SCV Seminar

Thursday Feb. 4th. 4-5pm 4096 EH	Thomas Bloom (University of Toronto) Voiculescu's entropy and potential theory : That the (negative of) the logarithmic energy of a planar measure is the triple limit of volumes originated in work of Voiculescu in the 90's. Subsequently this result was put in the context of large deviations by Ben Arous and A. Guionnet. We will give a new proof of this result and give versions of these results in several variables. This talk will deal only with developments in potential theory and pluripotential theory related to this result(and not those in operator theory)
Monday, Feb. 8th.	Crystal Zeager (University of Michigan) The Azukawa pseudometric and the pluricomplex Green function : The Azukawa pseudometric can be defined in terms of the pluricomplex Green function. I will discuss regularity of the Green function and its implications for regularity of the Azukawa metric. I will also discuss some results of Pflug and Zwonek about how the Green function relates to Bergman completeness.
Monday, Feb. 15th.	Kazuo Azukawa (University of Toyama, Japan) Lupe magic squares of order A^2+3B^2 odd (I) : Click for abstract
Monday, Mar. 8th.	Mats Andersson (Chalmers University) A fine resolution of the sheaf of holomorphic functions on an analytic space (joint with H Samuelsson) : Let $X$ be an analytic space. We introduce fine sheaves $A_k$ of $(0,k)$-currents that are smooth on the regular part of $X$ and have "mild" singularities at the singular set, and such that $0\to \O_X\to A_0\to A_1\to \cdots$ is exact, where the mappings are $\dbar$. In particular we get an extension of the Dolbeault isomorphism to the singular case.
Monday, Mar. 15th.	Alexander Izzo (Bowling Green State University) A tetrachotomy for certain algebras containing the disc algebra : We will answer a question raised by Joseph Cima. Let $D$ denote the open unit disc in the plane, and let $A(D)$ denote the disc algebra. A theorem of E. M. \v Cirka asserts that if $f$ is a function in ${C(\overline D)}$ and $f$ is harmonic but not holomorphic on $D$, then the uniformly closed subalgebra $A(D)[f]$ of ${C(\overline D)}$ generated by $A(D)$ and $f$ is equal to ${C(\overline D)}$. An analogous result for $H^\infty(D)$ was proved by S. Axler and A. Shields: If $f$ is a bounded function on $D$ that is harmonic but not holomorphic, then the uniformly closed subalgebra $H^\infty(D)[f]$ of $L^\infty(D)$ generated by $H^\infty(D)$ and $f$ contains ${C(\overline D)}$. These two theorems suggest that perhaps the inclusion ${A(D)} [f] \supset {C(\overline D)}$ holds whenever $f$ is a bounded harmonic nonholomorphic function on $D$. However, this is not the case; it is not even true that ${A(D)} {[f, \overline f\,]} \supset {C(\overline D)}$ whenever $f \in H^\infty (D)$. This led Cima to raise the question of which continuous functions are in ${A(D)} [f]$ or ${A(D)} {[f, \overline f\,]}$ when the inclusion ${A(D)} [f] \supset {C(\overline D)}$ or ${A(D)} {[f, \overline f\,]} \supset {C(\overline D)}$ fails. We will answer this question for ${A(D)} {[f, \overline f\,]}$.
Monday, Mar. 22nd.	Zhou Zhang (University of Michigan) Stability of solutions to complex Monge-Ampere equations over closed manifolds : stability of solutions comes up naturally with the existence of solutions being available. In a joint work with Slawomir Dinew, we generalize Kolodziej's result considering the case of Kaehler background form to the degenerate situation. For the same equation, uniqueness is a direction application of stability, but there are also places where people can apply stability result without uniqueness consideration.
Monday, Apr. 5th.	Dan Burns (University of Michigan) Limit Currents and Value Distribution of Holomorphic Maps : Given a holomorphic map F, we find conditions which ensure the existence of limits of sequences of normalized image currents over increasing domains which are either d-closed or dd^c-closed. This is modeled on currents constructed in one dimension based on Ahlfors's length-area estimate. The conditions are based on characteristic functions in each dimension, and ratios between characteristic functions in successive dimensions. If the limit current is closed or dd^c-closed this gives uniform distribution results for the map F. This is joint work with N. Sibony.
Monday, Apr. 12th.	Al Taylor (University of Michigan) The local Phragmen-Lindelof Theorem on Varieties : I will explain a geometric condition that characterizes the algebraic varieties in C^n on which every plurisubharmonic function of linear growth, u(z) = O(\|z\|), and which is bounded above by 0 on the real points in fact satisfies a growth condition, $u(z) = O(\| Im z\|). The condition is that the variety is nearly hyperbolic, a condition that is stronger than having a full dimensional set of real points, and weaker than being hyperbolic.
Tuesday, May. 18th. 4-5pm 3096 EH	Per Manne (NSEBA, Bergen, visiting U Mich.) On Totally Real Sets and Carleman Approximation by Entire Functions : This lecture will consist of two parts. First, we will introduce the class of totally real Lipschitz sets and show that they admit similar approximation results as totally real sets. Then we will focus on different global conditions on totally real sets relevant for Carleman approximation by entire functions, in particular conditions related to plurisubharmonic convexity.