Local formulas integrate multiplicative forms on Lie groupoids.
problem Integrating multiplicative forms on local Lie groupoids.
method Explicit formulas using infinitesimal data.
result Concrete integrations of various geometric structures.
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We solve extending local Lie groupoids to global ones, linking integrability and associativity.
problem Extending local Lie groupoids to global Lie groupoids.
method We characterize integrable local Lie groupoids and establish a relationship between their integrability and associativity failure.
result A precise relationship between integrability of Lie algebroids and associativity failure in local integration.
Defines a new distance for integral current spaces and proves convergence criteria.
problem Defining a new distance metric for integral current spaces.
method Defines a new distance function and proves convergence criteria.
result Establishes the compactness theorem for integral current spaces using a new distance function.
Integrable symmetries of diffieties are studied, leading to local morphisms.
problem Understanding local integrable symmetries of diffieties.
method Integrable infinitesimal symmetries defined as a one-parameter pseudogroup of local diffiety morphisms. Reduction of computation to solving PDEs.
result Preliminary results and examples show integrable symmetries can be reduced to solving linear systems.
Surveying integrability of Lie algebroids and structures.
problem Integrability of Lie algebroids and structures.
method Survey and recent results on integrability.
result Recent findings on local and global integrability.
Lie group integrators improve global error estimates.
problem Global error estimates for Lie group integrators.
method Relate local error to global error, derive from bounds.
result Lie-Butcher theory proves global error estimates for Lie group integrators.
Study localizes integrals at isolated degenerate zeros.
problem Localization of Futaki-Morita integrals at isolated degenerate zeros.
method Streamlined exposition in the spirit of Bott, localization procedure for a holomorphic vector field on .
result Essentially unique formula for Futaki-Morita integral invariants.
Counterexample shows Ito integrand needn't be locally square integrable.
problem Ito integrand's square integrability condition is not always met.
method Provided a counterexample to Ito's Lemma's integrability condition.
result Ito integrand needn't be locally square integrable.
Extends Calabi operator to Riemannian locally symmetric spaces.
problem Local integrability conditions on Riemannian locally symmetric spaces.
method Generalizes Calabi operator to Riemannian locally symmetric spaces.
result Generalised operator works in irreducible case and fails in products.
The geodesic flow of a generic metric does not admit nontrivial integrals polynomial in momentamath.DG
Any smooth geodesic flow is locally integrable with smooth integrals. We show that generically this fails if we require, in addition, that the integrals are polynomial (or, more generally, analytic) in momenta. Consequently we obtain that a generic real-analytic metric does not admit, even locally, a real-analytic inte…
Formula for twisted orbital integrals using hypoelliptic Laplacian.
problem Evaluate equivariant trace of Laplacians on compact locally symmetric spaces.
method Using the hypoelliptic Laplacian method and twisted trace formula.
result Explicit geometric formula for twisted orbital integrals.
Study local system points on surfaces using group descent.
problem Understanding integral points on moduli of local systems.
method Mapping class group descent and boundedness results for systoles.
result Established structure theorem for integral points.
Directly constructs Lie groupoids from Lie algebroids.
problem Local integration of Lie algebroids.
method Explicit construction of Lie groupoids.
result Finite-dimensional proof of Lie theory equivalence.
The paper calculates the full asymptotics of analytic torsions for compact orbifolds.
problem Analytic torsions of compact locally symmetric orbifolds.
method Using Selberg's trace formula and geometric localization, the paper evaluates the heat trace and orbital integrals.
result Explicit formula for the asymptotic Ray-Singer analytic torsion of compact orbifolds.
Proves local bi-integrability of bi-Hamiltonian systems on real smooth manifolds.
problem Proving local bi-integrability of bi-Hamiltonian systems on real smooth manifolds.
method Proves bi-integrability by constructing a complete set of functions in bi-involution and showing differentials can realize any bi-Lagrangian subspace.
result Bi-Hamiltonian systems are locally bi-integrable on real smooth manifolds.
The paper studies cohomological Donaldson-Thomas theory for local systems on a 3-torus.
problem Cohomological Donaldson-Thomas theory for local systems on the 3-torus.
method Using exponential maps and the tripled Jordan quiver, the paper proves cohomological integrality for GL_n and SL_n local systems.
result The paper proves Langlands duality statements for SL_n and PGL_n cohomological Donaldson-Thomas invariants for prime n.
Proves local bi-integrability of bi-Hamiltonian systems via bi-Poisson reduction.
problem Local bi-integrability of bi-Hamiltonian systems.
method Bi-Poisson reduction to prove local bi-integrability.
result Constructs a complete set of functions in bi-involution for bi-Hamiltonian systems.
Method finds differential equations for integrable billiard tables.
problem Finding differential equations for integrable billiard tables.
method Introducing a method to find differential equations for functions defining tables.
result Illustrated method in three billiard systems.
Study on curves on moduli spaces of local systems, proving structure and integral point determination.
problem Arithmetic of algebraic curves on moduli spaces of local systems.
method Proved structure theorem for morphisms and effectively determined integral points on curves.
result Effective determination of integral points on nondegenerate algebraic curves on moduli space.
Local study of foliation deformation cohomology.
problem Understanding deformations of singular foliations.
method Introducing and studying local deformation cohomology.
result Local deformation cohomology for singular foliations and related structures.
Local high-degree polynomial integrals of geodesic flows and the generalized hodograph methodmath.DS
Study integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.
problem Integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.
method Semi-Hamiltonian systems of PDEs and generalized hodograph method.
result Construction of many local explicit and implicit integrable examples with polynomial first integrals of degrees 3, 4, 5.
We obtain a local Sobolev constant estimate for integral Ricci curvature, which enables us to extend several important tools such as the maximal principle, the gradient estimate, the heat kernel estimate and the Hessian estimate to manifolds with integral Ricci lower bounds, without the non-collapsing conditions.
We prove that a local Hamiltonian operator of hydrodynamic type K_1 is compatible with a nondegenerate local Hamiltonian operator of hydrodynamic type K_2 if and only if the operator K_1 is locally the Lie derivative of the operator K_2 along a vector field in the corresponding domain of local coordinates. This result …
A theorem proves integrability of Fréchet tangent distributions.
problem Integrability of Fréchet tangent distributions on manifolds.
method Introduced Condition W, applied variational approach, used differential forms.
result Existence and uniqueness of maximal foliations.
Ecker's and Huisken's quantities agree for ancient mean curvature flows.
problem Understanding the finiteness of integral quantities for ancient mean curvature flows.
method Comparison of Ecker's and Huisken's integral quantities.
result Finiteness of Ecker's integral quantity implies finiteness of entropy at infinity.
We study local normal forms for completely integrable systems on Poisson manifolds in the presence of additional symmetries. The symmetries that we consider are encoded in actions of compact Lie groups. The existence of Weinstein's splitting theorem for the integrable system is also studied giving some examples in whic…
Paper extends Poincaré's work to stochastic differential equations.
problem Existence of first integrals in stochastic differential equations.
method Introduce two definitions of local first integrals for SDEs.
result Stochastic version of Poincaré non-integrability theorem.
We prove a pseudolocality type theorem for compact Ricci Flow under local integral bounds of curvature. The main tool is Local Ricci Flow introduced by Deane Yang in [4] and Pseudolocality Theorem of Perelman in [3]. We also study L^p bounds for the derivatives of curvature and smooth extension of Local Ricci Flow.
We consider the problem of integration of L_\infty-algebroids (differential graded manifolds) to L_\infty-groupoids. We first construct a "big" Kan simplicial manifold (Fréchet or Banach) whose points are solutions of a (generalized) Maurer-Cartan equation. The main analytic trick in our work is an integral transformat…
In this paper, we focus our study on the ends of a locally conformally flat complete manifold with finite total -curvature. We prove that for such a manifold, the integral of the -curvature equals an integral multiple of a dimensional constant , where is the integral of the -curvature on the unit $n…
Paper constructs solutions for a class of overdetermined systems.
problem Constructing solutions for a class of overdetermined systems.
method Resolution of the solution sheaf, sufficient condition for global exactness, gluing techniques, local solvability of the Treves complex.
result Obtained a sufficient condition for global exactness, leading to gluing techniques for local solutions.
Study convexity in Hamiltonian systems with focus singularities.
problem Convexity of singular affine structures in Hamiltonian systems with focus singularities.
method Systematic study of symplectic convexity for integrable Hamiltonian systems with focus-focus singularities.
result Local and global convexity properties of singular integral affine base spaces, with conditions for convexity breakdown.
In this article we give a totally new proof of the integral localization formula for equivariantly closed differential forms (Theorem 7.11 in [BGV]). We restate it here as Theorem 2. This localization formula is very well known, but the author hopes to adapt this proof to obtain a more general result in the future.
Study integrability of generalized almost complex structures on S^6.
problem Integrability of generalized almost complex structures on the 6-dimensional sphere.
method Local coordinate criteria for integrability with respect to brackets and Courant integrability for strong structures.
result No nontrivial spherical combinations of the canonical structures are integrable with respect to the Levi-Civita connection.
The study bounds Hausdorff measure of flat singular points in area-minimizing currents.
problem Bounding Hausdorff measure of flat singular points in area-minimizing currents.
method Proving locally finite -dimensional Hausdorff measure and Minkowski content bounds.
result The set of flat singular points has locally finite -dimensional Hausdorff measure.
Recently, a new embedding/compactness theorem for integral currents in a sequence of metric spaces has been established by the second author. We present a version of this result for locally integral currents in a sequence of pointed metric spaces. To this end we introduce another variant of the Ambrosio--Kirchheim theo…
On asymptotically flat and asymptotically hyperbolic manifolds, by evaluating the total mass via the Ricci tensor, we show that the limits of certain Brown-York type and Hawking type quasi-local mass integrals equal the total mass of the manifold in all dimensions.
This is an expanded version of the lecture notes for a minicourse that I gave at a summer school called "Advanced Course on Geometry and Dynamics of Integrable Systems" at CRM Barcelona, 9--14/September/2013. In this text we study the following aspects of integrable non-Hamiltonian systems: local and semi-local normal …
This article gives a local answer to the coquecigrue problem. Hereby we mean the problem, formulated by J-L. Loday in \cite{LodayEns}, is that of finding a generalization of the Lie's third theorem for Leibniz algebra. That is, we search a manifold provided with an algebraic structure which generalizes the structure of…
Local coordinates for non-integrable Lie algebroids constructed.
problem Constructing local coordinates for non-integrable Lie algebroids.
method Reformulating bi-submersion algebraically and proving their existence for the Weinstein groupoid.
result Existence of -algebra attached to every Lie algebroid.
It is well known that in compact local Lipschitz neighborhood retracts in Euclidean space flat convergence for integer rectifiable currents amounts just to weak convergence. In the present paper we extend this result to integral currents in complete metric spaces admitting a local cone type inequality. These include in…
New Q-manifolds theory integrates Lie algebroids.
problem Integrating Lie algebroids over smooth manifolds.
method Introducing Q-groupoids and Q-bundles, proving Lie algebroids arise from Q-manifolds.
result Transitive Lie algebroids over second countable, smooth manifolds are integrated to locally trivial Q-groupoids.
We review localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low-dimensional gauge theories. These are the functional integral counterparts of the Mathai-Quillen formalism, the Duistermaa…
We formulate the theory of nearly autoparallel maps (generalizing conformal transforms) of locally anisotropic spaces and define the nearly autoparallel integration as the inverse operation to both covariant derivation and deformation of connections by nearly autoparallel maps. By using this geometric formalism we cons…
Symplectic classification for a specific type of singularity in integrable systems.
problem Symplectic classification of integrable systems near singular points of type .
method Real-analytic symplectic normal forms and classification of Lagrangian foliations.
result All integrable systems are symplectically equivalent near singular points of this type.
Surveying hermitian integral geometry, the paper describes new kinematic formulas for complex space forms.
problem Understanding curvature measures and valuations on complex spaces.
method Analyzing valuations and curvature measures on complex space forms, deriving kinematic formulas.
result New local kinematic formulas for hermitian geometry, containing more information than global formulas.
Integral points are potentially dense in character varieties of quasi-projective varieties.
problem Density of integral points in character varieties of quasi-projective varieties.
method Reduction to Riemann surfaces and use of Corlette-Simpson work.
result Integral points have Zariski-dense orbit under the mapping class group.
Formulae for divergent integrals with singularities, useful in string theory.
problem Handling divergent integrals with singularities.
method Formulae for finite part of divergent integrals, using local residue map.
result Formulae for finite part of divergent integrals expressed in terms of local residue map.