arXiv research

Constructs Lie groupoid integrating elliptic tangent bundles and Poisson structures.

Paper defines invariants for elliptic Weyl groups and connects them to Frobenius structures.

No projective structure found on foliations of elliptic curves.

Study the relationship between canonical polynomials and elliptic sequences for elliptic singularities.

We construct a toric generalised Kähler structure on $\mathbb{C}P^2$ and show that the various structures such as the complex structure, metric etc are expressed in terms of certain elliptic functions. We also compute the generalised Kähler potential in terms of integrals of elliptic functions.

Proves cohomology of elliptic structures on Lie groups can be algebraic.

The elliptic associator of Enriquez can be used to define an invariant of tangles embedded in the thickened torus, which extends the Kontsevich integral. This construction by Humbert uses the formulation of categories with elliptic structures. In this work we show that an extension of the LMO functor also leads to an e…

This paper studies symplectic structures on elliptic surfaces with positive Euler number.

Complexity results for recognizing elliptic 3-manifolds.

New topological obstructions found for elliptic and quasiregularly elliptic manifolds.

In this paper, we provide a systematic and constructive description of Vaisman structures on certain principal elliptic bundles over complex flag manifolds. From this description we explicitly classify homogeneous l.c.K. structures on compact homogeneous Hermitian manifolds using elements of representation theory of co…

We give a representation theoretical proof of Branson's classification of minimal elliptic sums of generalized gradients. The original proof uses tools of harmonic analysis, which as powerful as they are, seem to be specific for the structure groups SO(n) and Spin(n). The different approach we propose is based on the r…

Proves solutions to elliptic equations on Hermitian manifolds with optimal conditions.

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 …

We consider an almost complex structure J on CP2, or more generally an elliptic structure E which is tamed by the standard symplectic structure. An E-curve is a surface tangent to E (this generalizes the notion of J(holomorphic)-curve), and an E-line is an E-curve of degree 1. We prove that the space of E-lines is agai…

In this short note we prove that in the case of elliptic curves, the isomorphism of generalized complex structure between $T$-dual manifolds described by Cavalcanti-Gualtieri coincides with the mirror map for elliptic curves described by Polishchuk and Zaslow.

Study on K3 surfaces' collapsing and special Kähler structures.

In this paper, we show that the quotient space of the domain by the reflection group for an elliptic root system has a structure of Frobenius manifold for the case of codimension 1. We also give a characterization of this Frobenius manifold structure under some suitable condition.

Solves nonlinear problems on metric structures through eigenvalue counting.

Several types of generically-nondegenerate Poisson structures can be effectively studied as symplectic structures on naturally associated Lie algebroids. Relevant examples of this phenomenon include log-, elliptic, $b^k$-, scattering and elliptic-log Poisson structures. In this paper we discuss topological obstructions…

We find a new relation among right-handed Dehn twists in the mapping class group of a $k$-holed torus for $4 \leq k \leq 9$. This relation induces an elliptic Lefschetz pencil structure on the four-manifold \cp $#(9-k)$ \cpb $$ with $k$ base points and twelve singular fibers. By blowing up the base points we get an el…

We show that an oriented elliptic 3-manifold admits a universally tight positive contact structure iff the corresponding group of deck transformations on $S^3$ preserves a standard contact structure pointwise. We also relate univerally tight contact structures on 3-manifolds covered by $S^3$ to the exceptional isomorph…

Researchers find a method to construct projective structures on a specific surface.

Elliptic surfaces have unique Lefschetz pencils and Calabi-Yau diffeomorphisms.

Study fully nonlinear elliptic equations on complex manifolds.

Elliptic Chern characters and Atiyah-Witten formula generalized to double loop spaces.

Characterizes elliptic operators on singular foliations.

The paper studies elliptic surfaces and proves unique fibered structures.

Characterizes stably elliptic elements in Lie groups and their properties.

We show that the $G_{2}$ holonomy equation on a manifold with boundary, with prescribed 3-form on the boundary, is elliptic. The main point is to set up a suitable linear elliptic boundary value problem. This result leads to a deformation theory. In particular we establish the existence of certain $G_{2}$ cobordisms be…

New framework uses elliptic operators to study projective maps.

Real del Pezzo surfaces split real lines into elliptic and hyperbolic types.

A generalized complex structure is called stable if its defining anticanonical section vanishes transversally, on a codimension-two submanifold. Alternatively, it is a zero elliptic residue symplectic structure in the elliptic tangent bundle associated to this submanifold. We develop Gompf-Thurston symplectic technique…

We construct a global geometric model for complex analytic equivariant elliptic cohomology for all compact Lie groups. Cocycles are specified by functions on the space of fields of the two-dimensional sigma model with background gauge fields and $\mathcal{N} = (0, 1)$ supersymmetry. We also consider a theory of free fe…

The main result of this paper states that a symplectic s-cobordism of elliptic 3-manifolds is diffeomorphic to a product (assuming a canonical contact structure on the boundary). Based on this theorem, we conjecture that a smooth s-cobordism of elliptic 3-manifolds is smoothly a product if its universal cover is smooth…

As an application of the construction of open books on plumbed 3-manifolds, we construct elliptic open books on torus bundles over the circle. In certain cases these open books are compatible with Stein fillable contact structures and have minimal genus.

Proves elliptic operator images are closed on Hilbert bundles.

Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.

Researchers prove a complex geometric conjecture about certain manifolds.

We propose a data-driven approach to solve multiscale elliptic PDEs with random coefficients based on the intrinsic low dimension structure of the underlying elliptic differential operators. Our method consists of offline and online stages. At the offline stage, a low dimension space and its basis are extracted from th…

Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.

New algorithm learns value and advantage functions for continuous-time Markov processes without structural assumptions.

Elliptic curve governs Hopf linking in symmetric tensegrity.

Extends T-duality to non-principal torus actions with elliptic tangent bundles.

The study characterizes a complex curve of residueless meromorphic differentials on elliptic curves.

We introduce a new elliptic operator on null hypersurfaces of four-dimensional Lorentzian manifolds. This operator depends on the first and second fundamental forms of the sections of a foliation of the null hypersurface and its novelty originates from its covariant transformation under change of foliation. It thus pro…

A new Riemannian framework for robust covariance estimation.

We solve the Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds under essentially optimal structure conditions, especially with no restrictions to the curvature of the underlying manifold and the second fundamental form of its boundary. The main result (Theorem 1.1) includes a new (and opt…