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Algebraic Geometry for Infinite Genus: Foam Curves

Mathematical Physics & Probability

Speaker: I. Zakharevich, MSRI
Location: 693 Kerr
Start time: Tue, Jan 29 2002, 3:10PM

For analytic curves of infinite genus, to get a theory parallel to algebraic geometry one needs to restrict attention to holomorphic sections satisfying some ``conditions on growth at infinity''. Each such condition effectively attaches an ``ideal point'' to the curve; this process is similar to compactification. We discuss the algebraic geometry on curves with such ``ideal points''. Conditions on the ``lengths of handles'' of the curve are found which ensure the "standard theorems" hold. It turns out that these conditions give no restriction on the density of ideal points on the curve. In particular, such curves may have a dense set of ideal points; these curves have no smooth points at all, and have a purely fractal nature. (Such ``foam'' curves live near the ``periphery'' of the corresponding moduli space; one needs to study these curves too, since they may be included in the support of natural measures arising on the moduli spaces.)

Albert Schwarz is the host.