Smooth approximations for continuous functions on orbit spaces.
problem Approximating continuous functions on orbit spaces.
method Study of subcartesian spaces and proper Lie group actions.
result Continuous functions can be approximated by smooth functions.
Study Kleinian groups using orbital functions and heat kernels.
problem Understanding Kleinian groups through orbital functions and heat kernels.
method Using the heat kernel approach developed in \cite{artmoiheatcounting1}.
result Developed a method to study Kleinian groups using orbital functions and heat kernels.
New method uses Brownian motion to estimate Kleinian group orbital functions.
problem Estimating orbital functions of Kleinian groups.
method Developed a new method using Brownian motion.
result Gave estimates of orbital functions for nilpotent covers.
Counting orbits for Anosov groups with specific functionals.
problem Counting orbits for relatively Anosov groups with linear functionals.
method Equidistribution results and previous counting results for periods.
result Generalization of earlier work on Anosov groups.
The study counts periodic orbits on smooth manifolds, adding ghost orbits for completeness.
problem Counting periodic orbits of vector fields on smooth closed manifolds.
method Enlarging the space of orbits to include ghost orbits, defining weight functions, and showing constancy under deformation.
result The weight function remains constant as the vector field moves and Γ deforms. The article studies groups generated by products and wreath products, focusing on first Betti numbers of Morse function orbits.
problem Calculating first Betti numbers of orbit groups generated by products and wreath products.
method Analyzes algebraic properties of specific groups G, proving ranks of center and quotient by commutator subgroup coincide. result The rank of the quotient by commutator subgroup is a first Betti number of the orbit of Morse function.
Study shows mean action of periodic orbits in annuli is bounded by their Calabi invariant.
problem Understanding the average distortion of periodic orbits in area-preserving annuli.
method Analyzes action functions and Calabi invariants of diffeomorphisms near annulus boundaries.
result Infimum of mean action of periodic orbits is bounded by their Calabi invariant.
Study on Hermitian Calabi functional in complexified orbits of symplectic manifolds.
problem Analyzing the Hermitian Calabi functional on complexified orbits of symplectic manifolds.
method Explicit formula for Hessian of Hermitian Calabi functional, semi-positive definiteness proof, and weak parabolicity of Hermitian Calabi flow.
result Hessian of Hermitian Calabi functional is semi-positive definite on complexified orbits.
Locally maximizing orbits studied in twist maps and billiards.
problem Characterize orbits in locally maximizing class for twist maps.
method Geometric and variational analysis of orbits in the cotangent bundle of a torus or ball bundle over a sphere.
result Two generating functions for the Birkhoff billiard map have the same class of locally maximizing orbits.
We introduce a class of regularisable infinite dimensional principal fibre bundles which includes fibre bundles arising in gauge field theories like Yang-Mills and string theory and which generalise finite dimensional Riemannian principal fibre bundles induced by an isometric action. We show that the orbits of regulari…
Researchers describe Casimir functions for 3- and 4-step nilpotent Lie groups.
problem Understanding Casimir functions for free nilpotent Lie groups of steps 3 and 4.
method Construction of Casimir functions for free nilpotent Lie groups of steps 3 and 4.
result For 3-step groups, coadjoint orbits are fully described as affine subspaces or direct products of quadrics.
Let Ĝ be a complex semisimple Lie group, Q a parabolic subgroup and G a real form of Ĝ. The flag manifold Ĝ/Q decomposes into finitely many G-orbits; among them there is exactly one orbit of minimal dimension, which is compact. We study these minimal orbits from the point of view of CR geometry. In particular we charac…
The main theme of this paper is a relative version of the almost existence theorem for periodic orbits of autonomous Hamiltonian systems. We show that almost all low levels of a function on a geometrically bounded symplectically aspherical manifold carry contractible periodic orbits of the Hamiltonian flow, provided th…
The study examines deformations of functions on surfaces.
problem Understanding the homotopy types of stabilizers and orbits of smooth functions on surfaces.
method Review of recent progress with new direct proof for generic Morse maps.
result For generic Morse maps, connected components of orbits are homotopy equivalent to finite products of circles.
Study geodesic orbits on noncompact curved spaces, proving their distribution and counting.
problem Counting and equidistribution of periodic orbits on noncompact manifolds.
method Proved equidistribution in narrow topology, deduced exact asymptotic counting.
result Exact asymptotic counting of periodic orbits on noncompact manifolds.
The paper describes orbits of circle-valued functions on a 2-torus.
problem Understanding the fundamental groups of orbits of circle-valued functions.
method Algebraic description of fundamental groups of orbits of circle-valued smooth functions.
result An algebraic description of fundamental groups of orbits of circle-valued smooth functions.
The paper characterizes potential functions whose level sets are orbits in mechanical systems.
problem Characterizing smooth potential energy functions on the plane with specific level set properties.
method Analyzing inverse curvature flow and properties of level sets.
result Analytic or functions with totally path-disconnected critical sets must be radial, while every compact convex set is a critical set of a Levi potential.
For a stratified symplectic space, a suitable concept of stratified Kaehler polarization, defined in terms of an appropriate Lie-Rinehart algebra, encapsulates Kaehler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kaehler space. This notion es…
This thesis explores Hamiltonian systems and Kähler structures on complex coadjoint orbits.
problem Investigating Hamiltonian systems and Kähler structures on complex coadjoint orbits.
method Analyzes (pseudo)-holomorphic Hamiltonian systems, Lefschetz and almost toric fibrations, and introduces pseudo-holomorphic Hamiltonian systems.
result Complex coadjoint orbits exhibit both Hyperkähler and holomorphic Kähler structures, suggesting Kähler duality.
It is known that nilpotent orbits in a complex simple Lie algebra admit hyperKähler metrics with a single function that is a global potential for each of the Kähler structures (a hyperKähler potential). In an earlier paper the authors showed that nilpotent orbits in classical Lie algebras can be constructed as finite-d…
The paper studies time-optimal problems on specific Lie groups, describing orbits and integrals.
problem Time-optimal control problems on two-step Carnot groups.
method Description of co-adjoint orbits, Casimir functions, and integrals for the Hamiltonian system.
result Characterization of the flow and constancy of solutions for two-dimensional co-adjoint orbits.
Geometrically, the first Betti number of orbits is linked to the Kronrod-Reeb graph.
problem Understanding the first Betti number of orbits of smooth functions.
method Established a correspondence between the first Betti number of f-orbits and the number of orbits of S′(f,V) on the Kronrod-Reeb graph. result The first Betti number of f-orbits is equal to the number of orbits of S′(f,V) on the Kronrod-Reeb graph. Via a non degenerate symmetric bilinear form we identify the coadjoint representation with a new representation and so we induce on the orbits a simplectic form. By considering Hamiltonian systems on the orbits we study some features of them and finally find commuting functions under the corresponding Lie-Poisson brack…
This paper classifies geodesic orbit metrics on compact Lie group G2.
problem Classifying geodesic orbit metrics on compact Lie groups.
method Using representation theory of Lie subgroups, specifically weakly regular subgroups.
result Left-invariant geodesic orbit metrics on compact Lie group G2 are classified. Study of MHD equilibria with orientation-reversing symmetry, showing all orbits are periodic.
problem Understanding MHD equilibria with non-reflection symmetry.
method Topological techniques to analyze invariant 2-tori and their orbits.
result All orbits on tori are periodic under certain conditions.
We consider the flows generated by generic gradients of Morse maps of a closed connected manifold M to a circle. To each such flow we associate an invariant counting the closed orbits of the flow. Each closed orbit is counted with the weight derived from its index and homotopy class. The resulting invariant is called…
We consider a compact manifold of dimension greater than 2 and a differential form of degree one which is closed but non-exact. This form, viewed as a multi-valued function has a gradient vector field with respect to any Riemannian metric. After S. Novikov's work and a complement by J.-C. Sikorav, under some genericity…
In this paper we consider the Poisson algebraic structure associated with a classical r-matrix, i.e. with a solution of the modified classical Yang--Baxter equation. In Section 1 we recall the concept and basic facts of the r-matrix type Poisson orbits. Then we describe the r-matrix Poisson pencil (i.e the pair o…
Study of Anosov representations with Lipschitz limit set and applications to rigidity.
problem Characterizing Anosov representations with specific limit set properties.
method Introducing an unstable Jacobian and analyzing its orbit growth rate.
result Many higher rank representations belong to the studied class.
The paper classifies symplectic invariants of specific singularities in integrable Hamiltonian systems.
problem Classifying symplectic invariants of singularities in integrable Hamiltonian systems.
method Smooth C∞ symplectic classification of Lagrangian fibrations near singularities. result Action variables form complete C∞ symplectic invariants for parabolic orbits and cuspidal tori. New model calculates Wilson surfaces in higher gauge theory.
problem Calculating Wilson surfaces in higher gauge theory.
method Derived geometric framework, topological coadjoint orbit model, functional integral framework.
result Strong evidence that model underlies Wilson surfaces partition function.
This paper develops a geometric framework for Wilson surfaces in higher gauge theory.
problem Quantum field theory of Wilson surfaces in higher gauge theory.
method Higher coadjoint orbit theory and derived geometric framework.
result Identification of derived coadjoint orbits and their quantization.
In this paper we prove the following result: if two 2-dimensional 2-homogeneous rational vector fields commute, then either both vector fields can be explicitly integrated to produce rational flows with orbits being lines through the origin, or both flows can be explicitly integrated in terms of algebraic functions. In…
Study limits of adjoint orbits for Lie groups, describing nilpotent orbits.
problem Understanding limits of adjoint orbits for Lie groups.
method Systematic and topological study of limits of continuous families of adjoint orbits for non-compact simple Lie groups.
result Explicit description of nilpotent orbits in terms of Richardson orbits for hyperbolic semisimple elements.
We will show that the period T of a closed orbit of the planar circular restricted three-body problem (viewed on rotating coordinates) depends on the region it encloses. Roughly speaking, we show that, 2T=kπ+∫Ωg where k is an integer, Ω is the region enclosed by the periodic orbit and $g:\mathbb{R}^2\to \m…
Deep neural network predicts molecular wave functions in minimal basis.
problem Improving accuracy and efficiency in quantum chemistry calculations.
method Adapted SchNet for Orbitals (SchNOrb) model in quasi-atomic minimal basis.
result Model accurately predicts molecular orbital energies and wavefunctions for large molecules.
In this paper, we address the problem of determining a function in terms of its orbital integrals on Lorentzian symmetric spaces. It has been solved by S. Helgason for even-dimensional isotropic Lorentzian symmetric spaces via a limit formula involving the Laplace-Beltrami operator. The result has been extended by J. O…
Let (Ft) be a smooth flow on a smooth manifold M and h:M→M be a smooth orbit preserving map. The following problem is studied: suppose that for every point z of M there exists a germ of a smooth function fz at z such that near z we have that h(x)=Ffz(x)(x). Can the functions (fz) be glued …
We solve integrable systems to describe the motion of Kaleidocycles.
problem Existence and motion of Kaleidocycles.
method Elliptic theta functions and integrable systems.
result Existence and motion of Kaleidocycles for any number of tetrahedra greater than five.
We consider the closed orbit structure of generic gradient flows of Morse closed 1-forms. The torsion of a chain homotopy equivalence between the Novikov complex and the completed simplicial chain complex of the universal cover detects the eta function of the flow. We extend this result to arbitrary Morse closed 1-form…
Study on counting special-orthogonal Anosov orbits in geometric spaces.
problem Counting points in specific orbits of projective Anosov subgroups.
method Analysis of totally geodesic copies in Riemannian and pseudo-Riemannian spaces.
result Number of points in the orbit is finite and asymptotically exponential.
The main result of this paper is, that for convex billiards in higher dimensions, in contrast with 2D case, for every point on the boundary and for every n there always exist billiard trajectories developing conjugate points at the n-th collision with the boundary. We shall explain that this is a consequence of the…
The study proves curvature bounds for quotient spaces of isometric actions.
problem Proving curvature bounds for quotient spaces of isometric actions.
method Disintegrate absolutely continuous measures and define a functional to prove curvature bounds.
result Necessary and sufficient conditions for Ricci curvature to be bounded below.
The paper counts primes in complex orthospectra and polynomial orbits.
problem Counting primes in complex orthospectra and polynomial orbits.
method Analytic number theory and Thermodynamic Formalism.
result Equidistribution of holonomy in dynamical systems.
In this note we study numerically the combinatorics of curves and geodesics on the torus with one boundary component. A potential computational difficulty is avoided by counting inside specific orbits of the mapping class group up to a certain length, either geometric or combinatorial. Some cases are rigurolosly determ…
Darboux inverses explain Kepler orbits on curved surfaces.
problem Understanding orbits on curved surfaces.
method Analyzing the Darboux inverses of the Kepler problem.
result Kepler orbits are periodic on open sets of phase space.
Polynomial density theorem for specific subgroup orbits in quotient spaces.
problem Effective density of orbits in arithmetic quotients of SL2(C) and SL2(R)imesSL2(R). method Use of Margulis function, incidence geometry tools, and spectral gap of ambient space.
result Proved effective density theorems with polynomial error rate.
Let f:T2→R be a Morse function on 2-torus T2 such that its Kronrod-Reeb graph Γ(f) has exactly one cycle, i.e. it is homotopy equivalent to S1. Under some additional conditions we describe a homotopy type of the orbit of f with respect to the action of the group of diffeomorphism of T2. Thi…