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4:00pm to 5:00pm - ISEB 1200 - Differential Geometry Jeff Viaclovsky - (UC Irvine) Fibrations on the 6-sphere Let Z be a compact, connected 3-dimensional complex manifold with vanishing first and second Betti numbers and non-vanishing Euler characteristic. We prove that there is no holomorphic mapping from Z onto any 2-dimensional complex space. Combining this with a result of Campana-Demailly-Peternell, a corollary is that any holomorphic mapping from the 6-dimensional sphere S^6, endowed with any hypothetical complex structure, to a strictly lower-dimensional complex space must be constant. In other words, there does not exist any holomorphic fibration on S^6. This is joint work with Nobuhiro Honda. |
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3:30pm to 4:30pm - - Algebra Aaron Landesman - (Harvard University) The distribution of Selmer groups and ranks of abelian varieties in quadratic twist families over function fields The minimalist conjecture predicts that, in quadratic twist families of abelian varieties, half have rank 0 and half have rank 1. This fits into the larger picture of the Bhargava-Kane-Lenstra-Poonen-Rains heuristics, which predict the distribution of Selmer groups of these abelian varieties. In joint work with Jordan Ellenberg, we prove a version of these heuristics: over function fields over the finite field F_q, we show that the above heuristics are correct to within an error term in q, which goes to 0 as q grows. The main inputs are a new homological stability theorem in topology for a generalized version of Hurwitz spaces and an expression of average sizes of Selmer groups in terms of the number of rational points on these Hurwitz spaces over finite fields. |
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4:00pm to 4:50pm - RH 306 - Colloquium Claude LeBrun - (Stony Brook University) Einstein Metrics, 4-Manifolds, and Gravitational Instantons A Riemannian metric is said to be Einstein if it has constant Ricci curvature. Certain peculiar features of 4-dimensional geometry make dimension four into a “Goldilocks zone” for Einstein metrics, with just the right amount of local flexibility managing to coexist with strong global rigidity results. This talk will first describe some aspects of the interplay between Einstein metrics and smooth topology on compact symplectic 4-manifolds without boundary. We will see how ideas from Kähler and conformal geometry allow us to construct Einstein metrics on many such manifolds, while a complimentary tool-box shows that these existence results are optimal in certain specific contexts. The talk will then conclude with a brief discussion of analogous results concerning complete Ricci-flat 4-manifolds. |