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Topics in Geometry and Topology I
Short description: Toric varieties are a class of examples of varieties (complex algebraic manifolds) which have two main attractive features:
(1) They appear in many contexts, such as Codes, Algebraic Geometry, Symplectic Geometry, Number theory, String Theory, Combinatorics, Singularity theory, and numerous other contexts.
(2) Everything is calculable in terms of combinatorial data. So they give concrete examples to many geometric concepts.
It can be said that toric varieties are the tool of choice today for many fields in mathematics. Close to my heart is mirror symmetry and string theory, which makes heavy use of toric geometry. But they are of equal importance in classical geometry and number theory. Since they are described by combinatorial data, all the abstract concepts can be presented in detailed concrete calculations. This helps one (a) to understand the concepts and (b) to actually prove theorems by calculations.
Audience: Graduates with basic knowledge about topology, geometry and algebra.
Textbook: I will probably use more than one source, but the main source will be:
Introduction to Toric Varieties by William Fulton.
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