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Proofs are given of two theorems of Berezin and Karpelevič, which as far as we know never have been proved correctly. By using eigenfunctions of the Laplace-Beltrami operator it is shown that the spherical functions on a complex Grassmann manifold are given by a determinant of certain hypergeometric functions. By application of this result, it is proved that a certain system of operators, fow which explicit expressions are given, generates the algebra of radial parts of invariant differential operators.

We study the effective Batalin-Vilkovisky quantization theory for chiral deformation of two dimensional conformal field theories. We establish an exact correspondence between renormalized quantum master equations for effective functionals and Maurer-Cartan equations for chiral vertex operators. The generating functions are proven to have modular property with mild holo- morphic anomaly. As an application, we construct an exact solution of quan- tum B-model (BCOV theory) in complex one dimension that solves the higher genus mirror symmetry conjecture on elliptic curves.

The case of dimension two was proved by Andreotti-Frankel [-7] and the case of dimension three by Mabuchi [18] using the result of Kobayashi-Ochiai [12]. Our method of proof uses harmonic maps and the characterization of the complex projective space obtained by Kobayashi-Ochiai [15]. According to the result of Kobayashi-Ochiai the complex projective space of dimension n is characterized by the fact that its first Chern class equals 2c1 (F) for some 2> n+ 1 and some positive holomorphic line bundle F over it. Since by the result of Bishop-Goldberg [-2] the second Betti number of a compact Kiihler manifold M of positive bisectional curvature is 1, for the Main Theorem it suffices to show that ca (M) is 2 times a generator of H2 (M, 7Z.) for some 2> 1+ dim M. This can be done by proving that a generator of the free part of H2 (M, Z) can be represented by a rational curve, because the tangent bundle of M restricted to the rational curve splits into a direct sum of holomorphic line bundles over the rational curve according to the result of Grothendieck [-11]. The existence of the rational curve is obtained in the following way. According to the result of Sacks-Uhlenbeck [22] and its improved formulation by Meeks-Yau [-19], the infimum of the energies of maps from S 2 to M representing the generator of rcz (M) can be achieved by a sum of stable harmonic maps f~ from S 2 to M (1< i< m). The key step in our proof is to show that each f~ is either holomorphic or conjugate holomorphic. The known methods of proving the complex-analyticity of a harmonic map use the formula for the Laplacian of the energy function [23, 25, 26] or a variation of it [24]. Here we use

Single-cell (sc)RNA-seq, together with RNA velocity and metabolic labeling, reveals cellular states and transitions at unprecedented resolution. Fully exploiting these data, however, requires kinetic models capable of unveiling governing regulatory functions. Here, we introduce an analytical framework dynamo (https://github.com/aristoteleo/dynamo-release), which infers absolute RNA velocity, reconstructs continuous vector fields that predict cell fates, employs differential geometry to extract underlying regulations, and ultimately predicts optimal reprogramming paths and perturbation outcomes. We highlight dynamo’s power to overcome fundamental limitations of conventional splicing-based RNA velocity analyses to enable accurate velocity estimations on a metabolically labeled human hematopoiesis scRNA-seq dataset. Furthermore, differential geometry analyses reveal mechanisms driving early megakaryocyte appearance and elucidate asymmetrical regulation within the PU.1-GATA1 circuit. Leveraging the least-action-path method, dynamo accurately predicts drivers of numerous hematopoietic transitions. Finally, in silico perturbations predict cell-fate diversions induced by gene perturbations. Dynamo, thus, represents an important step in advancing quantitative and predictive theories of cell-state transitions.