arXiv research

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

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

The paper proves convergence of discrete maps to Riemann mappings for polyhedral surfaces.

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

Combines Kleinian groups and polynomials into a dynamical system.

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

Proves classification of 4D complete intersections up to diffeomorphism.

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

Let $G, H$ be two Kleinian groups with homeomorphic quotients $\mathbb H^3/G$ and $\mathbb H^3/H$. We assume that $G$ is of divergence type, and consider the Patterson-Sullivan measures of $G$ and $H$. The measurable rigidity theorem by Sullivan and Tukia says that a measurable and essentially directly measurable equiv…

New algorithm proves most inflexible manifolds are not strongly inflexible.

Motivated by Bonahon's result for hyperbolic surfaces, we construct an analogue of the Patterson-Sullivan-Bowen-Margulis map from the Culler-Vogtmann outer space $CV(F_k)$ into the space of projectivized geodesic currents on a free group. We prove that this map is a topological embedding. We also prove that for every $…

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 …

Symplectic manifolds derived from Torelli groups, showing complex structure invariants without holomorphic structures.

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…

Develops Patterson-Sullivan theory for coarse cocycles.

Study quasisymmetric maps on hyperbolic plane boundaries.

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…

The paper studies obstructions to homotopy invariance of loop coproducts.

We give a sufficient condition for a metric (homology) manifold to be locally bi-Lipschitz equivalent to an open subset in $\rn$. The condition is a Sobolev condition for a measurable coframe of flat 1-forms. In combination with an earlier work of D. Sullivan, our methods also yield an analytic characterization for smo…

We study the homotopy type of the harmonic compactification of the moduli space of a 2-cobordism S with one outgoing boundary component, or equivalently of the space of Sullivan diagrams of type S on one circle. Our results are of two types: vanishing and non-vanishing. In our vanishing results we are able to show that…

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 connects function theory, dynamics, and ergodic theory via Thurston's theory.

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.

We study the asymptotic behavior of the sequence of the Nielsen numbers $\{N(f^k)\}$, the essential periodic orbits of $f$ and the homotopy minimal periods of $f$ by using the Nielsen theory of maps $f$ on infra-solvmanifolds of type $R$. We give a linear lower bound for the number of essential periodic orbits of such …

For a Poincare duality space X and a map X -> B, consider the homotopy fiber product X x^B X. If X is orientable with respect to a multiplicative cohomology theory E, then, after suitably regrading, it is shown that the E-homology of X x^B X has the structure of a graded associative algebra. When X -> B is the diagonal…

Overview of dynamics in algebraic correspondences and their connections.

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…

Geometric correspondence links flow metrics to reparameterizations.

For a convex cocompact subgroup $G<Mod(S)$, and points $x,y \in Teich(S)$ we obtain asymptotic formulas as $R\to \infty$ of $|B_{R}(x)\cap Gy|$ as well as the number of conjugacy classes of pseudo-Anosov elements in $G$ of dilatation at most $R$. We do this by developing an analogue of Patterson-Sullivan theory for the…

The paper proves rigidity and ergodicity of horospherical foliations.

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…

Study extreme values of stable random fields on geometric spaces.

New theory extends classical results to Anosov subgroups.

The paper extends the Manhattan curve concept to complex dynamics and studies its relation to multiplier spectra.

Discretizes diffusions and harmonic functions on covering spaces.

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

Residually finite groups found in manifold automorphisms.

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.

In 1985 D.Sullivan had introduced a dictionary between two domains of complex dynamics: iterations of rational functions on the Riemann sphere and Kleinian groups. The latters are discrete subgroups of the group of conformal automorphisms of the Riemann sphere. This dictionary motivated many remarkable results in both …

New theorem proves convergence of various discrete conformal structures to conformal maps.

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