Develops derived differential geometry theory.
problem Homotopy and intersection in smooth manifolds.
method Using -algebras and homotopy transfer.
result Derived manifolds form a category of fibrant objects.
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This paper proves equivalence between derived manifolds and differential graded manifolds.
problem Characterizing derived manifolds and their relationship to differential graded manifolds.
method Proving equivalence between the infinity categories of derived manifolds and differential graded manifolds.
result The infinity category of differential graded manifolds is equivalent to that of derived manifolds.
Quantization of (-1)-shifted derived Poisson manifolds via BV-infinity operators.
problem Quantizing -shifted derived Poisson manifolds.
method Using BV-infinity operators on the space of Berezinian half-densities, proving quantization via lifting of Maurer-Cartan elements.
result Quantization of -shifted derived Poisson manifolds is equivalent to the vanishing of the second Poisson cohomology group.
We approximate derivatives of functions on manifolds by embedding them and applying vector-valued operators.
problem Derivatives of manifold-valued functions are harder to approximate than vector-valued functions.
method Embed the manifold into a higher space, approximate the derivative of the vector-valued function, and project back.
result We provide error bounds for the approximation of manifold-valued function derivatives.
Study of Schwarzian derivative on Finsler manifolds with constant curvature.
problem Characterizing the role of the Schwarzian derivative in Finsler manifolds of constant curvature.
method Developed integrability conditions and rigidity results for Möbius equations on Finsler manifolds.
result Complete Finsler manifolds of positive constant Ricci curvature are homeomorphic to the n-sphere if they admit non-trivial Möbius mappings.
We establish variational formulas for Ricci upper and lower bounds, as well as a derivative formula for the Ricci curvature. As applications, constant curvature manifolds, Einstein manifolds and Ricci parallel manifolds are identified, respectively, with different integral-differential formulas and semigroup inequaliti…
Continues work on derived manifolds and symplectic schemes, constructing virtual classes.
problem Constructing virtual fundamental classes for derived manifolds and schemes.
method Cosection localization, reduced virtual fundamental classes, and applications to Donaldson-Thomas theory.
result Virtual fundamental classes for -shifted symplectic derived schemes are consistent with algebraic and differential geometric constructions.
Study virtual fundamental classes of derived manifolds, proving invariant vanishes.
problem Computing virtual fundamental classes for derived manifolds.
method Combining derived differential geometry and cosection localization.
result Stable pair invariants of hyperkähler fourfolds are zero.
In this short note we present local derivative estimates for heat equations on Riemannian manifolds following the line of W.-X. Shi. As an application we generalize a second derivative estimate of R. Hamilton for heat equations on compact manifolds to noncompact case.
Sharp bounds on heat kernel derivatives on incomplete manifolds.
problem Extending bounds on heat kernel derivatives to incomplete Riemannian manifolds.
method Analyzing heat kernels on incomplete Riemannian manifolds with conservative and non-conservative vector fields.
result Sharp bounds on all orders of heat kernel derivatives are established for incomplete manifolds.
In this short note, we propose an unified method to derive formulas for derivations conjugated by exponential functions on an almost complex manifold. In v3, we corrected some mistakes in previous versions.
The Schwarzian derivative is generalized to Finsler manifolds and its properties studied.
problem Generalizing the Schwarzian derivative to Finsler manifolds.
method Identifying a tensor field and defining Mobius mappings on Finsler manifolds.
result Mobius mappings preserve circles and are conformal on certain Finsler manifolds.
Let be a pseudo-Riemannian manifold. We propose a new approach for defining the conformal Schwarzian derivatives. These derivatives are 1-cocycles on the group of diffeomorphisms of related to the modules of linear differential operators. As operators, these derivatives do not depend on the rescaling of the…
Survey on smooth function and form density in Riemannian Sobolev spaces.
problem Density of smooth functions and forms in Sobolev spaces on Riemannian manifolds.
method Careful examination of weak covariant derivatives and partial derivatives.
result Equivalence of weak covariant derivatives to weak partial derivatives.
Formula derived for curvature on smooth manifolds.
problem Calculating curvature on smooth manifolds.
method Derived a formula for sectional curvature.
result Formula for sectional curvature on smooth manifolds.
Derives formulas from Green function Hessian assumption.
problem Deriving formulas from Green function Hessian assumption.
method Assumption on Hessian of Green function leads to monotonicity formulas.
result Explicit examples of manifolds satisfying assumption.
Derived differential manifolds are constructed using the usual homotopy theory of simplicial rings of smooth functions. They are proved to be equivalent to derived differential manifolds of finite type, constructed using homotopy sheaves of homotopy rings (D.Spivak), thus preserving the classical cobordism ring. This r…
Derives inequality for optimal transport on manifolds.
problem Optimal transport theory on manifolds.
method Five gradients inequality for cost functions on Lie groups and Riemannian manifolds.
result Derives inequality for optimal transport on specific manifolds.
A Complete Derivation of Complex Circle Manifold (CCM) Riemannian manifold Optimization Equationsmath.OC
Derives key CCM manifold equations for optimization.
problem Lack of rigorous derivation for CCM manifold equations.
method Systematic and rigorous proof of CCM properties.
result Unified reference for CCM Manifold Optimization.
Paper computes Atiyah class for DG manifolds of amplitude +1.
problem Computing the Atiyah class for DG manifolds of specific amplitude.
method Computed the Atiyah class by encoding the derived intersection of sections and zero sections of vector bundles.
result Atiyah class vanishes if and only if the intersection is clean.
We derive expressions for the Ricci curvature tensor and scalar in terms of intrinsic torsion classes of half-flat manifolds by exploiting the relationship between half-flat manifolds and non-compact holonomy manifolds. Our expressions are tested for Iwasawa and more general nilpotent manifolds. We also derive ex…
A model structure is defined on the category of derived differentiable schemes, and it is used to analyse the truncation 2-functor from derived manifolds to d-manifolds. It is proved that the induced 1-functor between the homotopy categories is full and essentially surjective, giving a bijection between the sets of equ…
Regulated curves on Banach manifolds with continuous projections and regulated derivatives are studied.
problem Regulated curves on Banach manifolds with continuous projections and regulated derivatives.
method Building a Banach manifold structure on the set of such curves.
result Existence of a 'local addition' on such a manifold for any Banach manifold.
After summarizing some necessary preliminaries and tools, including Berwald derivative and Lie derivative in pull-back formalism, we present ten equivalent conditions, each of which characterizes Berwald manifolds among Finsler manifolds. These range from Berwald's classical definition to the existence of a torsion-fre…
Derives a method to optimize high-dimensional functions on low-dimensional manifolds.
problem High-dimensional derivative-free optimization with high sample complexity.
method Online learning approach that learns the manifold while optimizing the function.
result Significantly reduces sample complexity compared to existing methods.
The paper derives inequalities on Finsler manifolds, influenced by their curvatures.
problem Deriving inequalities on Finsler manifolds.
method Local and global geometric inequalities on Riemannian and Finsler manifolds.
result Generalized Caffarelli-Kohn-Nirenberg and Hardy type inequalities on Finsler manifolds.
Generalizes neural network approximation to infinite-dimensional manifolds and derivatives.
problem Approximating differentiable maps on infinite-dimensional manifolds.
method Proves a weighted Nachbin theorem to establish universal approximation for differentiable maps, including derivatives.
result Linear functions of the signature can approximate path space functionals including their derivatives.
Volume renormalization for conformally compact asymptotically hyperbolic manifolds in dimension fourmath.DG
We derive a new renormalized volume formula for conformally compact asymptotically hyperbolic manifolds in dimension four. The formula generalizes the ones given by Anderson, Albin, and Chang-Qing-Yang for the case of Poincare-Einstein manifolds. We also derive variational formulas for the renormalized seen as a functi…
Study h-principles for non-integrable distributions on manifolds.
problem Existence and classification of maximally non-integrable distributions of derived length one.
method Introduced formal structures and used h-principles to discuss existence and classification.
result Discussed existence and classification of maximally non-integrable distributions of derived length one.
Extends differential geometry concepts to manifolds with super tangent bundles.
problem No specific problem stated; extending differential geometry to super tangent bundles.
method Introduces super tangent bundle and extends differential geometry concepts.
result Basic notions of differential geometry extended to manifolds with super tangent bundles.
Develops Riemannian geometry for optimization on manifolds with detailed derivations.
problem Abstract high-level optimization on nonlinear spaces like matrix manifolds.
method Systematic derivation of geometric structures and constructions in coordinates and matrix form.
result Unified treatment of Riemannian geometry for optimization on manifolds.
The paper shows objective derivatives are covariant derivatives on Riemannian metrics.
problem The definition and interpretation of objective derivatives in continuum mechanics.
method Demonstrates that objective derivatives correspond to covariant derivatives on the manifold of Riemannian metrics.
result Objective derivatives are unified as covariant derivatives on the manifold of Riemannian metrics.
Extends algebraic geometry to include spaces with corners.
problem Generalizing manifolds with corners.
method Defines and studies -rings and schemes with corners.
result New categories of -rings and schemes with corners.
Study moduli spaces of elliptic PDEs using derived -geometry.
problem Representability of moduli spaces of solutions of elliptic PDEs.
method Derived -geometry, stacks of relative jets, nonlinear Fredholm analysis.
result Moduli stack of solutions is relatively representable by quasi-smooth derived -schemes.
Derives integral formulae on weighted manifolds.
problem No specific problem stated; focuses on mathematical derivations.
method Introduces weighted mean sigma-r curvature and uses weighted Newton transformations.
result Derives integral formulae generalizing previous work.
We introduce a canonical outer vector field on a Poisson manifold, also due independently to A. Weinstein. We view it as a global section of the sheaf of Poisson vector fields modulo the subsheaf of hamiltonian vector fields. We study this outer derivation mostly in the case of holomorphic Poisson manifolds.
We study dg-manifolds which are R[2]-bundles over R[1]-bundles over manifolds, we calculate its symmetries, its derived symmetries and we introduce the concept of T-dual dg-manifolds. Within this framework we construct the T-duality map as a degree -1 map between the cohomologies of the T-dual dg-manifolds and we show …
Given a geometrically finite hyperbolic cone-manifold, with the cone singularity sufficiently short, we construct a one parameter family of cone-manifolds decreasing the cone angle to zero. We also control the geometry of this one parameter family via the Schwarzian derivative of the projective boundary and the length …
Formula derived for ALH manifolds, showing existence of specific 3D manifolds.
problem Energy calculation and existence of specific 3D conformally compact ALH manifolds.
method Derivation of energy formula and application to manifold existence.
result Existence of 3D conformally compact ALH manifolds with specific properties.
This is a survey of the author's book "D-manifolds and d-orbifolds: a theory of derived differential geometry", available at http://people.maths.ox.ac.uk/~joyce/dmanifolds.html We introduce a 2-category dMan of "d-manifolds", new geometric objects which are 'derived' smooth manifolds, in the sense of the 'derived algeb…
Derives spectral density function for symplectic manifolds.
problem Calculating spectral density functions on symplectic manifolds.
method Explicit local formula derivation for spectral density function.
result Explicit formula for spectral density function.
Motivated by the possible characterization of Sasakian manifolds in terms of twistor forms, we give the complete classification of compact Riemannian manifolds carrying a Killing vector field whose covariant derivative (viewed as a 2-form) is a twistor form.
We develop a family of infinite-dimensional (non-parametric) manifolds of probability measures. The latter are defined on underlying Banach spaces, and have densities of class with respect to appropriate reference measures. The case , in which the manifolds are modelled on Fréchet spaces, is included.…
Study of time-dependent metrics and connections in geometry.
problem Understanding geodesics and connections in time-dependent Riemannian manifolds.
method Examine connections on product manifolds, explore parallel transport, geodesics, and torsion.
result Define the derivative of a one-parameter family of connections.
Paper derives estimates for complex Hessian equations on Hermitian manifolds.
problem Estimating solutions to complex Hessian equations on Hermitian manifolds.
method Derives second order estimates for solutions in a specific cone.
result Establishes second order estimates for solutions in cone.
On an odd-dimensional oriented hyperbolic manifold of finite volume with strongly acyclic coefficient systems, we derive a formula relating analytic torsion with the Reidemeister torsion of the Borel-Serre compactification of the manifold. In a companion paper, this formula is used to derive exponential growth of torsi…
Study circle actions on 4-manifolds, deriving formulas and graphs.
problem Understanding circle actions on 4-dimensional manifolds.
method Derive the Atiyah-Hirzebruch formula and associate graphs to fixed point data.
result Show existence of 4D oriented S^1-manifolds from satisfying graphs.
This is a long summary of the author's book "D-manifolds and d-orbifolds: a theory of derived differential geometry", available at http://people.maths.ox.ac.uk/~joyce/dmanifolds.html . A shorter survey paper on the book, focussing on d-manifolds without boundary, is arXiv:1206.4207, and readers just wanting a general o…