arXiv research

Identifies quantum group representations for tangle Floer homology.

Study categorifies link invariants using Soergel bimodules.

Study connects bordered theory to Khovanov-Seidel quiver algebras.

Categorifies Hecke algebra subalgebra using sheaves and bimodules.

Let A be a dg algebra over F_2 and let M be a dg A-bimodule. We show that under certain technical hypotheses on A, a noncommutative analog of the Hodge-to-de Rham spectral sequence starts at the Hochschild homology of the derived tensor product of M with itself and converges to the Hochschild homology of M. We apply th…

We give a combinatorial proof of the quasi-invertibility of $\widehat{CFDD}(\mathbb{I}_\mathcal{Z})$ in bordered Heegaard Floer homology, which implies a Koszul self-duality on the dg-algebra $\mathcal{A}(\mathcal{Z})$, for each pointed matched circle $\mathcal{Z}$. This is done by giving an explicit description of a r…

Foams 2-equivalent to singular Soergel bimodules.

New algebras and maps defined in knot Floer homology for trivalent vertices.

Paper proposes BSP to find stable bimodules of cross-correlated features.

We analyse the structure of the first order operators in bimodules introduced by A. Connes. We apply this analysis to the theory of connections on bimodules generalizing thereby several proposals.

Researchers link tangle invariants for Khovanov and knot Floer homologies.

Study links using Soergel bimodules and Serre duality.

The paper solves a problem in constructing a bicategory of algebra bundles.

New invariant for links in handlebodies defined using braid groups and Soergel bimodules.

Decomposes Goldman-Turaev Lie bialgebra via cutting a surface.

We discuss a relationship between Khovanov- and Heegaard Floer-type homology theories for braids. Explicitly, we define a filtration on the bordered Heegaard-Floer homology bimodule associated to the double-branched cover of a braid and show that its associated graded bimodule is equivalent to a similar bimodule define…

Defines new Heegaard Floer invariants with actions of both E and F.

We remark some basic facts on homological aspects of involutive Lie bialgebras and their involutive bimodules, and present some problems on surface topology related to these facts.

We trade matrix factorizations and Koszul complexes for Hochschild homology of Soergel bimodules to modify the construction of triply-graded link homology and relate it to Kazhdan-Lusztig theory.

New 4-manifold invariant defined from trisection diagrams.

New connections found for quantum flag manifolds modules.

New stable homotopy refinement of quantum annular Khovanov homology.

Paper proves Koszul duality for weighted A-infinity algebras.

New deformation of link homology for colored diagrams.

The paper studies formal geometry of dg manifolds and proves isomorphism of their calculi.

Researchers define new algebraic structures for knot Floer homology.

Categorifies a skein relation for links colored by one-column Young diagrams.

Proves Serre duality in Khovanov-Rozansky homology.

We study dg-manifolds which are R[2]-bundles over R[1]-bundles over manifolds, we calculate its symmetries, its derived symmetries and we introduce the concept of T-dual dg-manifolds. Within this framework we construct the T-duality map as a degree -1 map between the cohomologies of the T-dual dg-manifolds and we show …

Bordered Heegaard Floer homology is a three-manifold invariant which associates to a surface F an algebra A(F) and to a three-manifold Y with boundary identified with F a module over A(F). In this paper, we establish naturality properties of this invariant. Changing the diffeomorphism between F and the boundary of Y te…

The paper constructs structures for Lie pairs and their Atiyah classes.

Paper computes Atiyah class for DG manifolds of amplitude +1.

Defines state sum models with defects in 3-manifolds.

The paper proves unique Levi-Civita connections on noncommutative forms.

We survey what is known about singularities of special Lagrangian submanifolds (SL m-folds) in (almost) Calabi-Yau manifolds. The bulk of the paper summarizes the author's five papers math.DG/0211294, math.DG/0211295, math.DG/0302355, math.DG/0302356, math.DG/0303272 on SL m-folds X with isolated conical singularities.…

Spectral sequence connects knot homologies via algebraic geometry.

We define and study the theory of derivation-based connections on a recently introduced class of bimodules over an algebra which reduces to the category of modules whenever the algebra is commutative. This theory contains, in particular, a noncommutative generalization of linear connections. We also discuss the differe…

Authors conjecture categorically diagonalizable complex for full twists.

Atiyah classes of DG manifolds of positive amplitude are invariant under weak equivalences.

New categories from TQFTs interpret skein relations.

Unique vertical isomorphisms between Fedosov dg manifolds are proven for Lie pairs.

The paper explores connections between dg manifolds and homotopy Lie algebras.

Constructs dg categories from surfaces using Khovanov homology.

New dg-algebras link graph colorings to sheaves.

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Develops theory of differential graded schemes for derived stacks.

In this article we show that some of the recent results of Marino, Moore, and Peradze (math.DG/9812042, hep-th/9812055) -- in particular their conjecture that all closed, smooth four-manifolds with b_2^+ > 1 (and Seiberg-Witten simple type) are of `superconformal simple type' -- can be understood using a simple mathema…

Study Hochschild cohomology of dg manifolds linked to integrable distributions.