arXiv research

Decategorifies higher actions in Heegaard Floer homology.

New method constructs proper affine actions of groups in higher dimensions.

In this thesis, we employ simplicial methods to study actions, principal bundles, and bibundles of higher groupoids. Roughly, we use Kan fibrations to model actions of higher groupoids, we use pairs of a Kan fibration and a special acyclic fibration to model principal bundles of higher groupoids, we use inner Kan fibra…

Extends Einstein-Hilbert action to higher-order spectral triples.

We show that sufficiently irreducible Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.

We show that polar actions of cohomogeneity two on simple compact Lie groups of higher rank, endowed with a biinvariant Riemannian metric, are hyperpolar. Combining this with a recent result of the second-named author, we are able to prove that polar actions induced by reductive algebraic subgroups in the isometry grou…

Study positive entropy actions by higher-rank lattices, proving rigidity and conjugacy results.

We classify all holomorphic actions of higher rank lattices on compact Kaehler manifolds of dimension 3. This provides a complete answer to Zimmer's program for holomorphic actions on compact Kaehler manifolds of dimension at most 3.

Affine maps reveal higher rank structures in certain spaces.

We show that most homogeneous Anosov actions of higher rank Abelian groups are locally smoothly rigid (up to an automorphism). This result is the main part in the proof of local smooth rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the Anosov…

Extends confining subset theory to describe hyperbolic actions of solvable groups with higher rank abelianizations.

Confirming a conjecture, new CAT(0) spaces of higher rank are rigid.

Innovates rotation index for matrix pairs, solving group action problems.

The study proves conditions for CAT(0) spaces with higher rank rigidity.

The (abelian bosonic) heterotic string effective action, equations of motion and Bianchi identity at order alpha prime in ten dimensions, are shown to be equivalent to a higher dimensional action, its derived equations of motion and Bianchi identity. The two actions are the same up to the gauge fields: the latter are a…

We prove that any action of a higher rank lattice on a Gromov-hyperbolic space is elementary. More precisely, it is either elliptic or parabolic. This is a large generalization of the fact that any action of a higher rank lattice on a tree has a fixed point. A consequence is that any quasi-action of a higher rank latti…

Defines and characterizes operators on Lie ∞-algebras with respect to actions.

Local-to-global principle for Morse actions on symmetric spaces.

The paper studies higher geometric structures and connections on manifolds, constructing moduli stacks and proving equivalence criteria.

We show that sufficiently irreducible totally non-symplectic Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.

New structural result classifies actions on symmetric spaces.

Extends Arnold's linking theory to higher dimensions and submanifolds.

Study equivariant isotopy in higher dimensions, finding exceptions.

We consider representations of the Cuntz algebras ${\mathcal O}_N$ as constructed by Bratteli-Jorgensen and use these to define a faithful action of the analytic loop group $L_{\text{an}}U(n)$ on ${\mathcal O}_N$ for $N \geq n$. This extends to a faithful action on the infinite Cuntz algebra ${\mathcal O}_{\infty}$, an…

This thesis extends Hamiltonian actions to multisymplectic geometry, classifying actions on spheres and constructing homotopy comomentum maps.

Paper generalizes spectral flow formulas for compact Lie group actions.

We show that J. Lott's equivariant higher analytic torsion for compact group actions depends only on the equivariant Euler characteristic.

We study higher rank Cartan actions on compact manifolds preserving an ergodic measure with full support. In particular, we classify actions by $\R ^k$ with $k \geq 3$ whose one-parameter groups act transitively as well as nondegenerate totally nonsymplectic $\Zk$-actions for $k \geq 3$.

We enhance the action of higher abelian gauge theory associated to a gerbe on an M5-brane with an action of a torus ${\mathbb T}^n (n\ge 2)$, by a noncommutative ${\mathbb T}^n$-deformation of the M5-brane. The ingredients of the noncommutative action and equations of motion include the deformed Hodge duality, deformed…

Survey recent constructions of cyclic cocycles for Lie groups.

We investigate conformal actions of cocompact lattices in higher-rank simple Lie groups on compact pseudo-Riemannian manifolds. Our main result gives a general bound on the real-rank of the lattice, which was already known for the action of the full Lie group by a result of Zimmer. When the real-rank is maximal, we pro…

Global rigidity theorem for certain lattice actions on manifolds.

This paper proves that there are no compact forms for a large class of homogeneous spaces admitting actions by higher-rank semisimple Lie groups. It builds on Zimmer's approach for studying such spaces using cocycle superrigidity. The proof involves cocycle superrigidity, measure rigidity for unipotent flows, technique…

We formulate a 4-dimensional higher gauge theoretic Chern-Simons theory. Its symmetry is encoded in a semistrict Lie 2-algebra equipped with an invariant non singular bilinear form. We analyze the gauge invariance of the theory and show that action is invariant under a higher gauge transformation up to a higher winding…

Paper calculates indices for group actions using cocycles.

We study the geometry and dynamics of discrete infinite covolume subgroups of higher rank semisimple Lie groups. We introduce and prove the equivalence of several conditions, capturing "rank one behavior'' of discrete subgroups of higher rank Lie groups. They are direct generalizations of rank one equivalents to convex…

Chern-Weil theory provides for each invariant polynomial on a Lie algebra g a map from g-connections to differential cocycles whose volume holonomy is the corresponding Chern-Simons theory action functional. Kotov and Strobl have observed that this naturally generalizes from Lie algebras to dg-manifolds and dg-bundles …

We prove the vanishing of higher A-hat-genera, in the sense of Browder and Hsiang, on smooth manifolds with effective circle actions and with finite second and fourth homotopy groups

We prove that closed symplectic four-manifolds do not admit any smooth free circle actions with contractible orbits, without assuming that the actions preserve the symplectic forms. In higher dimensions such actions by symplectomorphisms do exist, and we give explicit examples based on a construction of Fernandez, Gray…

We consider a simple instance of action up to homotopy. More precisely, we consider strict actions of DGLAs in degrees -1 and 0 on degree 1 NQ-manifolds. In a more conventional language this means: strict actions of Lie algebra crossed modules on Lie algebroids. When the action is strict, we show that it integrates to …

Canonical maps connect complex structures to Hitchin components.

The paper introduces a new method to create stable ergodic actions on higher-dimensional manifolds.

In this paper we study perturbations of constant cocycles for actions of higher rank semi-simple algebraic groups and their lattices. Roughly speaking, for ergodic actions, Zimmer's cocycle superrigidity theorems implies that the perturbed cocycle is measurably conjugate to a constant cocycle modulo a compact valued co…

We formulate and solve the analog of the universal Conformal Ward Identity for the stress-energy tensor on a compact Riemann surface of genus $g>1$, and present a rigorous invariant formulation of the chiral sector in the induced two-dimensional gravity on higher genus Riemann surfaces. Our construction of the action f…

A general Lefschetz formula for the geodesic action on locally symmetric spaces is proven.

Let $Γ$ be an irreducible lattice of $\Q$-rank $\geq 2$ in a semisimple Lie group of noncompact type. We prove that any action of $Γ$ on a $\CAT(0)$ cubical complex has a global fixed point.

We define and study the signature, A-hat genus and higher signatures of the quotient space of an $S^1$-action on a closed oriented manifold. We give applications to questions of positive scalar curvature and to an Equivariant Novikov Conjecture.

The paper proves actions of lattices in higher rank groups have cost one.