Abstracts

Lina Lee (U of M), Asymptotic behavior of invariant metrics

As a point inside a domain moves toward the boundary, biholomorphically invariant metrics behave according to the characteristics of the domain. If a domain is holomorphically convex, one naturally expects it to be complete with respect to invariant metrics and the metric blowing up near the boundary. We explore the asymptotic behavior of invariant metrics near the boundary of pseudoconvex domains, which are known to be holomorphically convex, and show that the metrics blow up at a rate that is determined by the geometry of the boundary.

Debraj Chakrabarti (U of Western Ontario), Holomorphic Extension of CR functions from Quadratic Cones

We consider holomorphic extension properties of CR functions defined on real submanifolds with singularities. It is proved that for a a quadratic cone M, a CR function admits one-sided extension iff it does not have two sided support, a geometric condition that generalizes minimality in the sense of Tumanov.

Alexander Izzo (Bowling Green State University), Function Algebras on the Sphere Invariant under Groups Actions

We will answer a question raised by Ronald Douglas in connection with his work on a conjecture in operator theory due to William Arveson. Let S denote the unit sphere in ℂⁿ. If A is a function algebra on S that contains the ball algebra A(S) and whose maximal ideal space is S, and if A is invariant under the action of the n-torus on S, does it follow that A = C (S) ? When n=1, Wermer's maximality theorem gives immediately that the answer is yes. In higher dimensions the answer depends on the dimension.

Dror Varolin (SUNY Stony Brook), Powers of Hermitian polynomials

A Hermitian polynomial P is a polynomial in (z₁,z₂...,z_n) and its conjugate. After a long introduction to the elementary and not so elementary features of such polynomials, I will consider the following basic question: When is a power of a Hermitian polynomial a sum of squared norms of holomorphic polynomials? I will discuss partial results and work in progress towards the solution of this question.

Jean Ruppenthal (U of Bonn), About the ∂-equation at isolated singularities with regular exceptional set

This talk is about solvability and regularity of the ∂-equation on a singular analytic variety Y. One method to attack the ∂-problem on singular Stein spaces is to use Hironaka's desingularization in order to blow up the singularities, and to investigate the ∂-equation on the resulting regular complex manifold. If Y has only isolated singularities, then the exceptional set X of a desingularization π: M → Y is an exceptional set in the sense of Grauert. In the case where X is regular, we can apply Grauert's formal principle and obtain continuous solutions, L²-estimates, and optimal Hölder-estimates if Y is a cone.

Andrei Iordan (Institut de Mathématiques de Jussieu), On the non existence of smooth Levi flat real hypersurfaces in the complex projective space

In 1993 D. Cerveau conjectured the non-existence of smooth Levi-flat real hypersurfaces in the complex n-dimensional projective space, n ≥2, i.e., of real hypersurfaces admitting a local foliation by complex analytic hypersurfaces. If n ≥3, this problem was solved for real analytic hypersurfaces by A. Lins Neto and for smooth hypersurfaces by Y.-T. Siu. In this lecture we will discuss this problem and we will give an improvement of the regularity in Siu's theorem.

Jeff Diller (Notre Dame), On the dynamics of meromorphic maps with low topological degree

I will discuss geometry and ergodic of rational self-maps of the projective plane. In particular, there is a fairly well-established scheme for constructing and analyzing so-called equilibrium measures for such maps. The work I present, which is joint with V. Guedj and R. Dujardin is devoted to implementing this scheme for maps whose topological degree is relatively small. By comparing and contrasting with the more extensively considered case of birational maps, I will outline our main results and then discuss some of the particular points that are new and more difficult in the non-invertible case.

Al Taylor (U of M), Convergence of fixed point iteration for a class of matrix-valued analytic functions

In the study of discrete random walks on strips, one solution of the matrix equation G = a(G) encodes important information about the long term behavior of the walk. Here, the function a(z) is an m x m matrix valued analytic function in the unit disk. For the functions a(z) that come from random walks, it is easy to see that the solution G that contains probabilistic informtion is obtained by making the initial estimate, G₀ = 0 and then taking the limit using the fixed point iteration G→ a(G). We will explain what happens in general for this fixed point iteration, at least for the a(z) that come from random walks. The results suggest several open problems about the iteration of maps on the space of matrices with spectral radius smaller than one.

Sebastien Boucksom (CNRS, Paris), Transfinite diameter and energy functionals on projective varieties

I will explain a recent joint work with R. Berman in which we define and investigate several natural generalizations of classical notions of capacity to a the setting of quasi-psh functions on compact complex manifolds. This enables us to recover by purely complex analytic means and on an arbitrary projective manifold a recent generalization by Rumely of Robin's formula to ℂⁿ.

Željko Čučković (U of Toledo), Special Toeplitz operators that improve integrability

We study mapping properties of Toeplitz operators on strongly pseudoconvex domains, whose symbol is a positive power of the distance to the boundary. (Joint work with J. McNeal)

Elizabeth Wulcan (Chalmers University), Decomposition of residue currents

I will discuss a joint work with Mats Andersson. In a previous work we have defined, given an ideal J⊂O₀ and a free resolution of J, a vector valued residue current R whose annihilator ideal ann R is precisely J. We show that R admits a natural decomposition R=∑ R^p, where the sum runs over the associated primes {p}of J, such that R^p has support on the variety V_p of p and J=∩_pann R^p is a minimal primary decomposition of J. The currents R^p are defined as certain restrictions (in a sense that will be made clear) of R to varieties and more generally constructible sets; in particular, if J has no embedded primes R ^p is the the restriction of R to V_p.

Joël Merker (ENS, Paris), Demailly-Semple jets of orders 4 and 5 in dimension 2

Siu and Demailly's strategy for Kobayashi's hyperbolicity conjecture involves the construction of global holomorphic jet differentials wich are (locally) invariant under reparametrization. For jets of orders 2, 3 and 4 in dimension 2; for jets of of order 2 and 3 in dimension 3, the so-called bracket procedure generates all the invariants (3, 5, 9; 6, 16 respectively). For jets of order 5 in dimension 2, the 36 bracket invariants share 210 syzygies ; for jets of order 6 in dimension 2, there would exist 210 bracket invariants sharing 14 950 syzygies. However, already for jets of order 5, we show that bracketing is not enough, and maybe, infinitely many invariants exist, as in Nagata's counterexample to Hilbert's 14th problem-conjecture. Strikingly, 5 is also the minimal expected degree for Kobayashi-hyperbolicity of surfaces of ℙ₃(ℂ).

Araceli Bonifant (U of Rhode Island), Schwarzian derivatives and cylinder maps

In this talk we will describe the way in which the sign of the Schwarzian derivative for a family of diffeomorphisms of the interval I affects the dynamics of an associated many-to-one skew product map of the cylinder (ℝ∖ℤ)×I.