Nevanlinna theory and Diophantine approximation


William Cherry: Superficial connections between the value distribution of complex meromorphic functions of zero order and that of non-Archimedean meromorphic functions

Abstract: There are a number of similarities among the  value distribution of rational functions, complex meromorphic functions of zero order, and non-Archimedean meromorphic functions of arbitrary order. I will survey some of what is known and not known about these various types of functions and highlight some open questions.


Liang-Chung Hsia: On the GCD problem for sequences arising from iterates of polynomials

Abstract: We will discuss the greatest common divisors (GCD) of sequences arising from iterates of non-constant polynomials. Let f, g, and c be polynomials in C[x]. We are interested in giving ‘natural conditions’ on f and g under which there are at most finitely many λ in C with the property that there is an n such that (x - λ) divides gcd(fn(x)-c(x),gn(x)-c(x)). This can be viewed as a compositional analog of Ailon-Rudnick's theorem on the GCD of sequences arising from powers of polynomials.  Besides composition analogues of Ailon-Rudnick's results, if time permits, we will  also discuss similar questions for elliptic sequences.


Nguyễn Thị Nhung: A non-integrated defect relation and uniqueness problems for meromorphic mappings from complete Kähler manifolds into projective spaces

Abstract: Let M be a complete Kähler manifold, whose universal covering is biholomorphic to a ball B(R0)⊆Cm, where 0 < R0 ≤ ∞. We establish a truncated non-integrated defect relation for meromorphic mappings from M into Pn(C) intersecting hypersurfaces in subgeneral position. We also study uniqueness problems for meromorphic mappings from M into Pn(C) sharing hyperplanes in general position under a general condition that the intersections of inverse images of any k + 1 hyperplanes are of codimension at least two. This is joint work with S. D. Quang, N. Q. Phuong, and L. N. Quynh.


Min Ru: Recent developments on  holomorphic curves into projective varieties

Abstract: In this talk, I will report some of the recent progress on the study of  holomorphic curves into projective varieties, including joint with with Paul Vojta as well as joint work with Julie Wang.


Đỗ Đức Thái: On integral points off divisors in subgeneral position in projective algebraic varieties

Abstract: The purpose of this talk is to present the following in Diophantine Geometry.

  1. The first is to show the dimension of the set of integral points off divisors in subgeneral position in a projective algebraic variety. As a consequence, results of Ru-Wong, Ru, Noguchi-Winkelmann, and Levin are recovered.
  2. The second is to show the complete hyperbolicity in the sense of Kobayashi of the complement of divisors in subgeneral position in a projective algebraic variety.
  3. Let k be a number field and S a finite set of valuations of k containing the archimedean valuations. The third is to determine when there exists a Zariski-dense set R of S-integral points on the complement of a union of divisors in projective space over k.


Nguyễn Văn Thìn: A non-integrated defect relation and uniqueness problems for meromorphic

mappings from complete Kähler manifolds into projective spaces

Abstract: This talk contains two parts. In the first part, we give some applications of Nevanlinna theory to normal families and normal functions. More specifically, we give extensions of Montel’s criterion and Lappan’s five point theorem. In the second part, we present a second main theorem for holomorphic curves intersecting one hypersurface in projective space.


Paul Vojta: Birational Nevanlinna constants and an example of Faltings

Abstract: In 2002, Corvaja and Zannier obtained a new proof of Siegel's theorem (on integral points on curves) based on Schmidt's celebrated Subspace Theorem. Soon after that (and based on earlier work), Evertse and Ferretti applied Schmidt's theorem to give diophantine results for homogeneous polynomials of higher degree on a projective variety in Pn.  This has led to further work of A. Levin, P. Autissier, M. Ru, G. Heier, and others. In particular, Ru has defined a number, Nev(D), that concisely describes the best diophantine approximation obtained by this method, where D is an effective Cartier divisor on a projective variety X.  In this talk, I will give an overview of variants of these constants as developed by Ru and myself, and indicate how an example of Faltings can be derived using these constants.


Julie Tzu-Yueh Wang: Asymptotic GCD and divisible sequences for entire functions

Abstract: Let f and g be algebraically independent entire functions. We first give an estimate of the Nevanlinna counting function for the common zeros of fn-1 and gn-1 for sufficiently large n.  We then apply this estimate to study divisible sequences in the sense that fn-1 is divisible by gn-1 for infinitely many n under the assumption that f and g are multiplicatively independent. For the first part of establishing our gcd estimate, we need to formulate a truncated second main theorem for effective divisors by modifying  a theorem of Hussein and Ru explicitly computing the constants involved for a blow-up of  P1×P1 along a point with its canonical divisor and the pull-back of vertical and horizontal divisors of P1×P1. This is joint work with Ji Guo.


Gisbert Wüstholz: Periods & Elliptic Billiards

Abstract: On the triaxial ellipsoid there are various classical dynamical systems. In the talk we shall report on joint work of Ronald Garcia on curvature lines and geodesics. We are dealing with the question under which conditions these curves are closed or not. For geodesics,  using nice results, we were recently able to give a complete answer if the ellipsoid is defined over a number field. For curvature lines, we show that there is a countable set of so-called α-curvature lines which are not closed. These are classical problems in the theory of dynamical systems and there were almost no results in this direction. We also discuss the cases of ellipsoids of revolution and ellipsoids in Minkowski space. The proofs make basic use of the analytic subspace theorem in the case of elliptic periods, and periods on abelian surfaces come up naturally. In the elliptic case we solve an extended problem of Th. Schneider dealing with periods of differentials of the third kind.