Extends Euler calculus to continuous integrands using curvature.
problem Limitations of Euler calculus with simple functions.
method Integrates with respect to Gaussian curvature within O-minimal theories.
result Satisfies a Fubini theorem and extends to a functor.
Develops a graphical calculus for stable curvature invariants.
problem Calculating stable curvature invariants of Riemannian manifolds.
method Graphical calculus based on trivalent graphs with colored edges.
result Derives a curvature identity for compact Einstein manifolds.
Paper generalizes Bakry-Émery calculus for curvature and applies to Markov chains.
problem Formulating both Bakry-Émery and entropic curvature simultaneously.
method Generalization of Bakry-Émery calculus, new measure optimality criterion, dimension parameter in entropic curvature.
result Diameter estimates for Markov chains with strictly positive entropic curvature and spectral gap.
Study on stochastic mean curvature flow on networks using Ito calculus.
problem Understanding the dynamics of network structures under random influences.
method Application of Ito calculus to derive a stochastic differential equation (SDE) for network edges.
result New insights into the stability, long-term behavior, and pattern formation of complex networks under stochastic influences.
Develops calculus for tamed Dirichlet spaces using measure theory.
problem Defines calculus for measure spaces with Dirichlet forms.
method Introduces first and second order calculus on tamed Dirichlet spaces.
result Defines various geometric objects on tamed Dirichlet spaces.
Study volatility of forward-start options using Malliavin Calculus.
problem Implied volatility of Forward-Start options, focusing on ATM behavior.
method Closed-form expressions derived using Malliavin Calculus in Markovian models.
result Derives expressions for at-the-money, skew, and curvature of forward implied volatility.
We describe an elementary algorithm for expressing, as explicit formulae in tractor calculus, the conformally invariant GJMS operators due to C.R. Graham et alia. These differential operators have leading part a power of the Laplacian. Conformal tractor calculus is the natural induced bundle calculus associated to the …
Derives sub-Riemannian Ricci curvature for various manifolds.
problem Calculating Ricci curvature in sub-Riemannian geometry.
method Generalized Gamma z calculus and z--Bochner's formula. result Analytical bounds for sub-Riemannian curvature dimension and log-Sobolev inequalities.
Simplicial, piecewise-flat discretizations of manifolds provide a clear path towards curvature analysis on discrete geometries and for solutions of PDE's on manifolds of complex topologies. In this manuscript we review and expand on discrete exterior calculus methods using hybrid domains. We then analyze the geometric …
Introduces non-regular spacetime geometry without smooth calculus.
problem Defining gravity without smooth spacetime geometry.
method Discusses non-regular spacetime geometry and curvature without differential calculus.
result Curvature and gravity can be defined without smooth spacetime calculus.
We introduce a new version of a curvature-dimension inequality for non-negative curvature. We use this inequality to prove a logarithmic Li-Yau inequality on finite graphs. To formulate this inequality, we introduce a non-linear variant of the calculus of Bakry and Émery. In the case of manifolds, the new calculus and …
A quaternionic calculus for surface pairs in the conformal 4-sphere is elaborated. This calculus is then used to discuss the relation between curved flats in the symmetric space of point pairs and Darboux and Christoffel pairs of isothermic surfaces. A new viewpoint on relations between surfaces of constant mean curvat…
Study on skew and curvature of implied and local volatilities using Malliavin calculus.
problem Relationship between short-end of local and implied volatility surfaces.
method Malliavin calculus techniques
result Recover the $ rac{1}{H+3/2}$ rule for rough volatilities and relationships between skew and curvature.
The paper computes CR GJMS operators and CR tractor calculus for pseudo-Einstein forms.
problem Computing CR GJMS operators and CR tractor calculus for pseudo-Einstein forms.
method Algorithm using CR tractors to compute CR GJMS operators, P' operator, and Q' curvature.
result Explicit factorization of CR GJMS operators and P' operator, and constant Q' curvature.
Survey on Finsler manifolds with weighted Ricci curvature, focusing on geometric analysis.
problem Analysis of Finsler manifolds with weighted Ricci curvature.
method Nonlinear geometric analysis based on the Bochner inequality.
result Gradient estimates, functional inequalities, and isoperimetric inequalities.
Modular method simplifies curvature computation in neural nets.
problem Efficient computation of curvature matrices for training neural nets.
method Modular backpropagation for block-diagonal approximations.
result Compact notation and easy integration into machine learning libraries.
Study proves Lp properties for Hodge-Dirac operator on curved spaces.
problem Analyzing Lp-behavior of Hodge-Dirac operator on curved manifolds. method Proves R-bisectoriality and boundedness of H∞-functional calculus in Lp. result Establishes Lp-properties for Hodge-Dirac operator on manifolds with non-negative Ricci curvature. First we express the holonomy along a boundary curve as the integral on the domain, of an expression which is linear in the curvature. Then we provide a rigorous justification of the definition of curvature in Regge calculus.
Proposes a new nonlocal curvature tensor concept.
problem Various nonlocal curvature concepts in literature.
method Generalizes classical curvature tensor representation and uses fractional differential operator analogies.
result Introduces a new nonlocal curvature tensor.
Develops a new calculus for volumes of singular metrics.
problem Calculating anomalies and divergences for volumes of singular metrics.
method Distributional calculus and boundary calculus for conformally compact manifolds.
result Explicit formulae for divergences and anomaly of regulated volumes.
In this paper, we extend the DC Calculus introduced by Perelman on finite dimensional Alexandrov spaces with curvature bounded below. Among other things, our results allow us to define the Hessian and the Laplacian of DC functions (including distance functions as a particular instance) as a measure-valued tensor and a …
Introduces a new tensor calculus for tensors dependent on direction.
problem Handling tensors that vary with direction in a consistent manner.
method Develops anisotropic tensor calculus with applications in manifold derivations and Finsler metrics.
result Defines anisotropic linear connections and curvature tensors.
Study of multidifferential operators and Dorfman connections on Courant algebroids.
problem Exploring multidifferential operators and Dorfman connections on Courant algebroids.
method Construction of an algebra and complex of multidifferential operators, study of Dorfman connections.
result Cartan calculus, curvatures of induced connections and basic differential geometric identities make sense in the constructed algebra.
Master thesis proves Bergman kernel asymptotics for positive line bundles.
problem Proving asymptotic expansion of Bergman kernel for positive line bundles.
method Introduced a semi-classical symbol space and symbolic calculus.
result Established pointwise asymptotic expansion on positive parts of certain semi-positive line bundles.
In Finsler geometry, we use calculus to study the geometry of regular inner metric spaces. In this note I will briefly discuss various curvatures and their geometric meanings from the metric geometry point of view, without going into the forest of tensors.
We discuss in which sense general metric measure spaces possess a first order differential structure. Building on this, we then see that on spaces with Ricci curvature bounded from below a second order calculus can be developed, permitting to define Hessian, covariant/exterior derivatives and Ricci curvature.
Abstract: Develops universal formulae for Q-curvatures and related operators in conformal geometry.
problem Q-curvatures and related operators in conformal geometry of embedded manifolds with boundary.
method Holographic construction using singular Yamabe problem and minimal hypersurface with boundary.
result Universal formulae for extrinsic Q-curvatures and associated operators.
The paper extends curvature concepts to a broader class of noncommutative manifolds.
problem Developing a good notion of intrinsic curvature in noncommutative geometry.
method Extending pseudo-differential calculus and modular geometry to toric noncommutative manifolds.
result Deriving a general expression for modular curvature on toric noncommutative manifolds.
Extends Hess-Schrader-Uhlenbrock inequality for 1-forms in tamed Dirichlet spaces.
problem Establishing a new inequality for 1-forms in tamed Dirichlet spaces.
method Developed a vector calculus for tamed Dirichlet spaces and applied it to establish the inequality.
result Established the Hess-Schrader-Uhlenbrock inequality for 1-forms in L2-cotangent module. We provide a quick overview of various calculus tools and of the main results concerning the heat flow on compact metric measure spaces, with applications to spaces with lower Ricci curvature bounds. Topics include the Hopf-Lax semigroup and the Hamilton-Jacobi equation in metric spaces, a new approach to differentiati…
The paper explores self-adjointness of Laplace-Beltrami operator on special geometric manifolds.
problem Characterizing self-adjoint extensions of the Laplace-Beltrami operator on α-Grushin manifolds. method Introducing an exotic calculus of pseudodifferential operators adapted to the geometry of the singularity.
result Criterion for essential self-adjointness and determination of several self-adjoint extensions.
Generalizes beam models to include curvature and torsion.
problem Modeling curvature and torsion in unidimensional structures.
method Generalized Euler-Bernoulli and Timoshenko beam models using non-holonomic kinematics.
result Generalized beams exhibit curvature and torsion, with torsion in Timoshenko model.
We consider a periodic problem for the motion of a charged particle in a magnetic field. Introducing a notion of Ricci curvature for such Lagrangian systems and using the methods of the calculus of variations in the large, we prove the existence of periodic motions for such particles under a condition of positivity of …
We develop a frame and dyad gauge-independent formalism for the calculus of variations of functionals involving spinorial objects. As part of this formalism we define a modified variation operator which absorbs frame and spin dyad gauge terms. This formalism is applicable to both the standard spacetime (i.e. SL(2,C)) 2…
The study develops inequalities for Riemannian foliations without bundle-like assumptions.
problem Developing inequalities for Riemannian foliations without restrictive conditions.
method Bochner theory and Bakry-Emery calculus for horizontal Laplacians, derived explicit Bochner formulas, generalized curvature dimension inequalities.
result Established generalized curvature dimension inequalities for Riemannian foliations.
New framework for diffusion geometry simplifies complex calculations.
problem Challenges in applying calculus and geometry to real data.
method Reformulates calculus and geometry via diffusion processes.
result Improves precision, robustness, and computational efficiency.
CMS formulation solves Poincare conjecture for all dimensions.
problem Poincaré Conjecture in higher dimensions.
method Calculus of moving surfaces (CMS) for evolving hypersurfaces.
result Compact simply connected hypersurfaces relax to constant mean curvature (CMC) manifolds.
Geometric calculus on probability simplex using Wasserstein metric.
problem Calculus on the probability simplex with Wasserstein metric.
method Embedding probability simplex in positive measure space with nonlinear metric tensor, deriving Christoffel symbols, connections, curvature tensors, and operators.
result Established geometric computations on probability manifold and density space, connecting Fisher-Rao and optimal transport metrics.
On conformally compact manifolds of arbitrary signature, we use conformal geometry to identify a natural (and very general) class of canonical boundary problems. It turns out that these encompass and extend aspects of already known holographic bulk-boundary problems, the conformal scattering description of boundary con…
Develops connections and characteristic classes for Courant algebroids.
problem Developing a theory for Courant algebroids.
method Explicit description of cochain complex, Courant algebroid connections, special class of connections.
result Construction of secondary characteristic classes for Courant algebroids.
Study of curves in Lie sphere geometry using moving frames and variational principles.
problem Characterize curves in Lie sphere geometry using Lie curvatures.
method Moving frames, exterior differential systems, and calculus of variations.
result Critical curves are uniquely determined by Lie curvatures.
The elastica is a curve in R3 that is stationary under variations of the integral of the square of the curvature. Elastica is viewed as a dynamical system that arises from the second order calculus of variations, and its quantization is discussed.
I classify the Finsler structures on the 2-sphere that have constant Finsler-Gauss curvature and whose geodesics are the great circles. Modulo diffeomorphism, there is a 2-parameter family of such Finsler structures, only one of which is homogeneous or symmetric, namely the Riemannian one. I discuss the history of the …
The algebra of differential geometry operations on symmetric tensors over constant curvature manifolds forms a novel deformation of the sl(2,R) [semidirect product] R^2 Lie algebra. We present a simple calculus for calculations in its universal enveloping algebra. As an application, we derive generating functions for t…
Global calculus on Wasserstein space defined for Riemannian manifolds.
problem Developing a global differential calculus on Wasserstein spaces.
method Derivations of cylinder functions, Levi-Civita connection, extended Otto metric.
result Global differential approach to Wasserstein spaces reveals intrinsic and extrinsic curvature.
Study extends convexity in curved spaces using fractional integrals.
problem Extending convexity to curved spaces with nonpositive curvature.
method Introducing (geodesically) h-convex functions and using Katugampola's fractional integrals. result Essentially sharp estimate involving squared distance mappings.
Study functional inequalities on non-reversible Finsler manifolds.
problem Functional inequalities on non-reversible Finsler manifolds.
method Application of Bochner inequality and Γ-calculus.
result Dimensional versions of Poincare--Lichnerowicz, logarithmic Sobolev, and Sobolev inequalities hold for non-reversible metrics.
We construct the generalized version of covariant Z_3-graded differential calculus introduced by one of us (R.K.), and then extended to the case of arbitrary Z_N grading. Here our main purpose is to establish the recurrence formulae for the N-th power of covariant q-differential D_q = d_q + A and to analyze more closel…