Constructs Lie groupoid integrating elliptic tangent bundles and Poisson structures.
problem Existence and integration of elliptic tangent bundles and Poisson structures.
method Explicit construction of Lie groupoids and local models for symplectic integration.
result Necessary and sufficient topological condition for integration of elliptic tangent bundles.
Researchers create a Kähler structure on complex projective plane using elliptic functions.
problem Constructing a toric generalised Kähler structure on CP2. method Expressed various structures in terms of elliptic functions and computed the generalised Kähler potential.
result Various structures on CP2 are described using elliptic functions. Paper defines invariants for elliptic Weyl groups and connects them to Frobenius structures.
problem Defining invariants for elliptic Weyl groups.
method Defines a set of good basic invariants and shows their connection to Frobenius structures.
result Good basic invariants give flat invariants and structure constants of Frobenius structures.
No projective structure found on foliations of elliptic curves.
problem Existence of transversely projective structures on foliations.
method Analyzing foliations on the product of two elliptic curves.
result Generic nonsingular turbulent foliations do not have transversely projective structures.
Study the relationship between canonical polynomials and elliptic sequences for elliptic singularities.
problem Understanding the relationship between canonical polynomials and elliptic sequences for elliptic singularities.
method An inductive setup of elliptic germs and comparison of their canonical polynomials.
result The exponents of the canonical polynomial determine the elliptic sequence and vice versa under certain conditions.
Proves cohomology of elliptic structures on Lie groups can be algebraic.
problem Computing cohomology of elliptic structures on compact semisimple Lie groups.
method Used spectral sequences to construct an isomorphism between left-invariant and usual differential complexes.
result Reduced analytical problem to algebraic computation.
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.
problem Determining symplectic representatives for cohomology classes on elliptic surfaces.
method Analyzes the symplectic cone for elliptic surfaces with positive Euler number.
result Characterizes the symplectic cone for elliptic surfaces with positive Euler number.
Complexity results for recognizing elliptic 3-manifolds.
problem Deciding if a 3-manifold is elliptic and its homeomorphism type.
method Using triangulations and barycentric subdivisions to find Heegaard solid tori.
result Conditions for a lens space to be elliptic and a method to find Heegaard solid tori.
New topological obstructions found for elliptic and quasiregularly elliptic manifolds.
problem Connection between ellipticity and quasiregular ellipticity.
method Proved new topological obstructions.
result Closed manifolds of both types must have virtually abelian fundamental group.
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.
problem Solving elliptic equations on Hermitian manifolds with boundary conditions.
method Derives quantitative boundary estimates and proves existence of solutions.
result Proves existence of solutions under almost optimal structural 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.
problem Understanding the structure of K3 surfaces' collapsing metrics.
method Analyzing M2 and establishing connections to SKSs and Jacobian elliptic K3 surfaces. result Established a bijection between integral singular SKSs on P1 and Jacobian elliptic K3 surfaces. 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.
problem Nonlinear equations on metric structures
method Counting large eigenvalues of linearized operators
result Solves fully nonlinear Loewner-Nirenberg and Yamabe problems
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, bk-, 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≤k≤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 S3 preserves a standard contact structure pointwise. We also relate univerally tight contact structures on 3-manifolds covered by S3 to the exceptional isomorph…
Researchers compute cohomology of Lie groups using Lie algebras.
problem Computing cohomology of left-invariant elliptic and hypocomplex structures on compact Lie groups.
method Used the Leray spectral sequence connecting Lie algebras to Dolbeault cohomology of homogeneous manifolds.
result Cohomology can be computed purely algebraically.
Researchers find a method to construct projective structures on a specific surface.
problem Constructing projective structures with given holonomy and tameness conditions.
method Grafting circular triangles determined by a natural framing of the representation.
result All structures satisfying the conditions can be obtained through this method.
Elliptic surfaces have unique Lefschetz pencils and Calabi-Yau diffeomorphisms.
problem Understanding symplectic structures on elliptic surfaces.
method Analyzing R. Inanc Baykur's family of symplectic manifolds.
result Elliptic surfaces admit Lefschetz pencils and diffeomorphic to K3 surfaces.
Study fully nonlinear elliptic equations on complex manifolds.
problem Solving fully nonlinear elliptic equations on complex manifolds.
method Derive C2,α-estimate and prove existence theorems for solutions and Dirichlet problems. result Existence theorems for solutions on closed Hermitian manifolds with unbounded conditions.
Elliptic Chern characters and Atiyah-Witten formula generalized to double loop spaces.
problem Generalizing classical Atiyah-Witten formula to double loop spaces.
method Constructing elliptic Chern and Bismut-Chern characters, defining elliptic holonomy, and using equivariant twisted parallel transport.
result Established elliptic Atiyah-Witten formula on double loop space.
Characterizes elliptic operators on singular foliations.
problem Understanding elliptic operators on singular foliations.
method Using Nash algebroids and symplectic leaves of dual Lie algebroids.
result Characterization of longitudinally elliptic differential operators.
The paper studies elliptic surfaces and proves unique fibered structures.
problem Understanding the structure of elliptic fibrations and their mapping class groups.
method Analyzes smooth 4-manifolds and uses Néron-Lang Theorem. result Proves the uniqueness of fibered structures in elliptic fibrations.
Characterizes stably elliptic elements in Lie groups and their properties.
problem Understanding stably elliptic elements in Lie groups and their geometric and algebraic properties.
method Characterization through fixed point algebra and Weyl group action; relates to maximal invariant cones and compactness of order intervals.
result Connected components of stably elliptic elements can be described using Weyl group action on a compactly embedded Cartan subalgebra.
We show that the G2 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 G2 cobordisms be…
New framework uses elliptic operators to study projective maps.
problem Understanding projective structures on Riemannian manifolds.
method Develops two elliptic operators of second and fourth order.
result Establishes a natural correspondence between analytical and geometric properties.
Real del Pezzo surfaces split real lines into elliptic and hyperbolic types.
problem Understanding real lines on real del Pezzo surfaces of degree 1.
method Intrinsically defined Pin-structure on the real locus.
result The difference between hyperbolic and elliptic lines is always 16.
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 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.
problem Closedness of images of elliptic operators on Hilbert bundles.
method Analyzes tensor product of elliptic operators and compares images.
result Establishes closedness of images with respect to natural topology.
Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
problem Characterizing compact locally conformally Kähler manifolds of algebraic codimension one.
method Proving bimeromorphic equivalence to elliptic fibrations.
result Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
Researchers prove a complex geometric conjecture about certain manifolds.
problem Compact simply connected Riemannian manifolds with nonnegative sectional curvature.
method Assumption of entire Grauert tube and real analytic structure.
result Compact simply connected Riemannian manifolds with entire Grauert tube are rationally elliptic.
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.
problem Characterizing compact locally conformally Kähler manifolds of algebraic codimension one.
method Proving bimeromorphic equivalence to elliptic fibrations.
result 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.
problem Learning value and advantage functions for continuous-time Markov processes without structural assumptions.
method Proposes Sobolev-prox fitted q-learning algorithm based on Hilbert-space positive definiteness and boundedness properties of Bellman operators. result Identifies ellipticity as a key structural property enabling reinforcement learning for Markov diffusions.
Elliptic curve governs Hopf linking in symmetric tensegrity.
problem Understanding Hopf linking in A4-symmetric tensegrity.
method Parametrized by points on the elliptic curve 30a2.
result Only one rational point corresponds to a stable configuration.
Extends T-duality to non-principal torus actions with elliptic tangent bundles.
problem Classifying and understanding non-principal torus actions with singularities.
method Introduces elliptic tangent bundle to control singularities, uses it to define connections and transport generalized complex structures via T-duality.
result New insights into the classification of torus actions and transport of generalized complex structures.
The study characterizes a complex curve of residueless meromorphic differentials on elliptic curves.
problem Characterizing the locus of residueless meromorphic differentials on elliptic curves.
method Multi-scale compactification of strata, formulas for genus and degree of maps, distinguishing components.
result Complete classification of connected components of residueless loci in exceptional strata.
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.
problem Robust covariance estimation for elliptically distributed data with low-rank covariance structure.
method Original Riemannian geometry on quotient manifolds, new optimization framework, and divergence function.
result Derivation of intrinsic Cramér-Rao lower bounds for covariance and subspace estimation.