Math Department | Past Seminars

Seminars in Several Complex Variables
and Complex Dynamics, Winter 2011

Monday 4:00 pm ~ 5:00 pm (EH 3096)

 Monday, Feb. 14th. Rodrigo Parra (University of Michigan) Equidistribution to the Green current: In this talk I will describe the problem of equidistribution in holomorphic dynamics on the complex projective space. More precisely, given a holomorphic self map f on the k dimensional complex projective space of algebraic degree d bigger than 1 then there exists a positive closed (1,1)-current T which is invariant (i.e. f^*T=dT) and supported on the Julia set of f. I will try to address the following question: If S is a positive closed (1,1)-current of mass 1, when does the sequence d^{-n}(f^n)^*S converge to T? This is always true if S is smooth and is always false if S is the current of integration of a totally invariant hypersurface. This question has been answered in dimensions k=1 and 2 and I will describe some results I have obtained in higher dimensions. Monday, Feb. 21st. Elizabeth Wulcan (Chalmers University, Sweden) Effective membership problems on varieties: This talk is based on a joint work (in progress) with Mats Andersson, in which we use residue currents to bound the degrees of solutions to polynomial ideal membership problems on algebraic varieties. In particular, I will discuss a generalization to the non-smooth case of a global Briancon-Skoda theorem due to Hickel and Ein-Lazarsfeld. Monday, Feb. 28th. No seminar - Winter break :) Monday, Mar. 7th. Al Taylor (University of Michigan) Leon Ehrenpreis's proof of the strip theorem: I will present results from a preprint, Morera-like theorems for partial differential equations'', that I received from Leon Ehrenpreis shortly before his death last August. Among other things, the paper contains his proof of the strip theorem: If a continuous function on the strip |Im z| \le 1 in the complex plane has the property that its restriction to every circle, |z - t | = 1, t real, has an analytic extension to the interior of the circle, then the function must be analytic in the strip. The paper also gives a framework for extending such results to other partial differential operators. To my knowledge, the manuscript has not been published. Monday, Mar. 14th. Crystal Zeager (University of Michigan) Some applications of the Skwarczynski metric: The Bergman Space is the space of L^2 holomorphic functions. We will study some applications of the Skwarczynski metric to the Bergman Space and also to the Bergman metric. We show that a Skwarczynski complete domain has infinite dimensional Bergman space, which is joint work with Lina Lee. Monday, Mar. 21st. Alexander Izzo (Bowling Green State University) Nonlocal function algebras on manifolds: The first nonlocal function algebra was constructed by Eva Kallin in 1963. We construct nonlocal function algebras generated by smooth functions on manifolds and investigate the minimum dimension of a manifold on which there exists a nonlocal function algebra. Monday, Mar. 28th. No seminar - RTG Workshop Monday, Apr. 4th. Dusty Grundmeier (University of Illinois at Urbana-Champaign) Invariant Theory and CR Mappings: We consider group-invariant CR mappings from spheres to hyperquadrics. Given a finite subgroup $\Gamma \subset U(n)$, a construction of D'Angelo and Lichtblau yields a target hyperquadric $Q(\Gamma)$ and a canonical map $h_{\Gamma} : S^{2n-1}/ \Gamma \to Q(\Gamma)$. In this talk, we will study the target hyperquadrics associated to lens spaces. We will also use results from classical invariant theory to study the case of higher dimensional spherical space forms. This talk partially builds on the results of my Friday (Apr. 1st.) talk, but I will recall the essential aspects of the previous talk. Monday, Apr. 18th. Dan Burns (University of Michigan) Holomorphic extension into the complex domain and the homogenous complex Monge-Ampere equation: Given a real analytic Riemannian manifold (M, g), there is determined a unique solution u of the homogeneous complex Monge-Ampere equation (HCMA) on a neighborhood of M in its complexification, associated to g. A classic result of Boutet de Monvel from the late seventies says that a real analytic function h on M has a holomorphic extension to u < r if and only if h = exp (r\sqrt{\Delta}) f , for some f in L^2. This is valid for small values of r. I will give examples where this breaks down, i.e., where there is a maximal r < \infty for which this is valid, with a speculation as to why it breaks down there. I will also discuss related results in terms of semi-groups of elliptic Toeplitz operators and other examples of solutions of HCMA. This is joint work, part with Victor Guillemin and part with Zhou Zhang.

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