arXiv research

Analytic torsion form constructed for non-commutative spaces.

Smooth structures on infinite dimensional Grassmannians and non-commutative cross-ratios.

Constructs non-commutative modular vector fields for Poisson manifolds.

Reformulates divergence map for Turaev cobracket in non-commutative geometry.

Study introduces dynamical ideals for non-commutative rings and classifies knots and links.

New neural networks for non-commutative data.

In an earlier paper the first author defined a non-commutative A-polynomial for knots in 3-space, using the colored Jones function. The idea is that the colored Jones function of a knot satisfies a non-trivial linear q-difference equation. Said differently, the colored Jones function of a knot is annihilated by a non-z…

Study non-commutative function algebras using contact geometry.

We construct and identify star representations canonically associated with holonomy reducible simple symplectic symmetric spaces. This leads the a non-commutative geometric realization of the correspondence between causal symmetric spaces of Cayley type and Hermitian symmetric spaces of tube type.

Motivated by large-scale Collaborative-Filtering applications, we present a Non-Commuting Latent Factor (NCLF) tensor-completion approach for modeling three-way arrays, which is diagonal like the standard PARAFAC, but wherein different terms distinguish different kinds of three-way relations of co-clusters, as determin…

We outline the notions and concepts of the calculus of variational multivectors within the Poisson formalism over the spaces of infinite jets of mappings from commutative (non)graded smooth manifolds to the factors of noncommutative associative algebras over the equivalence under cyclic permutations of the letters in t…

Defines metrics and Einstein tensors on Riemannian manifolds, proving vanishing for non-commutative two-torus.

This paper gives a survey of the index theory of tangentially elliptic and transversally elliptic operators on foliated manifolds as well as of related notions and results in non-commutative geometry.

We study q-holonomic sequences that arise as the colored Jones polynomial of knots in 3-space. The minimal-order recurrence for such a sequence is called the (non-commutative) A-polynomial of a knot. Using the "method of guessing", we obtain this polynomial explicitly for the K_p = (-2, 3, 3+2p) pretzel knots for p = -…

In this summary of Habilitation Thesis, it is outlined author's 18 years research activity on mathematical physics, geometric methods in particle physics and gravity, modifications and applications (after defending his PhD thesis in 1994). Ten most relevant publications are structured conventionally into three "strateg…

Given a circle-valued Morse function of a closed oriented manifold, we prove that Reidemeister torsion over a non-commutative formal Laurent polynomial ring equals the product of a certain non-commutative Lefschetz-type zeta function and the algebraic torsion of the Novikov complex over the ring. This paper gives a gen…

In this paper we analyze the obstructions to the existence of global action-angle variables for regular non-commutative integrable systems (NCI systems) on Poisson manifolds. In contrast with local action-angle variables, which exist as soon as the fibers of the momentum map of such an integrable system are compact, gl…

If X is a full, finitely generated, projective module over a non-commutative torus, the Yang-Mills functional attains its minimum exactly on the flat connections on X. We classify the flat connections on modules admitting integrable connections.

We associate a non-commutative $C^*$-algebra with any locally finite simplicial complex. We determine the $K$-theory of these algebras and show that they can be used to obtain a conceptual explanation for the Baum-Connes conjecture.

New non-commutative geometric Nahm transform for ASD connections.

We define a "sutured topological quantum field theory", motivated by the study of sutured Floer homology of product 3-manifolds, and contact elements. We study a rich algebraic structure of suture elements in sutured TQFT, showing that it corresponds to contact elements in sutured Floer homology. We use this approach t…

Let G be a connected Lie group with Lie algebra g. The Duflo map is a vector space isomorphism between the symmetric algebra S(g) and the universal enveloping algebra U(g) which, as proved by Duflo, restricts to a ring isomorphism from invariant polynomials onto the center of the universal enveloping algebra. The Duflo…

The purpose of the paper is two-fold: to introduce a multivariable creative telescoping method, and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative $A$-polynomial of twist knots. Our multivariable creative telescoping method allows us to compute linear recursions for sums of …

Geometric non-commutative geometry proves non-existence of certain metrics.

Study on integrability of geodesic flows on Heisenberg group.

A non-commutative differential calculus on the $h$-superplane is presented via a contraction of the $q$-superplane. An R-matrix which satisfies both ungraded and graded Yang-Baxter equations is obtained and a new deformation of the $(1+1)$ dimensional classical phase space (the super-Heisenberg algebra) is introduced.

In a previous paper, the second author defined integer-valued functions delta_n on the first cohomology of a 3-manifold, generalizing McMullen's Alexander norm. It was shown that these functions give lower bounds on the Thurston norm. In this paper, we reformulate these invariants in terms of Reidemeister torsion over …

In commutative differential geometry the Frölicher-Nijenhuis bracket computes all kinds of curvatures and obstructions to integrability. In \cit!{3} the Frölicher-Nijenhuis bracket was developped for universal differential forms of non-commutative algebras, and several applications were given. In this paper this bracke…

Invariants for singular knots and links using non-commutative cocycles.

Study new symmetries in non-symmetric spaces and discontinuous groups.

Constructs a new field theory for 4-manifolds.

In the early 2000's Cochran and Harvey introduced non-commutative Alexander polynomials for 3-manifolds. Their degrees give strong lower bounds on the Thurston norm. In this paper we make the case that the vanishing of a certain Novikov-Sikorav homology module is the correct notion of a monic non-commutative Alexander …

Characterizes algebraic integrability and minimality of Lie equations for non-commutative pseudogroups.

A cluster variety of Fock and Goncharov is a scheme constructed from the data related to the cluster algebras of Fomin and Zelevinsky. A seed is a combinatorial data which can be encoded as an $n\times n$ matrix with integer entries, or as a quiver in special cases, together with $n$ formal variables. A mutation is a c…

Study formalities on closed surfaces using connections.

A new formula connects supersymmetric path integrals to Chern-Simons theory.

The paper connects calculus, gauge theory, and noncommutative worlds.

J. Przytycki has established a connection between the Hochschild homology of an algebra $A$ and the chromatic graph homology of a polygon graph with coefficients in $A$. In general the chromatic graph homology is not defined in the case where the coefficient ring is a non-commutative algebra. In this paper we define a …

Lecture notes on singular foliations, smooth and holomorphic.

Study differential and integral calculus on noncommutative C*-algebras.

We extend Obata's rigidity theorem to free probability.

In his famous Princeton Notes, Thurston introduced the so-called gluing equations defining the deformation variety. Later, Kashaev defined a non-commutative ring from H-triangulations of 3-manifolds and observed that for trefoil and figure-eight knot complements the abelianization of this ring is isomorphic to the ring…

It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected components, and its representation should be the direct sum of three irreducible two-dim…

This work extends GNNs to handle multiple graphs with non-commuting operators, proving transferability.

New invariant distinguishes smooth structures in four-manifolds.

On the predual of a von Neumann algebra, we define a differentiable manifold structure and affine connections by embeddings into non-commutative L_p-spaces. Using the geometry of uniformly convex Banach spaces and duality of the L_p and L_q spaces for 1/p+1/q=1, we show that we can introduce the α-divergence, for αin (…

We identify branched coverings (continuous open surjections p:Y->X of Hausdorff spaces with uniformly bounded number of pre-images) with Hilbert C*-modules C(Y) over C(X) and with faithful unital positive conditional expectations E:C(Y)->C(X) topologically of index-finite type. The case of non-branched coverings corres…

The paper classifies topological holonomy groups in $SO(3)$.