This is a survey of the author's paper arXiv:1001.0023 on "Algebraic Geometry over C-infinity rings". If X is a smooth manifold then the R-algebra C^\infty(X) of smooth functions c : X --> R is a "C-infinity ring". That is, for each smooth function f : R^n --> R there is an n-fold operation Φ_f : C^\infty(X)^n --> C^\i…
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48 results for “C-infinity algebras”
Paper constructs characteristic classes using C-infinity algebras.
problem Characteristic classes of fiber bundles.
method Homotopy theory of C-infinity algebras.
result Chern-Weil-type construction of characteristic classes.
This is the first in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we study theories of supercommutative algebras for which infinitely differentiable functions can be evaluated on elements. Such…
Functional AD for Weil algebra computations.
problem Efficient computation of -structures on Weil algebras.
method Multivariate Tower Automatic Differentiation (AD) implementation.
result Functional AD for Weil algebra computation.
This is the second in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we extend the classical notion of a dg-algebra to define, in particular, the notion of a differential graded algebra in the wo…
A C-infinity ring is a set equipped with n-ary operations corresponding to smooth n-ary functions on the real line (satisfying natural axioms). We prove that the cosimplicial abelian group associated to the de Rham complex of Euclidean space has the structure of a cosimplicial C-infinity ring. We also analyse the notio…
Lie-Rinehart algebras over -rings defined and studied.
problem Defining and studying Lie-Rinehart algebras over -rings.
method Defining Lie-Rinehart algebras over -rings and showing their relationship with Poisson -rings.
result A natural Poisson bracket on the -ring associated with a Lie-Rinehart algebra over a -ring.
Defines vector fields and differential forms on local C-infinity-ringed spaces.
problem No specific problem stated; focuses on mathematical definitions.
method Defines tangent sheaf, contractions, Lie derivatives, and proves Cartan equations.
result Standard Cartan calculus equations hold for local C-infinity-ringed spaces.
A pair of points (x,y) in a Riemannian manifold (M,g) is said to have the finite blocking property if there is a finite set P contained in M\{x,y} such that every geodesic segment from x to y passes through a point of P. We show that for every closed C-infinity manifold M of dimension at least two and every pair (x,y) …
The paper proposes a method to compute higher infinitesimals in numerical and symbolic analysis.
problem Computing higher-order derivatives with higher infinitesimals.
method Automatic differentiation in terms of C-infinity rings and Weil algebras.
result A unifying theoretical framework for multivariate higher-order derivatives.
Let X be a C-infinity manifold. We construct a microlocalization functor from the derived category of bounded complexes of ind-sheaves on X to the one on the cotangent bundle of X. This functor generalizes the classical theory of microlocalization.
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.
In this paper, we consider a new length preserving curve flow for convex curves in the plane. We show that the global flow exists, the area of the region bounded by the evolving curve is increasing, and the evolving curve converges to the circle in C-infinity topology as t goes to infinity.
Smooth contact maps are always smooth in rigid Carnot groups.
problem Smoothness of contact maps in rigid Carnot groups.
method Analyzing -rigid Carnot groups to show -contact maps are smooth.
result Smooth contact maps are always smooth in rigid Carnot groups.
Uniform estimates for collapsing Calabi-Yau metrics on degenerating spaces.
problem Uniform estimates for Calabi-Yau metrics on degenerating spaces.
method Blowup arguments and Liouville theorems on cylinders; Schauder estimates for Laplacian on cylinders.
result Uniform C^alpha and C^infinity estimates for collapsing Calabi-Yau metrics.
In the first chapter, we give a precise and general description of gerbes valued in arbitrary crossed module and over an arbitrary differential stack. We do it using only Lie groupoids, hence ordinary differential geometry, by considering differential stacks as being Lie groupoids up to Morita equivalence. We prove the…
New invariants define the rational and real homotopy types of closed manifolds.
problem Defining invariants for the rational and real homotopy types of closed manifolds.
method Introducing isotopy modulo k and minimal unital cyclic C-infinity-algebras.
result A complete set of invariants uniquely defines the rational and real homotopy types of closed simply connected manifolds.
Vector fields on schemes have flows if rings are finitely generated.
problem Understanding vector fields and flows on schemes.
method Analyzing vector fields on affine -schemes with finitely generated rings.
result Vector fields on affine -schemes with finitely generated rings have flows and are groupoid internal maps.
We consider curvature flows in hyperbolic space with a monotone, symmetric, homogeneous of degree 1 curvature function F. Furthermore we assume F to be either concave and inverse concave or convex. For compact initial hypersurfaces, which are strictly convex by horospheres, we show the long time existence of mixed volu…
Constructs differential forms on -ringed spaces.
problem Developing a theory of differential forms for non-manifold spaces.
method Functorial construction of differential forms on local -ringed spaces.
result Stokes' theorem holds for integrated forms on simplices.
In the model of synthetic differential geometry consisting of sheaves (with respect to open covers) over the opposite category of the category of closed finitely generated C-infinity rings, any morphism from S, the zeroes of the "amazing right adjoint" of dx, to the real line R extends to a morphism from R to R. This s…
The study reveals scaling limits of spectral projectors on Riemannian manifolds.
problem Understanding spectral function scaling limits on Riemannian manifolds.
method New off-diagonal estimates and Weyl law application.
result Scaling limit of spectral projector is a normalized Bessel function.
Paper shows a geometry construction has universal limits property.
problem Defining universal limits in geometry.
method Adjoins finite homotopical limits to manifold categories.
result Construction produces geometry with universal limits property.
We define an simple invariant of an embedded nullhomologous Lagrangian torus and use this invariant to show that many symplectic 4-manifolds have infinitely many pairwise symplectically inequivalent nullhomologous Lagrangian tori. We further show that for a large class of examples that lambda(T) is actually a C-infinit…
We introduce a holomorphic sheaf E on a Sasaki manifold and study two new notions of stability for E along the Sasaki-Ricci flow related to the `jumping up' of the number of global holomorphic sections of E at infinity. First, we show that if the Mabuchi K-energy is bounded below, the transverse Riemann tensor is bound…
In this paper, we discuss the recognition problem for A_k-type singularities on wave fronts. We give computable and simple criteria of these singularities, which will play a fundamental role in generalizing the authors' previous work "the geometry of fronts" for surfaces. The crucial point to prove our criteria for A_k…
New algebraic structure derived from Hopf algebra and Drinfel'd twist.
problem Developing a new algebraic structure from existing mathematical concepts.
method Extending -algebra to a Hopf algebra, twisting with Drinfel'd twist, and identifying Hopf morphisms and braided morphisms.
result Braided -algebra is derived from the process.
Study of cluster and skein algebras for surfaces, showing their connection.
problem Understanding algebraic structures of curve algebras on surfaces.
method Generalization and explicit definition of maps between cluster and skein algebras.
result Explicit maps between cluster and skein algebras, showing their close relationship.
Study on pseudo-Riemannian algebraic Ricci solitons in 4D Lie groups.
problem Investigating conditions for pseudo-Riemannian algebraic Ricci solitons on 4D Lie algebras.
method Analyzing the algebraic Ricci soliton equation for each 4D Lie algebra.
result Complete description of pseudo-Riemannian algebraic Ricci solitons in dimension four.
Characterizes Lie groups with specific structures and finds a correspondence between carrollian and galilean Lie algebras.
problem Understanding Lie groups with specific structures.
method Using structure theory of metric Lie algebras and defining new Lie algebras with skew-symmetric derivations.
result A canonical correspondence between carrollian and galilean Lie algebras mediated by bargmannian Lie algebras.
Study on Lie algebras from Clifford modules, focusing on specific types.
problem Characterizing Lie algebras from Clifford modules and their graded structures.
method Analysis of pseudo -type Lie algebras and their representations through Clifford algebras.
result Different types of Lie algebras have varying possibilities of containing pseudo -type Lie algebras in their negative part.
We extend the classical characterization of a finite-dimensional Lie algebra g in terms of its Maurer-Cartan algebra-the familiar differential graded algebra of alternating forms on g with values in the ground field, endowed with the standard Lie algebra cohomology operator-to sh Lie-Rinehart algebras. To this end, we …
The paper studies prolongations of Lie algebras associated with pseudo -type Lie algebras.
problem Investigating prolongations of Lie algebras associated with pseudo -type Lie algebras.
method Analyzing prolongations of associated fundamental graded Lie algebra and associated conformal pseudo-subriemannian fundamental graded Lie algebra.
result The prolongation of the associated conformal pseudo-subriemannian fundamental graded Lie algebra coincides with that of the associated fundamental graded Lie algebra under certain conditions.
Relative cellular algebrasmath.RT
The paper extends cellular algebras with relative orderings and explores their properties.
problem Generalizing cellular algebras with different partial orderings.
method Classification and construction of simple modules, characterizations, and examples.
result Examples of relative cellular algebras that are not cellular.
New tools for studying Hsiang algebras discovered, linking them to known algebraic structures.
problem Classifying Hsiang algebras and understanding their properties.
method Introducing quasicomposition and tripling constructions to study Hsiang algebras.
result The triple of a quasicomposition algebra is an exceptional Hsiang algebra.
A Lie-admissible algebra gives by anticommutativity a Lie algebra. In this work we study remarkable classes of Lie-admissible algebras such as Vinberg, PreLie algebras. We compute the corresponding binary quadratic operads and study their Koszul duality. Considering Lie algebras as Lie-admissible algebras we can define…
New grading on algebras of curves by winding number.
problem Understanding the structure of algebras of curves.
method Constructing a new grading on the Goldman Lie algebra and related algebras by winding number.
result Induces a new grading on the HOMFLY-PT skein algebra and related algebras.
Study biderivations in complete Leibniz algebras, extending Lie algebra results.
problem Defining and studying biderivations in complete Leibniz algebras.
method Analyze biderivations according to two definitions, provide conditions for biderivations, and compare symmetric and skew-symmetric biderivations.
result Necessary and sufficient conditions for biderivations in Leibniz algebras are provided.
Universal enveloping algebras of Lie-Rinehart algebras: crossed products, connections, and curvaturemath.RA
The paper extends a theorem to Lie-Rinehart algebras and provides new decompositions of universal enveloping algebras.
problem Understanding universal enveloping algebras of Lie-Rinehart algebras.
method Extending a theorem to left Hopf algebroids and applying it to universal enveloping algebras of Lie-Rinehart algebras.
result Provides a crossed product decomposition of universal enveloping algebras for curved and flat connections.
The paper classifies Lie algebras with special operators.
problem Classifying 3D Lie algebras with regular semisimple algebraic Nijenhuis operators.
method Described all Nijenhuis eigenbases for each 3D Lie algebra.
result Different answers in real and complex cases, some Lie algebras admit operators, others do not.
New theory disassembles Lie algebras into dyons and triadons.
problem Understanding the structure of Lie algebras through disassembly.
method Modular disassembly of Lie algebras into dyons and triadons.
result Any Lie algebra can be assembled from dyons and triadons.
Symmetric spaces' connections form Lie admissible triple algebras.
problem Understanding the algebraic structure of symmetric spaces' connections.
method Analyzing the connection as a binary operator on tangent bundle sections, identifying Lie admissibility constraints.
result Connection algebra of symmetric spaces is a Lie admissible triple algebra.
The paper generalizes para-Kähler Lie algebras to k-para-Kähler Lie algebras and explores their structures.
problem Characterizing and understanding k-para-Kähler Lie algebras.
method Generalization of para-Kähler Lie algebras to k-para-Kähler Lie algebras, introduction of new structures, determination of Lie algebras.
result Determination of all k-symplectic Lie algebras of dimension (k+1) and six-dimensional 2-para-Kähler Lie algebras.
Abstract defines and identifies a planar algebra with spin properties.
problem Identifying a specific planar algebra.
method Generators and relations defined, structure studied, identified with Jones' spin planar algebra.
result Identified a specific planar algebra with spin properties.
The paper investigates gradings of complex simple Lie algebras, focusing on -gradings and their algebraic structures.
problem Investigating the algebraic structure of -gradings of complex simple Lie algebras.
method Completely determining the possible reductive algebras and proving the uniqueness of a specific free nilpotent Lie algebra.
result The only free nilpotent Lie algebra of step 3 that appears as the negative part of a -grading is the usual -grading of the exceptional Lie algebra .
Study on pre-Lie structures for semisimple Lie algebras over C.
problem Admissibility of pre-Lie structures in semisimple Lie algebras.
method Examined properties of anti-flexible algebras (AFAs), computed Lie-admissibility criteria, and provided examples.
result Explicit counterexample of an AFA admissible by sl(2, C).
Study resolves conjecture linking two algebraic structures on surfaces.
problem Compatibility of skein and cluster algebra structures on surfaces.
method Established compatibility between skein and cluster algebras of surfaces.
result Cluster algebra of positive genus surfaces is not finitely generated.
The pong algebramath.GT
New algebra pong algebra computed for knot Floer homology.
problem Computing A-infinity structure on knot Floer homology.
method Introduced differential graded algebra, pong algebra.
result Computed A-infinity structure on pong algebra's homology.