We survey the mathematical literature on umbral calculus (otherwise known as the calculus of finite differences) from its roots in the 19th century (and earlier) as a set of "magic rules" for lowering and raising indices, through its rebirth in the 1970’s as Rota’s school set it on a firm logical foundation using operator methods, to the current state of the art with numerous generalizations and applications. The survey itself is complemented by a fairly complete bibliography (over 500 references) which we expect to update regularly.

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A selected survey of umbral calculus

Given a rational map R, we consider the complement of the postcritical set $$S_R$$
 . In this paper we discuss the existence of invariant Beltrami differentials supported on an R invariant subset X of $$S_R$$
 . Under some geometrical restrictions on X, we show the absence of invariant Beltrami differentials with support intersecting X. In particular, we show that if X has finite hyperbolic area, then X cannot support invariant Beltrami differentials except in the case where R is a Lattes map.

On Hyperbolic Metric and Invariant Beltrami Differentials for Rational Maps

Given a rational map $R$, we consider the complement of the postcritical set $S_R$. In this paper we discuss the existence of invariant Beltrami differentials supported on a $R$ invariant subset $A$ of $S_R$. Under some geometrical restrictions, either on the hyperbolic geometry of $A$ or on the asymptotic behavior of infinitesimal geodesics of the Teichm\"uller space of $S_R$, we show the absence of invariant Beltrami differentials supported on $A$. In particular, we show that if $A$ has finite hyperbolic area, then $A$ can not support invariant Beltrami differentials except in the case where $R$ is a Latt\`es map.

On hyperbolic metric and asymptotically finite invariant differentials in holomorphic dynamics

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A canonical decomposition of automorphic forms which vanish on an invariant measurable subset

We generalize Earle-Li's polydisk theorem and embedding theorem, and study isometries from the unit disk to infinite dimensional Teichmuller spaces. We also give a simple proof that for any non-Strebel point τ, there exist infinitely many real analytic geodesic disks through τ and the basepoint in infinitely dimensional Teichmuller spaces.

Hiromi Ohtake

Papers

A canonical decomposition of automorphic forms which vanish on an invariant measurable subset

On the isometries from the unit disk to infinite dimensional Teichmüller spaces