Event - Math

November 22, 2018 3:30 PM - 4:30 PMCAB 657

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For complexprojective manifolds, the sign of the curvature of
the tangent bundle gives a basic trichotomy: positively curved (Fano), zerocurvature (Calabi-Yau), and negatively curved ("general type"). I will explain two recent theorems where positive curvature implies existenceand abundance of solutions of the defining polynomials of the projectivemanifold over fields more general than the complex numbers. The first theorem,joint with Chenyang Xu, proves existence of at least one solution over thefield $Z/pZ(t)$ of rational functions with coefficients in $Z/pZ$ in thepresence of "higher curvature". The second theorem, joint withZhiyu Tian and Runhong Zong, proves "abundance" of solutions over thefield $\overline{Z/pZ}(t)$ for Fano complete intersections (the"simplest" projective manifolds).  I will focus on examples.


Forthose attending the Colloquium,

areception will be held at 4:30 pm in CAB 649.