Lectures on topological field theories and differential cohomology.
problem Exploring topological field theories and their connections to differential cohomology.
method Introduction to topological field theory and generalized Abelian gauge theories.
result Explains the relationship between topological field theories and differential cohomology.
Characterizes differential forms and vector fields with constant coefficients on manifolds.
problem Understanding constant coefficient differential forms and vector fields on manifolds.
method Analyzes differential forms and vector fields of specific degrees, proving obstructions and characterizing solutions to partial differential systems.
result Characterizes differential forms and vector fields with constant coefficients of various degrees on smooth manifolds.
Study of differential forms and vector fields on orbit spaces.
problem Understanding vector fields and differential forms on orbit spaces.
method Defined differential forms and vector fields as multilinear maps on infinitesimal diffeomorphisms.
result Intrinsic view of vector fields and differential forms on orbit spaces.
Formulates Lagrangian field theory using covariant differentials.
problem Formulating Lagrangian field theory using partial derivatives.
method Introduces exterior covariant differentials of vector-valued forms.
result Natural Lagrangian can be written as a density on a covariant prolongation bundle.
Study natural operators transforming tensor fields, proving all bilinear ones are of order one.
problem Understanding natural differential operators on tensor fields.
method Proved all bilinear operators are of order one, then classified operators in specific cases.
result Full classification of natural differential operators on tensor fields.
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.
An ordinary differential field (F,d) of characteristic zero, a subgroup H of affine group GL(n,C)∝Cn with respect to its identical representation in Fn and the following two fields of differential rational functions in x=(x1,x2,...,xn)-column vector, $$C< x, d>^H=\{f^d< x> \in C< x, d> : f^d< hx+…
In our previous paper entitled "Axiomatic differential geometry -towards model categories of differential geometry-, we have given a category-theoretic framework of differential geometry. As the first part of our series of papers concerned with differential-geometric developments within the above axiomatic scheme, this…
Study on the smoothness of solutions to a specific type of stochastic differential equation.
problem Regularity of solutions to mean-field G-SDEs. method Analysis of first and second order Fréchet differentiability in the random initial condition.
result Established the Fréchet differentiability of the solution and specified the corresponding equations.
Study vector fields and derivations on differentiable stacks.
problem Understanding structures on differentiable stacks.
method Introduced module structures on dgla of multiplicative vector fields and graded algebra of functions on Lie groupoids.
result Associated structure of a graded Lie-Rinehart algebra on vector fields is Morita invariant.
New framework for RR fields using twisted differential K-theory.
problem Describing Ramond-Ramond fields mathematically.
method Systematic approach to twisted differential K-theory, using AHSS.
result Characterization of RR fields and their quantization.
We formulate differential cohomology and Chern-Weil theory -- the theory of connections on fiber bundles and of gauge fields -- abstractly in the context of a certain class of higher toposes that we call "cohesive". Cocycles in this differential cohomology classify higher principal bundles equipped with cohesive struct…
A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric shapes with finite number of elements (such as polyhedra)…
Einstein's philosophy uses differential identities to derive GR field equations.
problem Constructing field theories in alternative geometries.
method Explains differential identities and their role in GR and PAP-geometry.
result Derived a more general differential identity in PAP-geometry.
This paper establishes three relations between the Toda field theory associated to a simple Lie algebra and the integral curves of the standard differential system on the corresponding complete flag variety. The motivation comes from the viewpoint on the Toda field theories as Darboux integrable differential systems as…
Study geodesics on algebraic varieties over differential fields.
problem Characterize geodesics on pseudo-Riemannian algebraic varieties.
method Model-theoretic approach to algebraic differential equations.
result Algebraic differential equation describing geodesics is absolutely irreducible.
Given a triangulated region in the complex plane, a discrete vector field Y assigns a vector Yi∈C to every vertex. We call such a vector field holomorphic if it defines an infinitesimal deformation of the triangulation that preserves length cross ratios. We show that each holomorphic vector field can b…
The note answers a question about Betti numbers for 1D Euclidean space.
problem Understanding Betti numbers for vector fields and differential forms in 1D Euclidean space.
method Using Euler vector field and Lie superalgebra structure.
result The Betti numbers are 1 for the case where primary and secondary weights are equal.
Differentiable spaces derived from Lie group actions have vector fields and forms.
problem Understanding the differential structure of orbit spaces of Lie group actions.
method Analyzing the differential structure of orbit spaces of proper Lie group actions on smooth manifolds.
result Orbit spaces of Lie group actions are differentiable spaces with exterior algebra of differential forms.
Abstract formulates PDEs in synthetic geometry for enhanced contexts.
problem Solving open problem of covariant geometric pre-quantization of locally variational field theories.
method Abstract formulation in synthetic differential geometry.
result Generalization to super-geometry, higher differential geometry.
We propose definitions of homogeneity and projective equivalence for systems of ordinary differential equations of order greater than two, which allow us to generalize the concept of a spray (for systems of order two). We show that the Euler-Lagrange fields of parametric Lagrangians of order greater than one which are …
A new cohomology, induced by a vector field, is defined on pairs of differential forms (1--differentiable forms) in a manifold. It is proved a link with the classical de Rham cohomology and an 1-differentable cohomology of Lichnerowicz type associated to an one form. Also, the case when the manifold is complex and …
We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed. Special attention is paid to the case of mappings between topological vector spaces ov…
The higher gauge field in 11-dimensional supergravity -- the C-field -- is constrained by quantum effects to be a cocycle in some twisted version of differential cohomology. We argue that it should indeed be a cocycle in a certain twisted nonabelian differential cohomology. We give a simple and natural characterization…
This paper introduces the notions of vector field and flow on a general differentiable stack. Our main theorem states that the flow of a vector field on a compact proper differentiable stack exists and is unique up to a uniquely determined 2-cell. This extends the usual result on the existence and uniqueness of flows o…
Defines discrete differential geometry concepts in homotopy type theory.
problem No existing definition of Euler characteristic for comparison.
method Type families on higher inductive types, simplicial complexes, principal bundles, connections, curvature, vector fields, index.
result Theorem relating total curvature and total index, key to proving Gauss-Bonnet and Poincaré-Hopf theorems.
Defines quaternionic k-vector fields on quaternionic Kähler manifolds.
problem No specific problem stated; focuses on definition and properties.
method Introduced a modified Dirac operator to define quaternionic k-vector fields.
result Calculated the dimension of quaternionic k-vector fields on HPn. Reconstructing signature features from randomized vector fields in differential equations.
problem Reconstructing signature features from controlled differential equations with random vector fields.
method Using controlled ordinary differential equations driven by continuous bounded variation curves, the study explores the extent to which signature features can be reconstructed from the non-linear flow of these equations.
result The number of signature features that can be reconstructed from the non-linear flow of controlled ordinary differential equations with random vector fields is exponential in the hidden dimension, under certain conditions.
New differential geometry perspective on orthogonal RNNs.
problem Mitigating exploding and vanishing gradients in RNNs.
method Using tools from differential geometry, parameterizing vector fields via directional derivatives of scalar functions.
result Our approach achieves comparable or better results on benchmark tasks.
Unified physics field theories through a general conservation law.
problem Unified physics field theories.
method Introduced general field as a formal sum of differential forms, defined conservation law using action principle.
result Physics field theories become instances of the general conservation law.
Develops derived differential geometry for supermanifolds.
problem Handling non-transverse intersections and singular moduli problems in geometry and physics.
method Extends existing work on derived manifolds to supergeometric and infinite-dimensional contexts.
result Establishes foundational results relating derived differential geometry to differential operators and PDE theory.
Stability of PDEs linked to vector fields on manifolds.
problem Stability of solutions to differential equations involving vector fields.
method Analyzes Hyers-Ulam stability for global differential equations.
result Stability of solutions to the differential equation Vy=λy+f. Develops Poisson algebras for field theories using synthetic differential geometry.
problem Constructing Poisson algebras for non-linear field theories.
method Synthetic differential geometry and Cahiers topos model.
result Formulates a Poisson algebra for field theories, showing it forms a family of observables.
Study characterizes 2-Killing vector fields on complex spacetimes.
problem Characterize 2-Killing vector fields on multiply twisted product spacetimes. method Determine nonlinear differential equations, find twisted functions, provide solutions, and construct examples.
result Completely describe 2-Killing vector fields and twisted functions on multiply twisted product spacetimes. Differentiable pipeline replaces non-differentiable CAE components for shape optimization.
problem Gradient-based optimization is limited by non-differentiable components in CAE workflows.
method Surrogate models replace non-differentiable pipeline components, enabling gradient-based optimization.
result Gradient-based shape optimization possible without differentiable solvers.
Local description of solvable Lie algebras of vector fields.
problem Understanding solvable Lie algebras of vector fields.
method Local and constructive differential geometric description.
result Implication of Lie's conjecture for solvable Lie algebras.
Study linear differential operators on special manifolds.
problem Analyzing elliptic differential operators on specific types of manifolds.
method Examining a linear elliptic differential operator of the form Δ + V - λ on quasi-asymptotically conical manifolds.
result Established an isomorphism theorem for these operators.
An differential field (F;∂1,...,∂m) of characteristic zero, a subgroup H of affine group GL(n,C)∝Cn with respect to its identical representation in Fn and the following two fields of differential rational functions in x=(x1,x2,...,xn)-column vector, $$C< x, \partial >^H=\{f^{\part…
Three types of equations of mathematical physics, namely, the equations, which describe any physical processes, the equations of mechanics and physics of continuous media, and field-theory equations are studied in this paper. In the first and second case the investigation is reduced to the analysis of the nonidentical …
A parametric manifold is a manifold on which all tensor fields depend on an additional parameter, such as time, together with a parametric structure, namely a given (parametric) 1-form field. Such a manifold admits natural generalizations of Lie differentiation, exterior differentiation, and covariant differentiation, …
Spinor fields depending on tensor fields and other spinor fields are considered. The concept of extended spinor fields is introduced and the theory of differentiation for such fields is developed.
We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the second order. A result on the characterization of classes of these equations by t…
Projective geometry aids in analyzing fields near compact manifolds.
problem Analyzing fields near compact manifolds.
method Developed a projective exterior differential tractor calculus.
result Analogous calculus for projectively compact manifolds.
Paper connects differential geometry with geometric calculus.
problem None explicitly stated in the abstract.
method Introduces a more general Laplacian for multivector-valued functions on manifolds.
result Formulated a higher codimensional analog of Jacobi's field equation.
Tensor fields depending on other tensor fields are considered. The concept of extended tensor fields is introduced and the theory of differentiation for such fields is developed.
The space of differential operators acting on skewsymmetric tensor fields or on smooth forms of a smooth manifold are representations of its Lie algebra of vector fields. We compute the first cohomology spaces of these representations and show how they are related to the cohomology with coefficients in ther space of sm…
A tensor field generates separation of variables for certain metrics.
problem Finding metrics with specific tensor field properties.
method Constructing differential invariants for a (1,1)-tensor field. result Explicit system of invariants for metrics generating separation of variables.
Reviews connections in fiber and jet bundles, linking to PDEs and multivector fields.
problem Exploring different interpretations of connections in fiber and jet bundles.
method Analyzes connections in fiber and jet bundles, relating them to PDEs and multivector fields.
result Relates connections in fiber and jet bundles to PDEs and multivector fields.