arXiv research

Study cohomology of homeomorphisms and diffeomorphisms of manifolds.

Study of low dimensional representations of mapping class groups of surfaces, focusing on genus ≥ 7.

New insights into Coxeter groups' $L^2$-cohomology via weighted analysis.

This paper extends group cohomology to bounded cohomology using quasihomomorphisms.

We obtain formulas for the first and second cohomology groups of a general current Lie algebra with coefficients in the "current" module, and apply them to compute structure functions for manifolds of loops with values in compact Hermitian symmetric spaces.

Introduces $θ$-almost twisted Poisson structures and their cohomology.

We propose an approach to study non-Abelian Iwasawa theory, using the idea of Johnson homomorphisms in low dimensional topology. We introduce arithmetic analogues of Johnson homomorphisms/maps, called the p-Johnson homomorphisms/maps, associated to the Zassenhaus filtration of a pro-p Galois group over a Z_p-extension …

Introduces new algebraic structures and their cohomology.

We review localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low-dimensional gauge theories. These are the functional integral counterparts of the Mathai-Quillen formalism, the Duistermaa…

The paper studies deformations of Filippov algebroids using cohomology and DGLA.

We initiate the study of holomorphically convex groups: groups that can be realized as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers. If $G$ is a holomorphically convex group of cohomological dimension two, we show that $G$ is isomorphic to the fundamental group …

In earlier joint work with A. Connes on transverse index theory on foliations, cyclic cohomology adapted to Hopf algebras has emerged as a decisive tool in deciphering the total index class of the hypoelliptic signature operator. We have found a Hopf algebra H(n), playing the role of a `quantum structure group' for the…

Paper proves non-vanishing of index map for low-degree cohomology classes.

Monograph introduces bounded cohomology of discrete groups and spaces.

We give explicit formulae for fringe lengths of the Calegari-Walker Ziggurats -- i.e. graphs of extremal rotation numbers associated to positive words in free groups. These formulae reveal (partial) integral projective self-similarity in ziggurat fringes, which are low-dimensional projections of characteristic polyhedr…

The paper explores the formality of low-dimensional manifolds using algebraic structures.

The theory of differential forms began with a discovery of Poincare who found conservation laws of a new type for Hamiltonian systems - The Integral Invariants. Even in the absence of non-trivial integrals of motion, there exist invariant differential forms: a symplectic two-form, or a contact one-form for geodesic flo…

The quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite cyclic group in the boundary operator. Theories of Alexander extensions of quandles in relation to low dimensional cocycles are developed in parallel to gr…

These are notes from lectures given at the Clay Institute Summer School on "Floer homology, gauge theory and low-dimensional topology" (Budapest, 2004). The first part describes as background some of the geometry of symplectic fibre bundles and their monodromy. The second part, overviewing joint work with Paul Seidel, …

The paper explores the dynamics of composite symplectic Dehn twists with nonuniform hyperbolicity.

In low dimensional topology, we have some invariants defined by using solutions of some nonlinear elliptic operators. The invariants could be understood as Euler class or degree in the ordinary cohomology, in infinite dimensional setting. Instead of looking at the solutions, if we can regard some kind of homotopy class…

Unique map connects curve pairs to abelian varieties.

This work refines Cover's theory for binary classification on low-dimensional data.

Diffusion models adapt to low-dimensional data regardless of coefficient choices.

Improved likelihood-free inference by localizing and refining low-dimensional approximations.

Deep networks can efficiently represent low-dimensional manifolds.

Deep ReLU networks estimate Hölder functions on low-dimensional manifolds with fast convergence.

New cohomology defined for Lie algebroids by odd cocycles.

Researchers twist Deligne cohomology for the first time.

Extends cohomology theory for infinite volume transformation groups.

New cohomology theory for Lie 2-algebras extends classical theory.

New metric space cohomology relates to Riemannian manifold cohomology.

Generics extended to new cohomologies.

The groups $Γ_{n,s}$ are defined in terms of homotopy equivalences of certain graphs, and are natural generalisations of $\mbox{Out}(F_n)$ and $\mbox{Aut}(F_n)$. They have appeared frequently in the study of free group automorphisms, for example in proofs of homological stability in [8,9] and in the proof that Out$(F_n…

Defines Courant pairs and studies their deformations via cohomology.

The paper studies cohomology of complex manifolds using Morse-Novikov and Dolbeault-Morse-Novikov theories.

The article examines twisted cohomologies on algebraic and analytic varieties.

Optimizes global optimization with random embeddings by defining a minimal low-dimensional domain.

New cohomology theories for heaps and ternary operations linked to group cohomology.

Study cohomology of hemistrict Lie 2-algebras, proving isomorphic results.

Study characterizes cohomology of Vaisman manifolds, linking Bott-Chern and Dolbeault numbers.

Proves a vanishing property for symplectic manifold cohomology.

In this paper we define a new cohomology of a smooth manifold called Lichnerowicz type cohomology attached to a function. Firstly, we study some basic properties of this cohomology as: a de Rham type isomorphism, dependence on the function, singular forms, relative cohomology, Mayer-Vietoris sequence, homotopy invarian…

De Rham theorem extended to Orlicz cohomology.

New proof of blow-up formula for Morse-Novikov cohomology.

Survey on equivariant cohomology of Lie group actions.

Researchers calculate cohomological dimensions of manifold configuration spaces, proving arithmeticity and providing bounds.

Rham cohomology of Lie groupoids over triangulated manifolds is isomorphic to piecewise cohomology.