RTG Working Seminar

Wednesday, Jan. 21, 09	Crystal Zeager (University of Michigan) Regularity of Solutions to the Monge-Ampere Equation: Finding the solution of a Monge-Ampere equation is related to finding the largest plurisubharmonic function within an admissible class, and thus also to a number of biholomorphically invariant metrics. The continuity properties of a solution can depend on conditions which include the geometry of the domain, the smoothness of the boundary values, and singularities in the interior. I will discuss a few theorems and counterexamples depending on these conditions.
Wednesday, Feb. 4, 09	Rodrigo Parra (University of Michigan) Laminar currents in P^2: Given a sequence of irreducible curves (C_n) in P^2 converging to T in the sense of currents, there is a geometric condition on the curves ensuring that the limit current T is laminar. We explore this condition and then applied this to meromorphic dynamical systems in P^2, and laminarity of the dynamical Green current T.
Wednesday, Feb. 11, 09	Al Taylor (University of Michigan) Extremal plurisubharmonic functions for linear growth (Part II): We will discuss properties of the extremal function $\Lambda_E(z)$, associated to plurisubharmonic functions of linear growth, i.e. the upper envelope of all psh functions $u$ that are bounded above by zero on the set $E$ and satisfy $u(z) \le \|z\| + o(\|z\|)$. Questions about this function arise naturally in trying to classify the algebraic varieties with sufficiently many real points that they satisfy the strong radial Phragmen-Lindelof condition. This extremal function is also an analogue of the Siciak-Zaharuta extremal psh function $L_E$ of logarithmic growth. However, we will show that it fails to have most of the properties that make $L_E$ such a useful function in pluripotential theory. We will present enough of the proofs to explain all the main techniques used in the work. All of the material is taken from the 1998 Ph.D. thesis of David Bainbridge.
Wednesday, Mar. 4, 09	Chris Hammond (University of Michigan) The Isoperimetric Problem for Fefferman Hypersurface Measure: Fefferman introduced a scaled-version of surface-area measure on real hypersurfaces in $\mathbb{C}^{n}$ which is invariant under volume-preserving biholomorphisms. We derive the Euler equation for the associated isopermetric problem. We then use volume-preserving invariants to characterize the solutions. We show that under "natural" assumptions, spheres are the only solutions.
Wednesday, Mar. 18, 09	Elizabeth Wulcan (University of Michigan) Sparse effective Nullstellensaetze via residue currents: Residue currents are generalizations of one complex variable residues and can be thought of as currents representing ideals of holomorphic functions or polynomials. I will discuss how residue currents on toric varieties can be used to obtain certain effective versions of Hilbert's Nullstellensatz.
Wednesday, Apr. 8, 09	Richard Lärkäng (Chalmers University of Technology) Residue currents associated with weakly holomorphic functions: Residue currents like the Coleff-Herrera product and Bochner-Martinelli type residue currents are certain currents associated with a set of holomorphic functions, or more generally an ideal of holomorphic functions on a complex manifold or an analytic space. I will discuss how to define analogous currents related to weakly holomorphic functions, and how the theory generalizes from holomorphic functions to the case of weakly holomorphic functions.
Wednesday, Apr. 15, 09