arXiv research

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…

We prove that structured vector bundles whose holonomies lie in GL(N,C), SO(N,C), or Sp(2N,C) have structured inverses. This generalizes a theorem of Simons and Sullivan.

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

In this paper we give explicit formulas of differential characteristic classes of principal $G$-bundles with connections and prove their expected properties. In particular, we obtain explicit formulas for differential Chern classes, differential Pontryagin classes and differential Euler class. Furthermore, we show that…

Develops differential KO-character to determine real vector bundles in multiples of 8.

The caloron correspondence is a tool that gives an equivalence between principal $G$-bundles based over the manifold $M \times S^1$ and principal $LG$-bundles on $M$, where $LG$ is the Fréchet Lie group of smooth loops in the Lie group $G$. This thesis uses the caloron correspondence to construct certain differential f…

Odd $K$-theory has the interesting property that it admits an infinite number of inequivalent differential refinements. In this paper we provide a bundle theoretic model for odd differential $K$-theory using the caloron correspondence and prove that this refinement is unique up to a unique natural isomorphism. We chara…

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…

Develops Patterson-Sullivan theory for coarse cocycles.

Sullivan discusses his contributions to math and physics.

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.

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.

We use Chern-Weil theory for Hermitian holomorphic vector bundles with canonical connections for explicit computation of the Chern forms of trivial bundles with special non-diagonal Hermitian metrics. We prove that every del-dellbar exact real form of the type (k,k) on an n-dimensional complex manifold X arises as a di…

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…

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.

We introduce certain relative differential characters which we call Cheeger-Chern-Simons characters. These combine the well-known Cheeger-Simons characters with Chern-Simons forms. In the same way as the Cheeger-Simons characters generalize Chern-Simons invariants of oriented closed manifolds, the Cheeger-Chern-Simons …

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.

This paper solves minimal surface equations near Hardt-Simon foliations.

Computing Chern-Simons action for perturbed Dirac triples

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…

New approach connects 3D Chern-Simons theory to spectral networks.

Quantifies how geodesic planes isolate in hyperbolic 3-manifolds.

Alternative proof of Michael-Simon-Sobolev inequality using optimal transport.

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

Geometrically constructs dilogarithm from Chern-Simons theory.

Researchers calculate alpha invariant for certain complex projective spaces.