Demetre Kazaras:The geometry and topology of positive scalar curvature
- Graduate/Faculty Seminar,Uploaded Videos ( 1597 Views )I will give an informal overview of the history and status of my field. Local invariants of Riemannian metrics are called curvature, the weakest of which is known as "scalar curvature." The study of metrics with positive scalar curvature is very rich with >100 year old connections to General Relativity and smooth topology. Does this geometric condition have topological implications? The answer turns out to be "yes," but mathematicians continue to search for the true heart of the positive scalar curvature conditions.
Rita Pardini : Linear systems on irregular varieties
- Algebraic Geometry ( 154 Views )
I will report on joint work M.A. Barja (UPC, Barcelona) and L. Stoppino (Universita' dell'Insubria, Como - Italy).
Given a generically finite map a:X--> A, where X is a smooth projective variety and A is an abelian variety, and given a line bundle L on X, we study the linear system |L+P|, where P is a general element of Pic^0(A). We prove that up to taking base change with a suitable multiplication map A-->A, the map given by |L+P| is independent of P and induces a factorization of the map a.
When L is the canonical bundle of X, this factorization is a new geometrical object intrinsically attached to the variety X.
The factorization theorem also allows us to improve the known Clifford-Severi and Castelnuovo type numerical inequalities for line bundles on X, under certain assumptions on the map a:X-->A.
A key tool in these proofs is the introduction of a real function, the continuous rank function, that also allows us to simplify considerably the proof of the general Clifford-Severi inequality.