arXiv subjects

Tomasz Maszczyk

Publications and source records attributed to Tomasz Maszczyk.

At least 19 recordsLinked to original sources

arXiv2026

Milnor meets Hopf and Toeplitz at the K-theory of quantum projective planes

We explore applications of the celebrated construction of the Milnor connecting homomorphism from the odd to the even K-groups in the context of Hopf--Galois theory. For a finitely generated projective module associated to any piecewise cleft principal comodule algebra, we provide an explicit formula computing the clutching $K_1$-class in terms of the representation matrix defining the module. Thus, the module is determined by an explicit Milnor idempotent. We apply this new tool to the K-theory of quantum complex projective planes to determine their $K_0$-generators in terms of modules associated to noncommutative Hopf fibrations. On the other hand, using explicit homotopy between unitaries, we express the $K_0$-class of the Milnor idempotents in terms of elementary projections in the Toeplitz C*-algebra. This allows us to infer that all our generators are in the positive cone of the $K_0$-group, which is a purely quantum phenomenon absent in the classical case.

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arXiv2026

Rank-two Milnor idempotents for the multipullback quantum complex projective plane

The $K_0$-group of the C*-algebra of multipullback quantum complex projective plane is known to be $\mathbb{Z}^3$, with one generator given by the C*-algebra itself, one given by the section module of the noncommutative (dual) tautological line bundle, and one given by the Milnor module associated to a generator of the $K_1$-group of the C*-algebra of Calow-Matthes quantum 3-sphere. Herein we prove that these Milnor modules are isomorphic either to the section module of a noncommutative vector bundle associated to the $SU_q(2)$-prolongation of the Heegaard quantum 5-sphere $S^5_H$ viewed as a $U(1)$-quantum principal bundle, or to a complement of this module in the rank-four free module. Finally, we demonstrate that one of the above Milnor modules always splits into the direct sum of the rank-one free module and a rank-one non-free projective module that is \emph{not} associated with $S^5_H$.

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arXiv2025

Pullback of quantum principal bundles

We introduce an abstract framework of Cartesian squares beyond the context of fiber products, and use it to extend the notion of pullback from classical to compact quantum principal bundles. Based only on our abstract notion of a Cartesian square, we extend key concepts of Equivariant Topology, such as the pullback of a family of group actions, orbit spaces, slices and global sections, change of base and structure group, free actions, and the groupoid of compact principal bundles. Finally, we embed the thus extended Equivariant Topology inside the 2-category of Grothendieck categories in such a way that our notion of a Cartesian square becomes the appropriate Beck-Chevalley condition.

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arXiv2025

Introduction to Quantum Combinatorics

We construct a topos of quantum sets and embed into it the classical topos of sets. We show that the internal logic of the topos of sets, when interpreted in the topos of quantum sets, provides the Birkhoff-von Neumann quantum propositional calculus of idempotents in a canonical internal commutative algebra of the topos of quantum sets. We extend this construction by allowing the quantum counterpart of Boolean algebras of classical truth values which we introduce and study in detail. We realize expected values of observables in quantum states in our topos of quantum sets as a tautological morphism from the canonical internal commutative algebra to a canonical internal object of affine functions on quantum states. We show also that in our topos of quantum sets one can speak about quantum quivers in the sense of Day-Street and Chikhladze. Finally, we provide a categorical derivation of the Leavitt path algebra of such a quantum quiver and relate it to the category of stable representations of the quiver. It is based on a categorification of the Cuntz-Pimsner algebra in the context of functor adjunctions replacing the customary use of Hilbert modules.

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arXiv2021

Inertial Hopf-cyclic homology

We construct, study, and apply a characteristic map from the relative periodic cyclic homology of the quotient map for a group action to the periodic Hopf-cyclic homology with coefficients associated with inertia of the action. This result admits, and in fact, comes from, its noncommutative-geometric, or quantized, counterpart. The crucial ingredient is the construction of the appropriate quantization of the cyclic nerve of the action groupoid, the cyclic object related to inertia, as the Connes-cyclic dual of a Hopf-cyclic object with coefficients in some stable anti-Yetter--Drinfeld module quantizing the Brylinski space. For the Hopf-Galois quantization of the case of trivial inertia, we find a non-trivial identification of our characteristic map with a well-known isomorphism of Jara--Ştefan. In presence of nontrivial inertia, for an analytic ramified Galois double cover, we show that the cokernel of our characteristic map, which can be interpreted as an invariant of inertia modulo topology of the maximal free action, is supported on the branch locus of the quotient map. Finally, we use our inertial Hopf-cyclic object to construct a new invariant of finite-dimensional algebras.

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arXiv2020

Quantum CW-complexes in a Waldhausen category for unital C*-algebras

Using the ring structure of the K-groups of finite CW-complexes, Atiyah and Todd unravelled the K-theory of complex projective spaces $CP^n$. Herein, in the realm of noncommutative topology, we develop a new framework of finite quantum CW-complexes using the language of Waldhausen categories, which allows us to enrich the class of standard morphisms between unital C*-algebras by adding inverses of *-homomorphisms that are isomorphisms in K-theory. Our concept of quantum CW-complexes subsumes earlier constructions and enjoys a plethora of examples. Moreover, the framework allows us to reduce problems concerning the multipushout quantum complex projective space $CP^n_H$ to the much more approachable setting of the Vaksman-Soibelman quantum complex projective space $CP^n_q$ enjoying the availability of graph-algebraic methods. In particular, these methods permit us to transport the ring structure from $K^0(CP^n)$ to $K^0(CP_q^n)$. Finally, we adapt the formalism of Waldhausen categories to determine a natural set of free generators of $K^0(CP_H^n)$ from a natural set of free generators of $K^0(CP_q^n)$, and to transport the ring structure from $K^0(CP_q^n)$ to $K^0(CP_H^n)$.

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arXiv2020

AVATAR -- Machine Learning Pipeline Evaluation Using Surrogate Model

The evaluation of machine learning (ML) pipelines is essential during automatic ML pipeline composition and optimisation. The previous methods such as Bayesian-based and genetic-based optimisation, which are implemented in Auto-Weka, Auto-sklearn and TPOT, evaluate pipelines by executing them. Therefore, the pipeline composition and optimisation of these methods requires a tremendous amount of time that prevents them from exploring complex pipelines to find better predictive models. To further explore this research challenge, we have conducted experiments showing that many of the generated pipelines are invalid, and it is unnecessary to execute them to find out whether they are good pipelines. To address this issue, we propose a novel method to evaluate the validity of ML pipelines using a surrogate model (AVATAR). The AVATAR enables to accelerate automatic ML pipeline composition and optimisation by quickly ignoring invalid pipelines. Our experiments show that the AVATAR is more efficient in evaluating complex pipelines in comparison with the traditional evaluation approaches requiring their execution.

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arXiv2019

Support Feature Machines

Support Vector Machines (SVMs) with various kernels have played dominant role in machine learning for many years, finding numerous applications. Although they have many attractive features interpretation of their solutions is quite difficult, the use of a single kernel type may not be appropriate in all areas of the input space, convergence problems for some kernels are not uncommon, the standard quadratic programming solution has $O(m^3)$ time and $O(m^2)$ space complexity for $m$ training patterns. Kernel methods work because they implicitly provide new, useful features. Such features, derived from various kernels and other vector transformations, may be used directly in any machine learning algorithm, facilitating multiresolution, heterogeneous models of data. Therefore Support Feature Machines (SFM) based on linear models in the extended feature spaces, enabling control over selection of support features, give at least as good results as any kernel-based SVMs, removing all problems related to interpretation, scaling and convergence. This is demonstrated for a number of benchmark datasets analyzed with linear discrimination, SVM, decision trees and nearest neighbor methods.

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arXiv2018

Pullbacks and nontriviality of associated noncommutative vector bundles

Our main theorem is that the pullback of an associated noncommutative vector bundle induced by an equivariant map of quantum principal bundles is a noncommutative vector bundle associated via the same finite-dimensional representation of the structural quantum group. On the level of $K_{0}$-groups, we realize the induced map by the pullback of explicit matrix idempotents. We also show how to extend our result to the case when the quantum-group representation is infinite dimensional, and then apply it to the Ehresmann-Schauenburg quantum groupoid. Finally, using noncommutative Milnor's join construction, we define quantum quaternionic projective spaces together with noncommutative tautological quaternionic line bundles and their duals. As a key application of the main theorem, we show that these bundles are stably non-trivial as noncommutative complex vector bundles.

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arXiv2017

The cyclic-homology Chern-Weil homomorphism for principal coactions

We view the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model. Then we define the cyclic-homology Chern-Weil homomorphism by extending the Chern-Galois character from the characters of finite-dimensional comodules to arbitrary cotraces. To reduce the cyclic-homology Chern-Weil homomorphism to a tautological natural transformation, we replace the unital coaction-invariant subalgebra by its certain natural H-unital nilpotent extension (row extension), and prove that their cyclic-homology groups are isomorphic. In the proof, we use a chain homotopy invariance of complexes computing Hochschild, and hence cyclic homology, for arbitrary row extensions. In the context of the cyclic-homology Chern-Weil homomorphism, a row extension is provided by the Ehresmann-Schauenburg quantum groupoid with a nonstandard multiplication.

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arXiv2015

Cyclic Homology and Quantum Orbits

A natural isomorphism between the cyclic object computing the relative cyclic homology of a homogeneous quotient-coalgebra-Galois extension, and the cyclic object computing the cyclic homology of a Galois coalgebra with SAYD coefficients is presented. The isomorphism can be viewed as the cyclic-homological counterpart of the Takeuchi-Galois correspondence between the left coideal subalgebras and the quotient right module coalgebras of a Hopf algebra. A spectral sequence generalizing the classical computation of Hochschild homology of a Hopf algebra to the case of arbitrary homogeneous quotient-coalgebra-Galois extensions is constructed. A Pontryagin type self-duality of the Takeuchi-Galois correspondence is combined with the cyclic duality of Connes in order to obtain dual results on the invariant cyclic homology, with SAYD coefficients, of algebras of invariants in homogeneous quotient-coalgebra-Galois extensions. The relation of this dual result with the Chern character, Frobenius reciprocity, and inertia phenomena in the local Langlands program, the Chen-Ruan-Brylinski-Nistor orbifold cohomology and the Clifford theory is discussed.

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arXiv2011

Deformations of monoidal functors

We point out that for Yetter's deformational Hochschild complex of a monoidal functor between abelian monoidal categories the Gerstenhaber-Voronov type operations can be defined making it a strong homotopy Gerstenhaber algebra. This encodes deformation theory of monoidal functors in an analogical way as deformation theory of associative algebras is described by the strong homotopy Gerstenhaber algebra structure on the corresponding Hochschild cochains. We describe a quasi-classical limit of deformations of a symmetric monoidal functor in terms of Poisson type structure.

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arXiv2009

Distributive lattices and cohomology

A resolution of the intersection of a finite number of subgroups of an abelian group by means of their sums is constructed, provided the lattice generated by these subgroups is distributive. This is used for detecting singularities of modules over Dedekind rings. A generalized Chinese remainder theorem is derived as a consequence of the above resolution. The Gelfand-Naimark duality between finite closed coverings of compact Hausdorff spaces and the generalized Chinese remainder theorem is clarified.

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arXiv2009

On splitting polynomials with noncommutative coefficients

It is shown that for every splitting of a polynomial with noncommutative coefficients into linear factors $(X-a_{k})$ with $a_{k}$'s commuting with coefficients, any cyclic permutation of linear factors gives the same result and all $a_{k}$ are roots of that polynomial. Examples are given and analyzed from Galois theory point of view.

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arXiv2007

Noncommutative geometry through monoidal categories I

After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasi-coherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, Hopf-Galois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions in commutative geometry (e.g. faithfully flat descent theory, principal fibrations, equivariant and infinitesimal geometry) can be interpreted as noncommutative geometric constructions applied to commutative objects. For such generalized geometry we define global invariants constructing cyclic objects from which we derive Hochschild, cyclic and periodic cyclic homology (with coefficients) in the standard way.

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arXiv2007

Noncommutative geometry through monoidal categories

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arXiv2006

Maximal commutative subalgebras, Poisson geometry and Hochschild homology

A Poisson geometry arising from maximal commutative subalgebras is studied. A spectral sequence convergent to Hochschild homology with coefficients in a bimodule is presented. It depends on the choice of a maximal commutative subalgebra inducing appropriate filtrations. Its E^{2}_{p,q}-groups are computed in terms of canonical homology with values in a Poisson module defined by a given bimodule and a maximal commutative subalgebra.

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arXiv2005

A pairing between super Lie-Rinehart and periodic cyclic homology

We consider a pairing producing various cyclic Hochschild cocycles, which led Alain Connes to cyclic cohomology. We are interested in geometrical meaning and homological properties of this pairing. We define a non-trivial pairing between the homology of a Lie-Rinehart (super-)algebra with coefficients in some partial traces and relative periodic cyclic homology. This pairing generalizes the index formula for summable Fredholm modules, the Connes-Kubo formula for the Hall conductivity and the formula computing the K0-group of a smooth noncommutative torus. It also produces new homological invariants of proper maps contracting each orbit contained in a closed invariant subset in a manifold acted on smoothly by a connected Lie group. Finally we compare it with the characteristic map for the Hopf-cyclic cohomology.

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