Thesis Defense

from Friday
June 01, 2018 to Monday
December 31, 2018
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Friday
August 10, 2018

4:30 PM
Math Tower 5-127
Cristian Minoccheri, Stony Brook University
On the Arithmetic of low degree Weighted Complete Intersections

A variety is rationally connected if two general points can be joined by a rational curve. A higher version of this notion is rational simple connectedness, which requires suitable spaces of rational curves through two points to be rationally connected themselves. We prove that smooth, complex, weighted complete intersections of low enough degree are rationally simply connected. This result has strong arithmetic implications for weighted complete intersections defined over the function field of a smooth, complex curve. Namely, it implies that these varieties satisfy weak approximation at all places, that $R$-equivalence of rational points is trivial, and that the Chow group of zero cycles of degree zero is zero.


Friday
August 10, 2018

2:30 PM
Math Tower 5-127
Joseph Thurman, Stony Brook University
Quaternionic Geometry and Special Holonomy

This thesis studies connections with special holonomy group arising from quaternionic manifolds. The focus is on two previously described constructions that produce such connections from positive quaternion-Kähler manifolds with an isometric and quaternionic circle action. The first, due to Hitchin [17], yields quaternionic connections with a preferred complex structure, while the second, due to Haydys [14], yields Kähler metrics. In particular, both constructions produce Kähler metrics in real dimension four, and therefore generalize a theorem of Pontecorvo [38] that produces scalar-flat Kähler metrics from anti-self-dual Hermitian surfaces in real dimension 4.

The goal of this work is to explore the relationship between these two constructions. Although they are superficially similar, the main result of this dissertation shows that they are in fact distinct. This result is obtained by describing a simplification of Haydys’s construction that allows for explicit computation of the Levi-Civita connection of the Kähler metric. Hitchin’s methods are also generalized to give a construction of quaternionic complex manifolds from quaternionic manifolds without a metric.


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