arXiv research

The isotropy action on certain symmetric spaces is shown to be equivariantly formal.

We prove that all generalised symmetric spaces of compact simple Lie groups are formal in the sense of Sullivan. Nevertheless, many of them, including all the non-symmetric flag manifolds, do not admit Riemannian metrics for which all products of harmonic forms are harmonic.

A metric is formal if all products of harmonic forms are again harmonic. The existence of a formal metric implies Sullivan formality of the manifold, and hence formal metrics can exist only in presence of a very restricted topology. We show that a warped product metric is formal if and only if the warping function is c…

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

This paper formalizes manifolds in positive characteristic varieties.

Using a cell model for the little discs operad in terms of spineless cacti we give a minimal common topological operadic formalism for three a priori disparate algebraic structures: (1) a solution to Deligne's conjecture on the Hochschild complex, (2) the Hopf algebra of Connes and Kreimer, and (3) the string topology …

For a simply connected solvable Lie group G with a cocompact discrete subgroup Γ, we consider the space of differential forms on the solvmanifold G/Γ with values in certain flat bundle so that this space has a structure of a differential graded algebra(DGA). We construct Sullivan's minimal model of this DGA. This resul…

The article confirms Joyce's examples of G2-holonomy are formal spaces.

Develops Patterson-Sullivan theory for coarse cocycles.

Sullivan discusses his contributions to math and physics.

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

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.

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.

Proves classification of 4D complete intersections up to diffeomorphism.

Let X be a pseudomanifold. In this text, we use a simplicial blow-up to define a cochain complex whose cohomology with coefficients in a field, is isomorphic to the intersection cohomology of X, introduced by M. Goresky and R. MacPherson. We do it simplicially in the setting of a filtered version of face sets, also cal…

Combines Kleinian groups and polynomials into a dynamical system.

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…

Discretizes diffusions and harmonic functions on covering spaces.

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.

The string bracket introduced by Chas and Sullivan [math.GT/9911159] is reinterpreted from the point of view of topological field theories in the Batalin-Vilkovisky or BRST formalisms. Namely, topological action functionals for gauge fields (generalizing Chern-Simons and BF theories) are considered together with genera…

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.

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.

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…

We determine an explicit presentation by generators and relations of the cohomology algebra $H^*(\mathbb P^2\setminus C,\mathbb C)$ of the complement to an algebraic curve $C$ in the complex projective plane $\mathbb P^2$, via the study of log-resolution logarithmic forms on $\mathbb P^2$. As a first consequence, we 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.

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 $X(\infty)$ of $X$, we show the existence and uniqueness of the Busemann density, which is realized via the Patterson-Sullivan measure. Based …

Method extends eigenfunction construction to non-symmetric spaces.

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…

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

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

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

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…

Let M be a compact Riemannian manifold equipped with a parallel differential form ω. We prove a version of Kaehler identities in this setting. This is used to show that the de Rham algebra of M is weakly equivalent to its subquotient $(H^*_c(M), d)$, called {\bf the pseudocohomology} of M. When M is compact and Kaehler…

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 …