely bounded algebras over finite signatures, all these known cases arise as special instances of our result. Furthermore, we present some additional applications concerning topological semirings and their modules, as well as distributive associative algebras. We also deduce that any affinely bounded simple compact algebra over a compact signature is either connected or finite. Towards proving the main result, we also establish that any topological algebra is profinite if and only if its underlying space is a Stone space and its translation monoid is equicontinuous.
Vladimir Pestov, Friedrich Martin Schneider

On amenability and groups of measurable maps

Journal of Functional Analysis 273 (2017), no. 12, 3859–3874.

Abs doi arXiv

We show that if G is an amenable topological group, then the topological group L⁰(G) of strongly measurable maps from ([0,1],λ) into G endowed with the topology of convergence in measure is whirly amenable, hence extremely amenable. Conversely, we prove that a topological group G is amenable if L⁰(G) is.
Friedrich Martin Schneider, Andreas Thom

Topological matchings and amenability

Fundamenta Mathematicae 238 (2017), no. 2, 167–200.

Abs doi arXiv

We establish a characterization of amenability for general Hausdorff topological groups in terms of matchings with respect to finite uniform coverings. Furthermore, we prove that it suffices to just consider two-element uniform coverings. We also show that extremely amenable as well as compactly approximable topological groups satisfy a perfect matching property condition—the latter even with regard to arbitrary (i.e., possibly infinite) uniform coverings. Finally, we prove that the automorphism group of a Fraïssé limit of a metric Fraïssé class is amenable if and only if the class has a certain Ramsey-type matching property.
Friedrich Martin Schneider

A uniform Birkhoff theorem

Algebra Universalis 78 (2017), no. 3, 337–354.

Abs doi arXiv

Birkhoff’s HSP theorem characterizes the classes of models of algebraic theories as those being closed with respect to homomorphic images, subalgebras, and products. In particular, it implies that an algebra B satisfies all equations that hold in an algebra A of the same signature if and only if B is a homomorphic image of a subalgebra of a (possibly infinite) direct power of A. The former statement is equivalent to the existence of a natural map sending term functions of the algebra A to those of B—the natural clone homomorphism. The study of continuity properties of natural clone homomorphisms has been initiated recently by Bodirsky and Pinsker for locally oligomorphic algebras.
Manuel Bodirsky, Friedrich Martin Schneider

A topological characterisation of endomorphism monoids of countable structures

Algebra Universalis 77 (2017), no. 3, 251–269.

Abs doi arXiv

A topological monoid is isomorphic to an endomorphism monoid of a countable structure if and only if it is separable and has a compatible complete ultrametric such that composition from the left is non-expansive. We also give a topological characterisation of those topological monoids that are isomorphic to endomorphism monoids of countable ω-categorical structures. Finally, we present analogous characterisations for polymorphism clones of countable structures and for polymorphism clones of countable ω-categorical structures.
Friedrich Martin Schneider, Daniel Borchmann

Topological entropy of formal languages

Semigroup Forum 94 (2017), no. 3, 556–581.

Abs doi arXiv

We introduce the notion of topological entropy of a formal language as the topological entropy of the minimal topological automaton accepting it. Using a characterization of this notion in terms of approximations of the Myhill–Nerode congruence relation, we are able to compute the topological entropies of certain example languages. Those examples suggest that the notion of a “simple” formal language coincides with the language having zero entropy.
Sebastian Kerkhoff, Friedrich Martin Schneider

Clones of (Continuous) Partial Cofunctions

Journal of Multiple-Valued Logic and Soft Computing 28 (2017), no. 1, 59–79.

Abs URL

We introduce and study clones of partial cofunctions on sets of arbitrary cardinality. We start by outlining a general Galois theory similar to Pol-Inv. We then show some elementary results about the essential arity of clones of partial cofunctions, and take a closer look at partial idempotent cofunctions. Furthermore, we characterize all minimal clones of partial cofunctions and show that the join of all minimal clones is the full clone (provided that the Axiom of Choice is assumed). Finally, we discuss how introducing a topology and requiring the partial functions to be continuous changes the scenario.
Mike Behrisch, Sebastian Kerkhoff, Reinhard Pöschel, Friedrich Martin Schneider, Stefan Siegmund

Dynamical systems in categories

Applied Categorical Structures 25 (2017), no. 1, 29–57.

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In this article we establish a bridge between dynamical systems, including topological and measurable dynamical systems as well as continuous skew product flows and nonautonomous dynamical systems; and coalgebras in categories having all finite products. We introduce a straightforward unifying definition of abstract dynamical system on finite product categories. Furthermore, we prove that such systems are in a unique correspondence with monadic algebras whose signature functor takes products with the time space. We discuss that the categories of topological spaces, metrisable and uniformisable spaces have exponential objects w.r.t. locally compact Hausdorff, strongly σ-compact or arbitrary time spaces as exponents, respectively. Exploiting the adjunction between taking products and exponential objects, we demonstrate a one-to-one correspondence between monadic algebras (given by dynamical systems) for the left-adjoint functor and comonadic coalgebras for the other. This, finally, provides a new, alternative perspective on dynamical systems.

2016

Friedrich Martin Schneider, Jens Zumbrägel

Every simple compact semiring is finite

Topology and its Applications 206 (2016), 305–310.

Abs doi arXiv

A Hausdorff topological semiring is called simple if every non-zero continuous homomorphism into another Hausdorff topological semiring is injective. Classical work by Anzai and Kaplansky implies that any simple compact ring is finite. We generalize this result by proving that every simple compact semiring is finite, i.e., every infinite compact semiring admits a proper non-trivial quotient.
Sebastian Kerkhoff, Friedrich Martin Schneider

Dualities induced by topological semirings

Applied Categorical Structures 24 (2016), no. 4, 315–329.

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In this paper we establish a general duality theorem for compact Hausdorff spaces being recognizable over certain pairs consisting of a commutative unital topological semiring and a closed proper prime ideal. Indeed, we utilize the concept of blueprints and their localization to prove that the category of compact Hausdorff spaces generated by such a pair can be dually embedded into the category of commutative unital semirings if the pair possesses sufficiently many covering polynomials.

2014

Sebastian Kerkhoff, Friedrich Martin Schneider

Directed Tree Decompositions

In: Formal Concept Analysis - 12th International Conference, ICFCA 2014, Cluj-Napoca, Romania, June 10-13, 2014. Proceedings, C. Glodenau, M. Kaytoue, C. Sacarea (eds.), 80–95. Springer (2014).

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In the problem session of the ICFCA 2006, Sándor Radeleczki asked for the meaning of the smallest integer k such that a given poset can be decomposed as the union of k directed trees. The problem also asks for the connection of this number to the order dimension. Since it was left open what kind of decomposition might be used, there is more than one reading of this problem. In the paper, we discuss different versions and give some answers to this open problem.
Friedrich Martin Schneider

Chaotic actions of locally compact Hausdorff topological groups

In: Proceedings of the Workshop on Algebra, Coalgebra and Topology, WACT 2013, Bath, UK, March 1, 2013, J. Power, C. Wingfield (eds.), 181–195. Elsevier Sci. B. V., Amsterdam (2014).

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In this paper we study continuous actions of topological groups. We introduce a parametrized notion of periodicity – relative to a fixed class of compactifications of the acting group. This yields a natural generalization of Devaney’s well-recognized concept of chaos. As our main result, we establish a geometric characterization of those classes of compactifications of a locally compact Hausdorff topological group for which the group admits a faithful chaotic continuous action on some (compact) Hausdorff space.
Sebastian Kerkhoff, Reinhard Pöschel, Friedrich Martin Schneider

A short introduction to clones

In: Proceedings of the Workshop on Algebra, Coalgebra and Topology, WACT 2013, Bath, UK, March 1, 2013, J. Power, C. Wingfield (eds.), 107–120. Elsevier Sci. B. V., Amsterdam (2014).

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In universal algebra, clones are used to study algebras abstracted from their signature. The aim of this paper is to give a brief introduction to the theory thereof. We give basic definitions and examples, and we present several results and open problems, selected from almost one hundred years of ongoing research. We also discuss what is arguably the most important tool to study clones – the Galois connection between operations and relations built on the notion of preservation. We conclude the paper by explaining the connection between clones and the closely related category theoretic notion of Lawvere theory.

2013

Sebastian Kerkhoff, Friedrich Martin Schneider

Clones of Partial Cofunctions

In: 43rd IEEE International Symposium on Multiple-Valued Logic, ISMVL 2013, Toyama, Japan, May 22-24, 2013, (eds.), 186–191. IEEE Computer Society (2013).

Abs doi

We introduce and study clones of partial cofunctions on sets of arbitrary cardinality. In the process, we outline a general Galois theory similar to Pol-Inv, show some elementary results about the essential arity of clones of partial cofunctions, take a closer look at partial idempotent cofunctions, characterize all minimal clones of partial cofunctions, and show that the join of all minimal clones is the full clone (provided that the Axiom of Choice is assumed).
Friedrich Martin Schneider, Sebastian Kerkhoff, Mike Behrisch, Stefan Siegmund

Chaotic actions of topological semigroups

Semigroup Forum 87 (2013), no. 3, 590–598.

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In this paper we propose and explore a general notion of chaos in the abstract context of continuous actions of topological semigroups and show that any chaotic action on a Hausdorff uniform space is sensitive to initial conditions.
Friedrich Martin Schneider, Sebastian Kerkhoff, Mike Behrisch, Stefan Siegmund

Locally compact groups admitting faithful strongly chaotic actions on Hausdorff spaces

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 23 (2013), no. 9, article 1350158, 8 pages.

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In this paper we provide a geometric characterization of those locally compact Hausdorff topological groups which admit a faithful strongly chaotic continuous action on some Hausdorff space.

2011

Friedrich Martin Schneider

Weak homomorphisms between functorial algebras

Demonstratio Mathematica 44 (2011), no. 4, 801–818.

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In universal algebra, homomorphisms are usually considered between algebras of the same similarity type. Different from that, the notion of a weak homomorphism, as introduced by E. Marczewski in 1961, does not depend on a signature, but only on the clones of term operations generated by the examined algebras. We generalize this idea by defining weak homomorphisms between F₁- and F₂-algebras, where F₁ and F₂ denote not necessarily equal endofunctors of the category of sets. The aim is to show that, in many respects, weak homomorphisms behave very similarly to proper homomorphisms—without restricting the scope of considerations by the necessity of a common type. For instance, concerning a set ℱ of Set-endofunctors that weakly preserve kernels, the class of all algebras of types from ℱ equipped with the class of all weak homomorphisms between these algebras forms a category which admits a canonical factorization structure for morphisms. Furthermore, we treat two product constructions from which the notion of a weak homomorphism naturally arises.