Develops Patterson-Sullivan theory for coarse cocycles.
problem None explicitly stated in the abstract.
method Theory of Patterson--Sullivan measures for coarse cocycles of convergence groups.
result Existence, uniqueness, and ergodicity results for Patterson-Sullivan measures under geometric assumptions.
Sullivan discusses his contributions to math and physics.
problem None explicitly stated in the abstract.
method Personal overview of Dennis Sullivan's work.
result Sullivan's work spans mathematics 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…
Method extends eigenfunction construction to non-symmetric spaces.
problem Constructing eigenfunctions on harmonic manifolds.
method Applying Sullivan's method to non-compact harmonic manifolds.
result Eigenfunctions constructed for non-symmetric spaces.
Paper develops a theory for Patterson-Sullivan measures in higher rank symmetric spaces.
problem Establishing existence and uniqueness of Patterson-Sullivan measures in higher rank symmetric spaces.
method Develops theory for vector-valued horofunction boundaries and shadows.
result Proves existence and uniqueness of Patterson-Sullivan measures for transverse groups.
Extends Hopf-Tsuji-Sullivan dichotomy to higher rank groups and applies to Anosov subgroups.
problem Understanding discrete subgroups of semisimple real algebraic groups.
method Establishes an extension of the Hopf-Tsuji-Sullivan dichotomy and applies it to Anosov subgroups.
result Anosov subgroups exhibit different phenomena depending on the rank of the group.
Paper proves finite BMS measure for SPR groups in higher rank Lie groups.
problem Finite measure for certain groups in higher rank Lie groups.
method Developed SPR property and proved finite BMS measure.
result Finite Bowen-Margulis-Sullivan 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.
problem Understanding the rigidity of Patterson-Sullivan systems and their applications.
method Generalization of Tukia's measurable boundary rigidity theorem for Patterson-Sullivan systems.
result Entropy rigidity for Anosov groups with Lipschitz limit sets.
Unique entropy measure found for convex projective manifolds.
problem Entropy measure for convex projective manifolds.
method Developed Patterson--Sullivan densities and mixing theory.
result Unique mixing measure of maximal entropy exists.
The paper connects geodesic flows and limit sets on visibility manifolds.
problem Understanding dynamics and ergodic properties on non-compact visibility manifolds.
method Analyzing geodesic flows and Patterson-Sullivan measures on visibility manifolds without conjugate points.
result The positivity of the Patterson-Sullivan measure of the Myrberg limit set is equivalent to the conservativity of the geodesic flow.
We give a finite dimensional approach to the Chas-Sullivan product on the free loop space of a manifold, orientable or not.
Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
problem Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
method Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
result Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
Proves classification of 4D complete intersections up to diffeomorphism.
problem Classifying 4-dimensional complete intersections up to diffeomorphism.
method Uses Hambleton-Madsen theory of degree-d normal maps and connects Segal Conjecture for S1 to Sullivan Conjecture. result Proves the Sullivan Conjecture for 4-dimensional complete intersections.
Combines Kleinian groups and polynomials into a dynamical system.
problem Connecting Kleinian groups and rational dynamics.
method Framework for combining Fuchsian groups with complex polynomials.
result Establishes a new dynamical system on the Riemann sphere.
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…
Proves finite measure implies product structure for certain discrete subgroups.
problem Classifying discrete subgroups with finite Bowen-Margulis-Sullivan measure.
method Product structure of leafwise measures and high entropy method.
result Proves virtually a product structure for certain subgroups.
New findings on geometric flows and equidistribution in Hilbert geometry.
problem Characterizing dynamical and counting results in Hilbert geometry.
method Study of dynamical and counting results in rank-one properly convex projective structures with Hilbert metrics.
result Hilbert geodesic flow is strongly mixing and orbits and primitive closed geodesics equidistribute.
Develops a chain-level model for Chas-Sullivan products using Morse theory with differential graded coefficients.
problem Chas-Sullivan products on homology of loop spaces.
method Morse theory with differential graded coefficients, functorial properties, K{ü}nneth formula, Pontryagin-Thom construction.
result Chain-level description of Chas-Sullivan products.
New findings show mapping class groups of certain high-dimensional manifolds are not residually finite.
problem Understanding the mapping class groups of simply connected high-dimensional manifolds.
method Provided a counterexample showing mapping class groups are not residually finite.
result Mapping class groups of some high-dimensional manifolds are not residually finite.
Anosov groups limit sets are Ahlfors regular, with applications in Teichmüller spaces.
problem Understanding Ahlfors regularity of limit sets for Anosov groups.
method Proving Ahlfors regularity for limit sets and Patterson-Sullivan measures.
result Patterson-Sullivan measures are Ahlfors regular if and only if associated linear forms are symmetric.
We show that uniform lattices in some semi-simple groups (notably complex ones) admit Anosov surface subgroups. This result has a quantitative version: we introduce a notion, called K-Sullivan maps, which generalizes the notion of K-quasi-circles in hyperbolic geometry, and show in particular that Sullivan maps are…
Extends Morse-Novikov Homology to include differential graded coefficients and fibration structures.
problem Extending Morse-Novikov Homology with differential graded coefficients.
method Constructs a Morse-Novikov complex and proves the existence of a Chas-Sullivan-like product for a fibration.
result Proves the existence of a Chas-Sullivan-like product on the Novikov completion of a fibration.
Study shows exact dimensionality and regularity of manifolds for specific groups.
problem Exact dimensionality and regularity of manifolds for relatively Anosov groups.
method Dynamical methods, including finite and mixing of Bowen–Margulis–Sullivan measures.
result Manifolds are C1-regular and growth indicator is strictly concave. 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 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…
The paper develops a theory of conformal density at infinity for groups with contracting elements.
problem Understanding conformal dynamics at infinity for groups with contracting elements.
method Introducing a class of convergence boundary and establishing the basic theory of conformal density on it.
result Unified theory of conformal density on various boundaries for different types of groups.
In this article, we consider the geodesic flow on a compact rank 1 Riemannian manifold M without focal points, whose universal cover is denoted by X. On the ideal boundary X(∞) of X, we show the existence and uniqueness of the Busemann density, which is realized via the Patterson-Sullivan measure. Based …
We extend several notions and results from the classical Patterson-Sullivan theory to the setting of Anosov subgroups of higher rank semisimple Lie groups, working primarily with invariant Finsler metrics on associated symmetric spaces. In particular, we prove the equality between the Hausdorff dimensions of flag limit…
The study examines the realizability of a 4-manifold invariant for homeomorphisms.
problem Realizing the Casson-Sullivan invariant for homeomorphisms of 4-manifolds.
method Investigation of the invariant's realizability and application to surface isotopy.
result The invariant can be realized fully after stabilizing with a single S2imesS2 for all orientable pairs of homeomorphic 4-manifolds. 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…
Develops measures for non-Borel Anosov groups on Furstenberg boundary.
problem Measuring non-Borel Anosov groups on the Furstenberg boundary.
method Theory of Patterson--Sullivan measures, strict convexity, entropy rigidity.
result Existence, uniqueness, and ergodicity of measures on Furstenberg boundary.
Study vanishing theorems for CR manifolds using contact forms and Laplacian formulas.
problem Vanishing theorems for Kohn-Rossi cohomology of spherical CR manifolds.
method Used a canonical contact form and Weitzenböck-type formulae for the Kohn Laplacian.
result Results are optimal in some cases and prove vanishing theorems.
Research confirms a conjecture about complex manifolds with total Betti number three.
problem Understanding the minimal total Betti number of closed almost complex manifolds.
method Analyzing properties of almost complex manifolds and using topological results.
result The only simply connected closed complex manifold with total Betti number three is the complex projective plane.
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…
The paper studies ergodicity of flows on subspaces, generalizing earlier work.
problem Ergodicity of flows on subspaces of higher rank groups.
method Analyzes one-parameter diagonalizable subgroups of connected semisimple groups acting on homogeneous spaces.
result Obtains an ergodicity criterion similar to Hopf-Tsuji-Sullivan for general Anosov subgroups.
Frame flows on certain symmetric spaces mix exponentially.
problem Exponential mixing of frame flows in convex cocompact locally symmetric spaces.
method Generalized local non-integrability and non-concentration properties to apply Dolgopyat's method.
result Exponential mixing of frame flows proved for convex cocompact locally symmetric spaces.
Sullivan showed that there exists K0 such that if Ω⊂C^ is a simply connected hyperbolic domain, then there exists a conformally natural K0-quasiconformal map from Ω to the boundary Dome(Ω) of the convex hull of its complement which extends to the identity on ∂Ω. Explicit …
Maximal representations show strong entropy rigidity.
problem Entropy rigidity for maximal representations.
method Measurable hypertransversality, Gromov product, Bowen-Margulis-Sullivan measure.
result Strong entropy rigidity proved for maximal representations.
New example solves topological dynamics problem.
problem Embedding compact metric space into cubical shift.
method Borsuk-Ulam theorem, p-adic completions, equivariant Sullivan conjecture.
result Existence of a compact metric space not embeddable into a cubical shift.
Confirms unique eigenfunction in hyperbolic packing has maximal spectral gap.
problem Sarnak's spectral gap question for hyperbolic packings.
method Analysis of Patterson-Sullivan base eigenfunctions and spectral gaps.
result Unique square-integrable eigenfunction has maximal spectral gap.
New measure of maximal entropy found for a class of geometrically finite groups.
problem Finding a measure of maximal entropy for relatively Anosov groups.
method Constructing reparameterizations and using exponential expansion along unstable foliations.
result The Bowen-Margulis-Sullivan measure is finite and unique for relatively Anosov groups.
Paper resolves decades-old problem about L-spectra.
problem Identifying L-spectra local information with geometric data. method Proved equivalence of L-orientations and characteristic classes. result Levitt-Ranicki's theory equivalent to Brumfiel-Morgan's classes.
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…
The paper extends geometric results from negatively-curved spaces to strictly convex Hilbert geometry.
problem Extending geometric results from negatively-curved spaces to strictly convex Hilbert geometry.
method Demonstrates dynamical and counting results for geometrically-finite strictly convex projective structures with Hilbert metric.
result Hilbert geodesic flow is strongly mixing and orbits and primitive closed geodesics equidistribute.
We give necessary and sufficient conditions for the existence of smooth Lyapunov 1-forms for the flow of a smooth vector field in terms of the behavior of certain locally finite invariant measures. The main statement generalizes a result of Schwartzman, whereas the methods are adapted from work of Sullivan.
The paper proves convergence of discrete maps to Riemann mappings for polyhedral surfaces.
problem Discrete conformal geometry of polyhedral surfaces.
method Establishing rigidity for hexagonal triangulations and estimating quasiconformal constants.
result Discrete conformal maps converge to Riemann mappings for Jordan domains.
In his 1985 paper Sullivan sketched a proof of his structural stability theorem for differentiable group actions satisfying certain expansion-hyperbolicity axioms. In this paper we relax Sullivan's axioms and introduce a notion of "meandering hyperbolicity" for group actions on geodesic metric spaces. This generalizati…
New examples of embeddings defy Anosov representation limits.
problem Examples of robust quasi-isometric embeddings not approximated by Anosov representations.
method Exhibited non-locally rigid, Zariski dense embeddings in SLm(K). result Higher rank Anosov representation theorems fail for m≥30.