Conformal Transformations and the SLE Partition Function Martingale

We present an implementation in conformal field theory (CFT) of local finite conformal transformations fixing a point. We give explicit constructions when the fixed point is either the origin or the point at infinity. Both cases involve the exponentiation of a Borel subalgebra of the Virasoro algebra. We use this to build coherent state representations and to derive a close analog of Wicks theorem for the Virasoro algebra. This allows to compute the conformal partition function in non trivial geometries obtained by removal of hulls from the upper half-plane. This is then applied to stochastic Loewner evolutions (SLE). We give a rigorous derivation of the equations, obtained previously by the authors, that connect the stochastic Loewner equation to the representation theory of the Virasoro algebra. We give a new proof that this construction enumerates all polynomial SLE martingales. When one of the hulls removed from the upper half-plane is the SLE hull, we show that the partition function reduces to a useful local martingale known to probabilists, thereby unraveling its CFT origin. Communicated by Vincent RivasseauSubmitted 13/06/03, accepted 21/10/03     

Cyclic cocycles on deformation quantizations and higher index theorems

We construct a nontrivial cyclic cocycle on the Weyl algebra of a symplectic vector space. Using this cyclic cocycle we construct an explicit, local, quasi-isomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof. We give a new proof of Nest-Tsygan's algebraic higher index theorem by computing the pairing between such cyclic cocycles and the K-theory of the formal deformation quantization. Furthermore, we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold. As an application, we obtain the analytic higher index theorem of Connes-Moscovici and its extension to orbifolds.     

Sur l'histoire des démonstrations analytiques du théorème fondamental de l'algèbre

The first proof of the fundamental theorem of algebra, proposed by D'Alembert in 1746 and practically unknown to this day, stimulated a series of “analytic” proofs which made essential use of the properties of polynomials as analytic functions and placed the theorem within complex analysis. One of the simplest and most widely known modern proofs is formed from the “analytic” proofs of Gauss, Argand, Legendre and Cauchy, which used and developed the ideas of D'Alembert.     

Théorème de Gabrielov et fonctions log-exp-algébriques

We give a new proof of Wilkie 's theorem on log-exp algebraic functions. The tools used here are “explicit” Gabrielov's theorem and our geometric presentation of the theorem of van den Dries, Macintyre, and Marker on log-exp analytic functions.     

半平面中有限阶解析函数的因子分解

与经典有限阶整函数的Hadamard因子分解定理和半平面中属于Hardy空间的解析函数的内外函数的因子分解类似,对右半平面中有限阶ρ解析函数f,可以分解为三个解析函数G,eQ和eg的乘积GeQeg,其中G是一个加权Blaschke乘积,Q是一个次数不超过ρ的多项式以及eg是一个加权外函数,log|G|,ReQ和Reg-log|f|在右半平面的边界恒为零．     

Spin geometry of Dirac and noncommutative geometry of Connes

The review is devoted to the interpretation of the Dirac spin geometry in terms of noncommutative geometry. In particular, we give an idea of the proof of the theorem stating that the classical Dirac geometry is a noncommutative spin geometry in the sense of Connes, as well as an idea of the proof of the converse theorem stating that any noncommutative spin geometry over the algebra of smooth functions on a smooth manifold is the Dirac spin geometry.     

Some new results on domains in complex space with non-compact automorphism group

We provide a new proof of the Wong-Rosay theorem, using the structure of the ring of holomorphic functions. As a byproduct, we provide an analogous theorem for classical bounded symmetric domains. The second main result of this article concerns a new existence theorem for holomorphic peaking functions at a hyperbolic orbit accumulation boundary point. Finally, we give a proof of a version of the Greene-Krantz conjecture using holomorphic vector fields and a strengthened Hopf lemma.     

Fundamental Theorem of Calculus

A simple but rigorous proof of the Fundamental Theorem of Calculus is given in geometric calculus, after the basis for this theory in geometric algebra has been laid out. Various classical examples of this theorem, such as the Green’s and Stokes’ theorem are discussed, as well as the theory of monogenic functions which generalizes analytic functions of a complex variable to higher dimensions.     

Zeta functions and analyticity of metric entropy for anosov systems

In this note we shall give an alternative proof, using generalized zeta functions, of a theorem of Contreras that the metric entropy of aC ^ω Anosov diffeomorphism or flow has a real analytic dependence on perturbations.     

Spectral flow is the integral of one forms on the Banach manifold of self adjoint Fredholm operators

One may trace the idea that spectral flow should be given as the integral of a one form back to the 1974 Vancouver ICM address of I.M. Singer. Our main theorem gives analytic formulae for the spectral flow along a norm differentiable path of self adjoint bounded Breuer-Fredholm operators in a semifinite von Neumann algebra. These formulae have a geometric interpretation which derives from the proof. Namely we define a family of Banach submanifolds of all bounded self adjoint Breuer-Fredholm operators and on each submanifold define global one forms whose integral on a norm differentiable path contained in the submanifold calculates the spectral flow along this path. We emphasise that our methods do not give a single globally defined one form on the self adjoint Breuer-Fredholms whose integral along all paths is spectral flow rather, as the choice of the plural ‘forms’ in the title suggests, we need a family of such one forms in order to confirm Singer's idea. The original context for this result concerned paths of unbounded self adjoint Fredholm operators. We therefore prove analogous formulae for spectral flow in the unbounded case as well. The proof is a synthesis of key contributions by previous authors, whom we acknowledge in detail in the introduction, combined with an additional important recent advance in the differential calculus of functions of non-commuting operators.     

Joint value-distribution theorems on Lerch zeta-functions. II

We give corrected statements of some theorems from [5] and [6] on joint value-distribution of Lerch zeta-functions (limit theorems, universality, functional independence). We also present a new direct proof of a joint limit theorem in the space of analytic functions and an extension of a joint universality theorem. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 332–350, July–September, 2006.     

A note on the extension of the null-additive set function on the algebra of subsets

In this work, we point out that the proof of Theorem 2 in [E. Pap, Extension of null-additive set functions on algebra of subsets, Novi Sad J. Math. 31 (2) (2001) 9–13] is incorrect and give a correct proof. Moreover, we also get a corresponding theorem on extension of the weakly null-additive set function.     

Strongly stable real infinitesimally symplectic mappings

We prove that a mapA εsp(σ,R), the set of infinitesimally symplectic maps, is strongly stable if and only if its centralizerC(A) insp(σ,R) contains only semisimple elements. Using the theorem that everyB insp(σ,R) close toA is conjugate by a real symplectic map to an element ofC(A), we give a new proof of the openness of the set of strongly stable maps. Then we prove that the set of strongly stable maps is the interior of the set of all infinitesimally symplectic maps with purely imaginary or zero eigenvalues, and the connected components of this set are described. Finally, we give a new proof of the analytic conjugacy theorem for an analytic curve through a given strongly stable map.     

Jensen measures and analytic multifunctions

This paper describes plurisubharmonic convexity and hulls, and also analytic multifunctions in terms of Jensen measures. In particular, this allows us to get a new proof of Słodkowski's theorem stating that multifunctions are analytic if and only if their graphs are pseudoconcave. We also show that multifunctions with plurisubharmonically convex fibers are analytic if and only if their graphs locally belong to plurisubharmonic hulls of their boundaries. In the last section we prove that minimal analytic multifunctions satisfy the maximum principle and give a criterion for the existence of holomorphic selections in the graphs of analytic multifunctions. The author was partially supported by an NSF Grant.     

Wakimoto modules, opers and the center at the critical level

Wakimoto modules are representations of affine Kac-Moody algebras in Fock modules over infinite-dimensional Heisenberg algebras. In this paper, we present the construction of the Wakimoto modules from the point of view of the vertex algebra theory. We then use Wakimoto modules to identify the center of the completed universal enveloping algebra of an affine Kac-Moody algebra at the critical level with the algebra of functions on the space of opers for the Langlands dual group on the punctured disc, giving another proof of the theorem of B. Feigin and the author.     

An additive formula for Samuel multiplicities on Hilbert spaces of analytic functions

We establish a short exact sequence to relate the germ model of invariant subspaces of a Hilbert space of vector-valued analytic functions and the sheaf model of the corresponding coinvariant subspaces. As a consequence we obtain an additive formula for Samuel multiplicities. As an application, we give a different proof for a formula relating the fibre dimension and the Samuel multiplicity which is first proved in Fang (2005) [11]. The feature of the new proof is that the analytic arguments in Fang (2005) [11] are now subsumed by algebraic machinery.     

Differentials of the second kind on mumford curves

We define ap-adic analytic Hodge decomposition for the cohomology of Mumford curves, with values in a local system. When the local system is trivial, we give a new proof of Gerritzen’s theorem, that this decomposition forms a variation of Hodge structure, in a family of Mumford curves.     

On the Bicomplex Gleason–Kahane–Żelazko Theorem

We prove that a bicomplex linear functional acting on a bicomplex Banach algebra (with a hyperbolic-valued norm) in such a way that invertible elements are transformed into invertible bicomplex numbers is, in fact, a multiplicative functional and thus, an algebra homomorphism. We give two proofs of this. The first of them is based on the theory of bicomplex holomorphic functions and we present here a number of previously not published facts; the second uses its complex antecedent (classic Gleason–Kahane–?elazko theorem).     

A note on the existence, uniqueness and symmetry of par-balanced realizations

We give a proof of the realization theorem of N.J. Young which states that analytic functions which are symbols of bounded Hankel operators admit par-balanced realizations. The main tool used in this proof is the induced Hilbert spaces and a lifting lemma of Kreîn-Reid-Lax-Dieudonné. Alternatively one can use the Loewner inequality. A short proof of the uniqueness of par-balanced realizations is included. As an application, it is proved that par-balanced realizations of real symmetric transfer functions areJ-self-adjoint.Research supported in part by the Romanian Academy grant GAR-6645/1996.This research was supported in part by NSF grant DMS-9501223.