Sponsored by Elizabethtown College, Franklin and Marshall College, Lebanon Valley College, and Millersville University
11 September Mahmoud Zeinalian, Long Island University 2 October James Hughes, Elizabethtown College 6 November David Futer, Temple University 4 December Barbara Nimershiem, Franklin and Marshall College, Ron Umble, Millersville University, and Merv Fansler, Millersvill University 5 February Terrance Napier, Lehigh University 4 March Thomas Hunter, Swarthmore College 1 April David Lyons, Lebanon Valley CollegeTGTS is a regional mathematics seminar/colloquium. We meet at 4:30 p.m. on the first Friday of each month during the academic year (with some exceptions, as noted in the schedule above). The public is cordially invited to attend.
Date: 5 February 2016, 4:30pm Location: Hempfield High School, Room 213 Speaker: Terrance Napier, Lehigh University Title: The Bochner-Hartogs dichotomy Abstract. In 1906, Friedrich Hartogs discovered that holomorphic (i.e., complex analytic) functions in several complex variables have the following extension property. Let K be a compact set with connected complement E in complex Euclidean space C^n of dimension n>1, and let f be a holomorphic function on E. Then there exists a unique holomorphic function g on all of C^n such that the restriction of g to E is equal to f. Clearly, this fails in dimension n=1 (take f(z)=1/z for example). Of course, it had long been known that in one variable, the geometry and topology of a domain (or more generally speaking, a Riemann surface) have strong implications for complex analysis on the domain. Hartogs’ discovery was probably the first indication of the ways in which geometry and topology, in the form of convexity and the ends structure, might enter in higher dimensions. This theme played a central role in the development of the theory of several complex variables. One may view this Hartogs property as the vanishing of the first compactly supported cohomology with coefficients in the structure sheaf. We will consider analogues for certain complete Kaehler manifolds. More precisely, we will look at conditions under which a connected noncompact complete Kaehler manifold X satisfies the Bochner-Hartogs dichotomy: Either the first compactly supported cohomology of X with coefficients in the structure sheaf vanishes, or X admits a proper holomorphic mapping onto a Riemann surface.