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Roughly talking, the theme of the course is to research the connection between differential geometry and algebraic geometry in Kahler geometry. The course will begin with some background material in advanced algebraic geometry and Riemannian geometry. We will discuss Yau’s proof of the existence of Kahler-Einstein metrics with adverse, or zero Ricci curvature. Possible additional subjects embrace features of collapsing degenerations, connections to real Monge Ampere equations and mirror symmetry, and full, non-compact Calabi-Yau manifolds. This course will develop advanced strategies in linear algebra and introduce the theory of optimization. On the linear algebra facet, we will study necessary matrix factorizations (e.g. LU, QR, SVD), matrix approximations , convergence of iterative methods, and spectral theorems.

This generalization is especially nicely suited to studying singularities on moduli spaces and has come to play an essential role in geometric representation principle and mathematical physics. This course shall be an introduction to the subject with a focus of deformation principle and functions to moduli areas. It might be largely self-contained, assuming just some familiarity with algebraic geometry and homological algebra. This course serves as an introduction to the concrete features of algebraic geometry in low dimensions, loosely following Hartshorne Chapters IV and V . The aim is to function a transition from the technical foundations of algebraic geometry to questions closer to precise analysis in the area. In this course we research partial differential equations showing in physics, material sciences, biology, geometry, and engineering.

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