arXiv research

The study shows ergodicity of unitary frame flows on Kähler manifolds with specific curvature conditions.

The Hull-Strominger system for supersymmetric vacua of the heterotic string allows general unitary Hermitian connections with torsion and not just the Chern unitary connection. Solutions on unimodular Lie groups exploiting this flexibility were found by T. Fei and S.T. Yau. The Anomaly flow is a flow whose stationary p…

On a compact Kähler manifold there is a canonical action of a Lie-superalgebra on the space of differential forms. It is generated by the differentials, the Lefschetz operator and the adjoints of these operators. We determine the asymptotic distribution of irreducible representations of this Lie-superalgebra on the eig…

The paper explores noncommutative geometry of frame bundles using C*-algebras.

Framed flow categories were introduced by Cohen-Jones-Segal as a way of encoding the flow data associated to a Floer functional. A framed flow category gives rise to a CW-complex with one cell for each object of the category. The idea is that the Floer invariant should take the form of the stable homotopy type of the r…

New progress on frame flow ergodicity for nearly pinched manifolds.

New flow generates surfaces with constant curvature.

Researchers compute de Rham cohomology of geodesic flow foliations on hyperbolic surfaces.

We classify all harmonic maps of finite uniton number from a Riemann surface into SU(n) in terms of certain pieces of the Bruhat decomposition of the subgroup of algebraic loops in SU(n). We give a description of the "Frenet frame data" for such harmonic maps in a given class.

Study connects spectral properties to frame flows on curved manifolds.

We pursue the analogy of a framed flow category with the flow data of a Morse function. In classical Morse theory, Morse functions can sometimes be locally altered and simplified by the Morse moves. These moves include the Whitney trick which removes two oppositely framed flowlines between critical points of adjacent i…

The paper studies the ergodicity of frame flow on even-dimensional manifolds.

Let $Y$ be a compact, oriented 3-manifold with a contact form $a$ and a metric $ds^2$. Suppose that $F\to Y$ is a principal bundle with structure group $U(2) = SU(2)\times_{\pm1}S^1$ such that $F/S^1$ is the principal SO(3) bundle of orthonormal frames for $TY$. A unitary connection $A_0$ on the Hermitian line bundle $…

In this paper, we characterize Riemannian 4-manifold in terms of its almost Hermitian twistor spaces $(Z,g_t,\mathbb{J}_{\pm})$. Some special metric conditions (including Balanced metric condition, first Gauduchon metric condition) on $(Z,g_t,\mathbb{J}_{\pm})$ are studied. For the first Chern form of a natural unitary…

A formula is given in terms of secondary characteristic classes for the leading order contribution to the spectral flow for a path of twisted Dirac operators on an odd dimensional, Riemannian manifold when the twisting is done by a path of unitary connections with large curvature.

This paper proves exponential mixing for frame flows on hyperbolic manifolds with cusps.

Researchers study injectivity of magnetic and thermostatic nonabelian ray transforms on compact surfaces.

New geometric quantisation scheme for hyper-Kähler manifolds.

Generalizes Hasimoto transformation to arbitrary flows on space curves.

In this second part of the `essay on the completion of quantum theory' we define the {\em unitary setting of completed quantum mechanics}, by adding as intrinsic data to those from Part I (arXiv:1711.08643) the choice of a north pole N and south pole S in the geometric space. Then we explain that, in the unitary settin…

We describe a calculus of moves for modifying a framed flow category without changing the associated stable homotopy type. We use this calculus to show that if two framed flow categories give rise to the same stable homotopy type of homological width at most three, then the flow categories are move equivalent. The proc…

Mackey showed that for a compact Lie group $K$, the pair $(K,C^{0}(K))$ has a unique non-trivial irreducible covariant pair of representations. We study the relevance of this result to the unitary equivalence of quantizations for an infinite-dimensional family of $K\times K$ invariant polarizations on $T^{\ast}K$. The …

Extends Kanai's result to higher dimensions for negatively curved manifolds.

Proves flows of two-convex Lagrangians are regular, global, and converge.

Frame flows on certain symmetric spaces mix exponentially.

Geodesic flow on orbifolds has a genus 1 Birkhoff section.

3D dual field theories for Virasoro minimal models constructed using Seifert fiber spaces.

The paper proves exponential mixing for hyperbolic manifolds, with applications to geodesic holonomy.

Harmonic maps link Teichmüller spaces to framed representations.

Study of Anosov flows using microlocal analysis for ergodicity and mixing properties.

We consider geodesic flows between hypersurfaces in $\R^n$. However, rather than consider using geodesics in $\R^n$, which are straight lines, we consider an induced flow using geodesics between the tangent spaces of the hypersurfaces viewed as affine hyperplanes. For naturality, we want the geodesic flow to be invaria…

Universal bi-Hamiltonian hierarchies of group-invariant (multicomponent) soliton equations are derived from non-stretching geometric curve flows $\map(t,x)$ in Riemannian symmetric spaces $M=G/H$, including compact semisimple Lie groups $M=K$ for $G=K\times K$, $H={\rm diag} G$. The derivation of these soliton hierarch…

Study inextensible flows of curves in 4D pseudo-Galilean space and defines energy functions.

New formula connects surface singularity zeta function to Reidemeister-Turaev torsion.

Extends magnetic flow theory results to higher dimensions.

We study the behavior of the Yang-Mills flow for unitary connections on compact and non-compact oriented surfaces with varying metrics. The flow can be used to define a one dimensional foliation on the space of SU(2) representations of a once punctured surface. This foliation universalizes over Teichmüller space and is…

Constructs a Morse-Bott function on symplectic Grassmannians.

Classifies horocycle flow closures in hyperbolic 3-manifolds.

Making predictions of future frames is a critical challenge in autonomous driving research. Most of the existing methods for video prediction attempt to generate future frames in simple and fixed scenes. In this paper, we propose a novel and effective optical flow conditioned method for the task of video prediction wit…

Study geometric mKdV flows for Legendrian curves in a 3-sphere.

Study of spectral invariants on CR contact manifolds with circle action.

Researchers prove a method to upgrade Morse-Bott homology to stable homotopy invariants.

Researchers find a Steenrod square for link Floer homology.

We formulate a statistical analogy of regular Lagrange mechanics and Finsler geometry derived from Grisha Perelman's functionals generalized for nonholonomic Ricci flows. There are elaborated explicit constructions when nonholonomically constrained flows of Riemann metrics result in Finsler like configurations, and inv…

Let M be a non-elementary convex cocompact hyperbolic 3 manifold and delta the critical exponent of its fundamental group. We prove that a one-dimensional unipotent flow for the frame bundle of M is ergodic for the Burger-Roblin measure provided that delta>1.

The execution flow drives market dynamics, validated on real data.

We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a unitary modular category. Many new unitary modular categories are obtained. We also sh…

Contact group retracts to unitary subgroup.