arXiv research

Study on homotopy type of Sullivan diagrams space.

Using intersection theory in the context of Hilbert manifolds and geometric homology we show how to recover the main operations of string topology built by M. Chas and D. Sullivan. We also study and build an action of the homology of reduced Sullivan's chord diagrams on the singular homology of free loop spaces, extend…

A note on the uniqueness of differential characters and K-theory via homological algebra.

We compare two combinatorial models for the moduli space of two-dimensional cobordisms: Bödigheimer's radial slit configurations and Godin's admissible fat graphs, producing an explicit homotopy equivalence using a "critical graph" map. We also discuss natural compactifications of these two models, the unilevel harmoni…

We show that the Malcev Lie algebra of the fundamental group of a compact $2n+1$-dimensional Sasakian manifold with $n\ge 2$ admits a quadratic presentation by using Morgan's bigradings of minimal models of mixed-Hodge diagrams. By using bigradings of minimal models, we also simplify the proof of the result of Cappelle…

This is the first of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co--chains of a Frobenius algebra. We also prove that a there is dg--PROP action of a version of Sullivan Chord diagrams which acts on the …

Let $Σ_{\infty, 1}$ be the inductive limit of compact oriented surfaces with one boundary component. We prove the center of the Goldman Lie algebra of the surface $Σ_{\infty,1}$ is spanned by the constant loop. A similar statement for a closed oriented surface was conjectured by Chas and Sullivan, and proved by Etingof…

Develops Patterson-Sullivan theory for coarse cocycles.

Sullivan discusses his contributions to math and physics.

We define the Simons-Sullivan differential analytic index by translating the Freed-Lott differential analytic index via explicit ring isomorphisms between Freed-Lott differential K-theory and Simons-Sullivan differential K-theory. We prove the differential Grothendieck-Riemann-Roch theorem in Simons-Sullivan differenti…

Paper develops a theory for Patterson-Sullivan measures in higher rank symmetric spaces.

Extends Hopf-Tsuji-Sullivan dichotomy to higher rank groups and applies to Anosov subgroups.

Paper proves finite BMS measure for SPR groups in higher rank Lie groups.

The paper introduces Patterson-Sullivan systems and proves their rigidity, with applications to random walks and entropy rigidity.

Unique entropy measure found for convex projective manifolds.

The paper connects geodesic flows and limit sets on visibility manifolds.

Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.

We give a finite dimensional approach to the Chas-Sullivan product on the free loop space of a manifold, orientable or not.

Proves classification of 4D complete intersections up to diffeomorphism.

Combines Kleinian groups and polynomials into a dynamical system.

Uniform lattices in certain semi-simple groups contain Anosov surface subgroups.

For a fixed closed manifold $P$, we construct a cobordism category of embedded manifolds with a single Baas-Sullivan singularity of type $P$. Our main theorem identifies the homotopy type of the classifying space of this cobordism category with that of the infinite loop-space of a certain spectrum related to the spectr…

New theory extends classical results to Anosov subgroups.

Discretizes diffusions and harmonic functions on covering spaces.

The article proves a unique invariant measure for geodesic flows on certain rank 1 manifolds.

This is the second of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co--chains of a Frobenius algebra. We also prove that a there is dg--PROP action of a version of Sullivan Chord diagrams which acts on the…

New findings on geometric flows and equidistribution in Hilbert geometry.

Develops a chain-level model for Chas-Sullivan products using Morse theory with differential graded coefficients.

New findings show mapping class groups of certain high-dimensional manifolds are not residually finite.

Anosov groups limit sets are Ahlfors regular, with applications in Teichmüller spaces.

Extends Morse-Novikov Homology to include differential graded coefficients and fibration structures.

Let f: P-->W be an embedding of a compact polyhedron in a closed oriented manifold W, let T be a regular neighborhood of P in W and let C:=closure(W-T) be its complement. Then W is the homotopy push-out of a diagram C<--dT-->P. This homotopy push-out square is an example of what is called a Poincare embedding. We study…

We define generalized currents associated with immersions of abstract solenoids with a transversal measure. We realize geometrically the full real homology of a compact manifold with these generalized currents, and more precisely with immersions of minimal uniquely ergodic solenoids. This makes precise and geometric De…

The paper develops a theory of conformal density at infinity for groups with contracting elements.

The fundamental group of a hyperbolic manifold acts on the limit set, giving rise to a cross-product C^* algebra. We construct nontrivial K-cycles for the cross-product algebra, thereby extending some results of Connes and Sullivan to higher dimensions. We also show how the Patterson-Sullivan measure on the limit set c…

Method extends eigenfunction construction to non-symmetric spaces.

Refines mixed Hodge structures using non-abelian Hodge theory.

Develops measures for non-Borel Anosov groups on Furstenberg boundary.

Study vanishing theorems for CR manifolds using contact forms and Laplacian formulas.

We prove that for a relatively hyperbolic group G there is a sequence of relatively hyperbolic proper quotients such that their growth rates converge to the growth rate of G. Under natural assumptions, the same conclusion holds for the critical exponent of a cusp-uniform action of G on a hyperbolic metric space. As a c…

Research confirms a conjecture about complex manifolds with total Betti number three.

Quantifies how geodesic planes isolate in hyperbolic 3-manifolds.

The paper studies ergodicity of flows on subspaces, generalizing earlier work.

Researchers calculate alpha invariant for certain complex projective spaces.

Sullivan showed that there exists $K_0$ such that if $Ω\subset \hat{\mathbb{C}}$ is a simply connected hyperbolic domain, then there exists a conformally natural $K_0$-quasiconformal map from $Ω$ to the boundary ${\rm Dome}(Ω)$ of the convex hull of its complement which extends to the identity on $\partialΩ$. Explicit …

Maximal representations show strong entropy rigidity.

New example solves topological dynamics problem.

Confirms unique eigenfunction in hyperbolic packing has maximal spectral gap.