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Quantum calculus I Victor Kac, Pokman Cheung. p. cm. -(Universitext) Includes bibliographical references and index.

Summary We propose a class of intrinsic Gaussian processes (GPs) for interpolation, regression and classification on manifolds with a primary focus on complex constrained domains or irregularly shaped spaces arising as subsets or submanifolds of double-struckR, double-struckR2, double-struckR3 and beyond. For example, intrinsic GPs can accommodate spatial domains arising as complex subsets of Euclidean space. Intrinsic GPs respect the potentially complex boundary or interior conditions as well as the intrinsic geometry of the spaces. The key novelty of the approach proposed is to utilize the relationship between heat kernels and the transition density of Brownian motion on manifolds for constructing and approximating valid and computationally feasible covariance kernels. This enables intrinsic GPs to be practically applied in great generality, whereas existing approaches for smoothing on constrained domains are limited to simple special cases. The broad utilities of the intrinsic GP approach are illustrated through simulation studies and data examples.

The first part of this paper provides a new description of chiral differential operators (CDOs) in terms of global geometric quantities. The main result is a recipe to define all sheaves of CDOs on a smooth cs-manifold; its ingredients consist of an affine connection ∇ and an even 3-form that trivializes p1(∇). With ∇ fixed, two suitable 3-forms define isomorphic sheaves of CDOs if and only if their difference is exact. Moreover, conformal structures are in one-to-one correspondence with even 1-forms that trivialize c1(∇).Applying our work in the first part, we construct what may be called "chiral Dolbeault complexes" of a complex manifold M , and analyze conditions under which these differential vertex superalgebras admit compatible conformal structures or extra gradings (fermion numbers). When M is compact, their cohomology computes (in various cases) the Witten genus, the two-variable elliptic genus and a spin c version of the Witten genus. This part contains some new results as well as provides a geometric formulation of certain known facts from the study of holomorphic CDOs and σ-models. §1. IntroductionIn physics, the study of a type of quantum field theory called σ-models has inspired many important insights in topology and geometry. The theory of elliptic genera is an example. In particular, associated to any compact, string manifold 1 M is a σ-model whose "partition function" equals, up to a constant factor, the formal power seriesknown as the Witten genus of M . [Wit87,Wit88] Similarly, associated to any compact, spin manifold M is another σ-model, which gives rise to the formal power seriesknown as the Ochanine elliptic genus of M . [Och87, Wit87] The physical interpretation of these topological invariants have led to predictions that are not immediately clear from the mathematical point of view. Even though many of them have since been verified, e.g. [Zag88, BT89], a complete, geometric understanding of elliptic genera has yet to emerge. The latter probably requires to some extent a mathematical framework for σ-models.Sheaves of vertex algebras provide a mathematical approach to σ-models. Important constructions along this line include the chiral de Rham complex and, more generally, sheaves of chiral differential operators, or CDOs. [MSV99, GMS00] In particular, a complex manifold M admits a sheaf of holomorphic 1 Let λ ∈ H 4 (BSpin; Z) ∼ = Z be the generator such that 2λ = p 1 . This defines a characteristic class λ(·) for spin vector bundles. A spin manifold M is said to be string if λ(T M ) = 0. Moreover, a string structure on M is a "trivialization of λ(T M )", i.e. a homotopy class of liftings of the classifying map M → BSpin along the homotopy fiber of λ : BSpin → K(Z, 4). 1CDOs D ch M with a conformal structure if and only if c hol 1 (T M ) = c hol 2 (T M ) = 0; 2 notice that M as a spin c manifold admits a string structure if and only if c 1 (suggesting a connection between D ch M and the σ-model underlying the Witten genus. In fact, physicists have recognized a connection between CDO...

Gaussian processes (GPs) are very widely used for modeling of unknown functions or surfaces in applications ranging from regression to classification to spatial processes. Although there is an increasingly vast literature on applications, methods, theory and algorithms related to GPs, the overwhelming majority of this literature focuses on the case in which the input domain corresponds to a Euclidean space. However, particularly in recent years with the increasing collection of complex data, it is commonly the case that the input domain does not have such a simple form. For example, it is common for the inputs to be restricted to a non-Euclidean manifold, a case which forms the motivation for this article. In particular, we propose a general extrinsic framework for GP modeling on manifolds, which relies on embedding of the manifold into a Euclidean space and then constructing extrinsic kernels for GPs on their images. These extrinsic Gaussian processes (eGPs) are used as prior distributions for unknown functions in Bayesian inferences. Our approach is simple and general, and we show that the eGPs inherit fine theoretical properties from GP models in Euclidean spaces. We consider applications of our models to regression and classification problems with predictors lying in a large class of manifolds, including spheres, planar shape spaces, a space of positive definite matrices, and Grassmannians.Our models can be readily used by practitioners in biological sciences for various regression and classification problems, such as disease diagnosis or detection. Our work is also likely to have impact in spatial statistics when spatial locations are on the sphere or other geometric spaces.