SCV Seminar

Monday, Oct. 4th	Crystal Zeager (U. Mich) Resolution of Singularities and the pluricomplex Taylor Green function: We consider a generalization of the pluricomplex Green function proposed by B. A. Taylor and using Hironaka's resolution of singularities, we show that it is plurisubharmonic. We also study conditions under which the Taylor Green function is the same as the pluricomplex Green function with multiple poles.
Monday, Oct. 11th	Elizabeth Wulcan (U. Mich) Non-proper intersection theory and positive currents: I will discuss an ongoing project with M. Andersson, H. Samuelsson, and A. Yger, which aims to present a current approach to non-proper intersection theory. Our main result (so far) is a generalization of the classical King's formula: Given a coherent ideal sheaf J on a reduced analytic space we construct a positive current whose Lelong numbers are precisely the Segre numbers of J.
Monday, Oct. 18th	No seminar - Fall break
Monday, Oct. 25th	Min Ru (University of Houston) Some recent development in Nevanlinna theory: In this talk, I'll give a brief introduction to the Nevanlinna theory and survey some recent works in this area.
Monday, Nov. 1st	Mattias Jonsson (U. Mich.) Archimedean and non-archimedean pluripotential theory: Pluripotential theory is the study of plurisubharmonic functions on complex manifolds, and has recently seen important applications to complex geometry and dynamics. For various reasons it is natural to develop pluripotential theory also in a non-archimedean context. I will explain what this means and how to solve the Monge-Ampere equation in a particular setting. The corresponding results in the usual complex settings will also be reviewed. This is joint work with Sebastien Boucksom and Charles Favre.
Monday, Nov. 8th	Dror Varolin (Stony Brook) Carleson Measures, Toeplitz operators, and off-diagonal decay of the Bergman kernel in generalized Bargmann-Fock spaces: We study the relationship between Carleson measures and the boundedness of their associated Toepiltz operators in the generalized Bargmann-Fock spaces of entire holomorphic functions in C^n having finite L^p norm with respect to a weight whose curvature (1,1)-form is comparable to the Euclidean form. The main tool is an exponential decay for the Bergman kernel function as one moves far from the diagonal. The results we establish are known in the case of the Gaussian weight, but in that case the off-diagonal decay is Gaussian (which is exceptional) and proofs use this fact.
Monday, Nov. 15th	Nessim Sibony (Université Paris-Sud) Speed of equidistribution for holomorphic endomorphisms of Pk.: Let f be a holomorphic endomorphism of algebraic degree d>1. It is conjectured that for a Zariski generic subvariety H of codimension p and degree s, the normalized pullback of H by the nth iterate of f converge exponentially fast towards the p-Green current of f. I will discuss recent results around this conjecture.
Monday, Nov. 22nd	Alan Sola (Oklahoma State University) Scaling limits of anisotropic HL(0) clusters: The Hastings-Levitov growth models $HL(\alpha)$, $\alpha \in [0,2]$, were introduced in theoretical physics about 15 years ago in an attempt to obtain a more rigorous description of random growth processes in the plane that involve aggregating particles. A single particle is modeled by a simple conformal map, and an aggregation process is defined in terms compositions of random rotations and scalings of the original map. In recent joint work with F. Johansson Viklund (Columbia U.) and A. Turner (Lancaster, UK) we analyze an anisotropic version of the simplest case (corresponding to $\alpha=0$) of the HL model. We prove that as the number of particles tends to infinity, and the particle size tends to zero, the resulting random sets (called clusters) can be described in terms of solutions to certain Loewner equations. We also obtain a description of how the harmonic measure on the cluster boundary evolves in this limit. In my talk, I plan to focus on the complex-analytic aspects of our study, such as continuity properties of Loewner equations.
Monday, Nov. 29th	Dong Wang (U. Mich.) Hermitian random matrix model with spiked external source, via orthogonal polynomials: Hermitian random matrices are studied extensively in random matrix theory. Adding a "spiked" external source, we can observe interesting phase transition phenomenon. I will talk about my current work with Jinho Baik on this model. Many asymptotic results of Hermitian random matrices are derived via orthogonal polynomials. I will introduce the asymptotic analysis of orthogonal polynomials by Riemann-Hilbert Problem, and show how to apply it in our model.
Monday, Dec. 6th	Alexander Izzo (Bowling Green State University) Function algebras invariant under every self-homeomorphism: We answer, in suitably generalized form, the following question which could be posed in a beginning analysis course: If A is a uniformly closed algebra of continuous complex-valued functions on a closed ball in Euclidean space such that A contains the constants and separates points, and if for each self-homeomorphism h of the closed ball and each function f in A the composition of h and f is also in A, must then A contain every continuous complex-valued function on the closed ball?
Monday, Dec. 13th	Jan-Li Lin (Indiana University) Dynamical degrees of monomial maps: Applying the intersection theory on toric varieties, one can compute the dynamical degrees of monomial maps. In the talk, I will introduce the related terminologies, then give a sketch of the proof.