arXiv research

Abstract not provided enough details, focusing on vector bundles and orbits.

Proves existence of elliptic Reeb orbit on real projective 3-space using ECH.

Paper finds conditions for finite-dimensional vector spaces of cross-sections.

The paper calculates the full asymptotics of analytic torsions for compact orbifolds.

The study counts closed elliptic curves and Reeb orbits on Vaisman and Sasakian manifolds.

Paper proves ellipticity of certain Reeb orbits and estimates ECH spectrum on lens spaces.

We list up all the possible local orbit types of hyperbolic or elliptic orbits for the isotropy representations of semisimple pseudo-Riemannian symmetric spaces. It is key to give a recipe to determine the local orbit types of hyperbolic principal orbits by using three kind of restricted root systems and Satake diagram…

The application of equivalence method to classify Monge-Ampère system leads to three orbits, parabolic case, hyperbolic case and elliptic case wich correspond to three types of Monge-Ampère systems. In this paper we will study the elliptic case and give a presentation of the group as a complex group.

Rational ellipticity proven for $G$-manifolds with specific quotient properties.

Study limits of adjoint orbits for Lie groups, describing nilpotent orbits.

We solve integrable systems to describe the motion of Kaleidocycles.

Paper calculates indices for group actions using cocycles.

A long-standing conjecture in Hamiltonian Dynamics states that the Reeb flow of any convex hypersurface in $\mathbb{R}^{2n}$ carries an elliptic closed orbit. Two important contributions toward its proof were given by Ekeland in 1986 and Dell'Antonio-D'Onofrio-Ekeland in 1995 proving this for convex hypersurfaces satis…

This thesis studies the symplectic structure of holomorphic coadjoint orbits, and their projections. A holomorphic coadjoint orbit O is an elliptic coadjoint orbit which is endowed with a natural invariant Kählerian structure. These coadjoint orbits are defined for a real semi-simple connected non-compact Lie group G w…

On a Riemannian compact manifold, we give existence and multiplicity results for solutions of elliptic PDE by introducing isometry invariances. When the groups we used have finite orbits, we get multiplicity results for equations with the classical critical Sobolev exponent, for instance the Yamabe equation. When there…

Study shows conditions for rational ellipticity of manifolds with symmetries.

We give a sharp lower bound for the number of geometrically distinct contractible periodic orbits of dynamically convex Reeb flows on prequantizations of symplectic manifolds that are not aspherical. Several consequences of this result are obtained, like a new proof that every bumpy Finsler metric on $S^n$ carries at l…

A functional ansatz is developed which gives certain elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equation. This is based on the elliptic trilogarithm function introduced by Beilinson and Levin. For this to be a solution results in a number of purely algebraic conditions on the set of vectors …

Study of separatrix configurations in holomorphic flows with real time.

Let M be a compact, connected and simply-connected Riemannian manifold, and suppose that G is a compact, connected Lie group acting on M by isometries. The dimension of the space of orbits is called the cohomogeneity of the action. If the direct sum of the higher homotopy groups of M, tensored with the field of rationa…

The MICZ-Kepler orbits are the non-colliding orbits of the MICZ Kepler problems (the magnetized versions of the Kepler problem). The oriented MICZ-Kepler orbits can be parametrized by the canonical angular momentum $\mathbf L$ and the Lenz vector $\mathbf A$, with the parameter space consisting of the pairs of 3D vecto…

M. F. Atiyah proved that the index of a transversally elliptic operator relative to a free action can be computed by using indices of elliptic operators on the orbit manifold. In this paper, we derive an explicit formula for the transversal indices on S^1-bundles over complex projective spaces. Using this explicit form…

We use the equivalence between embedded contact homology and Seiberg-Witten Floer homology to obtain the following improvements on the Weinstein conjecture. Let Y be a closed oriented connected 3-manifold with a stable Hamiltonian structure, and let R denote the associated Reeb vector field on Y. We prove that if Y is …

We construct geometric generators of the effective $S^1$-equivariant Spin- (and oriented) bordism groups with two inverted. We apply this construction to the question of which $S^1$-manifolds admit invariant metrics of positive scalar curvature. It turns out that, up to taking connected sums with several copies of the …

Study global geometry of dynamical systems with entire vector fields.

The paper conjectures and proves fixed points for certain group actions on nonpositively curved spaces.

About 15 years ago, Bismut gave a natural construction of a Hodge theory for a hypoelliptic Laplacian acting on the total space of the cotangent bundle of a Riemannian manifold. This operator interpolates between the classical elliptic Laplacian on the base and the generator of the geodesic flow. We will describe recen…

We study nearly-Kahler 6-manifolds equipped with a cohomogeneity-two Lie group action for which the principal orbits are coisotropic. If the metric is complete, then we show that this last condition is automatically satisfied, and both the acting Lie group and the principal orbits are finite quotients of $S^3 \times S^…

Let $x\in\mathbb{R}^{n}$. For $φ:\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ and $t\in\mathbb{R}$, we put $φ^{t}=t^{-1}φ(xt)$. A projective flow is a solution to the projective translation equation $φ^{t+s}=φ^{t}\circφ^{s}$, $t,s\in\mathbb{R}$. Previously we have developed an arithmetic, topologic and analytic theory of $2$-d…

Egorov's theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first order transversally elliptic operator with the (classical) evolution of it…

We give a description of asymptotic quadratic growth rates for geodesic segments on covers of Veech surfaces in terms of the modular fiber parameterizing coverings of a fixed Veech surface. To make the paper self contained we derive the necessary asymptotic formulas from the Gutkin-Judge formula. As an application of t…

Study perturbed Dirac operators on non-compact manifolds using KK-theory.

We study modular fibers of elliptic differentials, which are roughly spaces of torus-coverings over a fixed base torus. For genus 2 torus covers with fixed degree we show, that the modular fibers F_d(1,1) are itself connected torus covers with Veech group SL_2(Z). Using results of Eskin, Masur and Schmoll we calculate …

Let $S$ be an oriented surface of finite type, $\mathcal{MCG}(S)$ its mapping class group, and $\mathcal{T}(S)$ its Teichmüller space with the Teichmüller metric. Let $H \leq \mathcal{MCG}(S)$ be a finite subgroup and consider the subset of $\mathcal{T}(S)$ fixed by $H$, $\mathrm{Fix}(H) \subset \mathcal{T}(S)$. For an…

The paper explores the geometry of holomorphic flows and orbits.

We investigate solutions of the elliptic sinh-Gordon equation of spectral genus g<3. These solutions are parametrized by complex matrix-valued polynomials called potentials. On the space of these potentials there act two commuting flows. The orbits of these flows are called Polynomial Killing fields and are double peri…

Symplectic classification for a specific type of singularity in integrable systems.

In this paper we study non-negatively curved and rationally elliptic GKM$_4$ manifolds and orbifolds. We show that their rational cohomology rings are isomorphic to the rational cohomology of certain model orbifolds. These models are quotients of isometric actions of finite groups on non-negatively curved torus orbifol…

Origami graphs' Euler characteristics grow as origami complexity increases.

In the first part of this dissertation, we give a new definition of a Laplace operator for Finsler metric as an average, with regard to an angle measure, of the second directional derivatives. This operator is elliptic, symmetric with respect to the Holmes-Thompson volume, and coincides with the usual Laplace--Beltrami…

The fact that the modular template coincides with the Lorenz template, discovered by Ghys, implies modular knots have very peculiar properties. We obtain a generalization of these results to other Hecke triangle groups. In this context, the geodesic flow can never be seen as a flow on a subset of $S^3$, and one is led …

Study optimal partition problem for Q-curvature equations on Einstein manifolds.

Study on knots and dynamics on three-sphere, linking bounds, and upper action bounds.

In this paper we are interested in the stratum H^{hyp}(4) of translation surfaces, which consists of pairs (M,ω), where M is a hyper-elliptic Riemann surface of genus 3, and ωis a holopmorphic 1-form on M having only one zero. We first show that every surface in this stratum can be decomposed into parallelograms follow…

A principal toric bundle $M$ is a complex manifold equipped with a free holomorphic action of a compact complex torus $T$. Such a manifold is fibered over $M/T$, with fiber $T$. We discuss the notion of positivity in fiber bundles and define positive toric bundles. Given an irreducible complex subvariety $X\subset M$ o…

Researchers compute the index of a specific operator on contact manifolds.

We consider a consider the case of a compact manifold M, together with the following data: the action of a compact Lie group H and a smooth H-invariant distribution E, such that the H-orbits are transverse to E. These data determine a natural equivariant differential form with generalized coefficients J(E,X) whose prop…

Proves Birkhoff-Poritsky conjecture for centrally-symmetric billiards.