arXiv research

The paper proves an asymptotic development for weighted sums of multiplicity functions on a torus-manifold.

New construction of Atiyah and Todd classes for Lie pair pullback dg Lie algebroids.

We introduce the notions of Atiyah class and Todd class of a differential graded vector bundle with respect to a differential graded Lie algebroid. We prove that the space of vector fields on a dg-manifold with homological vector field $Q$ admits a structure of L-infinity algebra with the Lie derivative $L_Q$ as unary …

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

Atiyah and Todd classes of Lie algebroids respect their Atiyah sequence.

The paper studies Atiyah and Todd classes for DG manifolds derived from integrable distributions.

Simplified construction recovers Todd class using algebraic methods.

Simplified proof of complex manifold Todd genera invariance.

This paper proves positivity of Riemann-Roch polynomials for hyperkähler manifolds.

A process enumerates rack elements from a presentation.

This paper announces results on the behavior of some important algebraic and topological invariants --- Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. --- and their associated characteristic classes, under morphisms of projective algebraic varieties. The formulas …

The paper studies vanishing cycles in matrix singularities.

For a closed oriented 3-manifold Y, we define an absolute grading on the Heegaard Floer homology groups of Y by homotopy classes of oriented 2-plane fields. We show that this absolute grading refines the relative one and that it is compatible with the maps induced by cobordisms. We also prove that if ξ is a contact str…

New connections adapted to graded structures defined for a class of manifolds.

In this paper, we construct a canonical grading on bordered Heegaard Floer homology by homotopy classes of nonvanishing vector fields. This grading is a generalization of our construction of an absolute grading on Heegaard Floer homology and it extends the well-known grading with values in a noncommutative group define…

Formality theorem established for g-manifolds, relating cohomologies of polyvector fields.

The paper studies almost complex torus manifolds using graphs and Hirzebruch genera, proving properties of their fixed points and cohomology.

In joint work with Yang Huang, we defined a canonical absolute grading on Heegaard Floer homology by homotopy classes of oriented 2-plane fields. A similar grading was defined on embedded contact homology by Michael Hutchings. In this paper we show that the isomorphism between these homology theories defined by Colin-G…

Study graded coverings for supermanifolds, proving their universal properties.

We prove that the isomorphism type of a large class of groups (containing finite groups, countable Artinian groups and mapping class groups of certain surfaces, among others) is determined by the set of differential graded $\mathbb Q$-algebras on which these groups act faithfully.

In this article, we start to recall the inversion formula for the convolution with the Box spline. The equivariant cohomology and the equivariant K-theory with respect to a compact torus G of various spaces associated to a linear action of G in a vector space M can be both described using some vector spaces of distribu…

Graded Transformers embed algebraic structure in neural networks through graded transformations.

New vanishing theorems for genera derived under almost nonnegative Ricci curvature.

Chevalley's theorem and it's converse, the Sheppard-Todd theorem, assert that finite reflection groups are distinguished by the fact that the ring of invariant polynomials is freely generated. We show that in the Euclidean case, a weaker condition suffices to characterize finite reflection groups, namely that a freely-…

The abstract generalizes a construction for splitting supermanifolds and studies Lie supergroup cases.

A formula connects two algebraic structures derived from a category.

Geometric structures on $\mathbb N Q$-manifolds, i.e.~non-negatively graded manifolds with an homological vector field, encode non-graded geometric data on Lie algebroids and their higher analogues. A particularly relevant class of structures consists of vector bundle valued differential forms. Symplectic forms, contac…

Paper proposes a fair grading method for randomized exams.

The paper introduces an inference algorithm for graded Bayesian networks.

Graded bundles are a class of graded manifolds which represent a natural generalisation of vector bundles and include the higher order tangent bundles as canonical examples. We present and study the concept of the linearisation of graded bundle which allows us to define the notion of the linear dual of a graded bundle.…

Given a supervector bundle $E = E_0\oplus E_1 \to M$, we exhibit a parametrization of Quillen superconnections on $E$ by graded connections on the Cartan-Koszul supermanifold $(M;Ω(M))$. The relation between the curvatures of both kind of connections, and their associated Chern classes, is discussed in detail. In parti…

In his 1990 doctoral thesis, Todd Drumm showed that proper affine deformations of free Fuchsian groups could be constructed as Schottky groups using a new family of hypersurfaces called "crooked planes." The existence of proper affine deformations of Fuchsian Schottky groups was demonstrated by Margulis in the early 19…

Study on characteristic classes for foliation deformations.

The paper finds obstructions to Lie algebroid representations up to homotopy.

Researchers find examples of real Bott manifolds with nonzero dual Stiefel-Whitney class wbar_{n-ahat(n)} for all n nonzero mod 4.

A Q-manifold is a graded manifold endowed with a vector field of degree one squaring to zero. We consider the notion of a Q-bundle, that is, a fiber bundle in the category of Q-manifolds. To each homotopy class of ``gauge fields'' (sections in the category of graded manifolds) and each cohomology class of a certain sub…

We study symplectic (contact) structures on nilmanifolds that correspond to the filiform Lie algebras - nilpotent Lie algebras of the maximal length of the descending central sequence. We give a complete classification of filiform Lie algebras that possess a basis e_1, ..., e_n, [e_i,e_j]=c_{ij}e_{i{+}j} (N-graded Lie …

Defines formal exponentials for graded manifolds and linearizes QP-manifolds.

A derivation of the Cesàro-Fedorov relation from the Selberg trace formula on an orbifolded 2-sphere is elaborated and extended to higher dimensions using the known heat-kernel coefficients for manifolds with piecewise-linear boundaries. Several results are obtained that relate the coefficients, $b_i$, in the Shephard-…

NPC tackles class imbalance in datasets by progressively considering more neighbors.

New calculus framework for vector bundles with metrics.

A rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers. In dimension at least three only multiples of the top Chern number, which is the Euler characteristic, are invarian…

Develops non-semisimple ETQFTs for 3-manifolds.

Develops Johnson-Morita theory for 3D handlebody groups.

We introduce the notions of geometric height and graded (geometric) relative hyperbolicity in this paper. We use these to characterize quasiconvexity in hyperbolic groups, relative quasiconvexity in relatively hyperbolic groups, and convex cocompactness in mapping class groups and $Out(F_n)$. Corrigendum: there is an u…

Study prolongations of nilpotent Lie algebras with specific structural subalgebras.

Study shows non-trivial knot Floer homology for specific knots.

Constructive approach to Lie algebra gradings, computing maximal and enumerating all gradings.