SCV Seminar

Monday, Sep. 14	Per Manne (NSEBA, Bergen, visiting U Mich.) Holomorpic Convexity and Carleman Approximation by Entire Functions on Stein Manifolds : We give necessary and sufficient conditions for totally real subsets of Stein manifolds to admit Carleman approximation of class C^k, k >= 1, by entire functions. T. Carleman (1927) showed that for any continuous function f on the real line and any strictly positive and continuous error function e, there exists an entire function h such that \|h(x)-f(x)\| < e(x) for all real x. We show that if M is a totally real subset of class C^k in a Stein manifold X such that M is holomorphically convex and has bounded exhaustion hulls in X, then M admits C^k Carleman approximation with interpolation. Moreover, if M is a totally real subset which admits C^1 Carleman approximation, then M is holomorphically convex and has bounded exhaustion hulls in X. This is a joint work with Erlend Fornaess Wold and Nils Ovrelid.
Monday, Sep. 28	Elizabeth Wulcan (U. Mich.) Stabilization of monomials maps : The construction of many dynamical objects associated with rational maps (such as invariant currents) requires the induced maps on cohomology to be compatible with iteration. I will discuss the problem of finding a model where (f^n)^=(f^^n) when f is a monomial map. This is joint work in progress with Mattias Jonsson.
Monday, Oct. 5	Tien-Cuong Dinh (Paris VI, visiting U. Mich) On the Hodge-Riemann theorem and the Beauville-Bogomolov theorem: We give an abstract version of the Hodge-Riemann theorem for compact Kaehler manifolds and discuss some applications. We will consider in particular the case of symplectic Kaehler manifolds. This is a joint work with Viet-Anh Nguyen (Orsay).
Monday, Oct. 12	No seminar - Midwest SCV Conference, Purdue
Monday, Oct. 19	No seminar - Fall break
Monday, Oct. 26	Debraj Chakrabarti (Notre Dame) The dbar-equation on Product Domains: It is well-known that the dbar-Neumann operator on a product domain does not preserve smoothness up to the boundary; however, the canonical solution operator does preserve smoothness up to the boundary. We try to understand this phenomenon, and derive some estimates for the canonical solution in Sobolev spaces. (Joint work with Mei-Chi Shaw.)
Monday, Nov. 2	Malgorzata Marciniak (University of Toledo) Hartogs type extensions in Toric Varieties : I will discuss the Hartogs phenomenon (holomorphic extensions from the connected complement of a compact set) and the Hartogs-Bochner phenomenon (holomorphic extensions of Cauchy-Riemann functions from a hypersurface to one of its sides) in the case of toric varieties.
Monday, Nov. 9	Steven Krantz (Washington University in St. Louis) Domains with Noncompact Automorphism Group: We survey results on domains with noncompact automorphism group, including some new results in a metric context. This is joint work with K. T. Kim. The talk will be accessible to graduate students.
Monday, Nov. 16	Crystal Zeager (U. Mich.) Comparison of Invariant Metrics: Metrics that are invariant under biholomorphism provide a tool for studying holomorphic maps. I will discuss some of these metrics on two explicit domains. First I will discuss some Picard type results that are related to the Kobayashi metric on C \ {0,1}. Then I will discuss an extension question for the Bergman metric, and given an answer for the ring domain in C^n. This is joint work with Lina Lee and Hyunsuk Kang.
Monday, Nov. 23	Dennis Eriksson (University of Gothenburg) Hilbert polynomials in line bundles: The determinant of the cohomology of smooth varieties is known by Knudsen-Mumford to satisfy certain Hilbert-polynomial type behaviour. I will discuss the "coefficients" of the Hilbert polynomial in line bundles and describe them in terms of generalized Deligne products as well as how they can be metrized, giving a relation to the Quillen metric. Parts of this talk will be work in progress.
Monday, Nov. 30	Lina Lee (U. Mich.) On supnorm estimate for \overline\partial on infinitne type convex domains in C^2: We study the \overline\partial-equation on some convex domains of infinite type in C^2 and show that supnorm estimates hold for infinite exponential type domains provided that the exponent is less than 1.
Monday, Dec. 7 3-4 pm (3096 EH)	Matteo Ruggiero (Scuola Normale Superiore, Pisa) On the classification of semi-superattratting germs in C^2: A semi-superattracting germ in C^2 is a germ whose differential at 0 has a null eigenvalue and a non-null eigenvalue. We shall study these germs by their induced action on the valuative tree, following the rigidification process studied by Favre and Jonsson. We shall first find some invariants and then give the formal classification of semi-superattracting germs, showing that the moduli space has infinite dimension. Eventually, the convergence of the conjugation map will be discussed.
Monday, Dec. 7	Lisa Nilsson (MSRI) Discriminant coamoebas in dimension two: The coamoeba of an algebraic variety is its image under coordinatewise argument mappings. Coamoebas have useful applications in complex analysis, algebraic geometry, string theory e.t.c. In this talk I will discuss coamoebas of A-discriminantal curves. I will give an explicit description of these coamoebas in dimension two, and show that they are intimately connected with a certain zonotope. If we consider the coamoeba and the zonotope as chains (in the sense of algebraic topology) on the torus (\mathbb{R}/2\pi\mathbb{Z})^2 they will together cover the torus an integer number of times, hence they form a 2-cycle.