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18.725 Algebraic Geometry (MIT)

Description

This is the first semester of a two-semester sequence on Algebraic Geometry. The goal of the course is to introduce the basic notions and techniques of modern algebraic geometry. It covers fundamental notions and results about algebraic varieties over an algebraically closed field; relations between complex algebraic varieties and complex analytic varieties; and examples with emphasis on algebraic curves and surfaces. This course is an introduction to the language of schemes and properties of morphisms.

Subjects

algebraic geometry | Zariski topology | Product Topology | Affine Varieties | Projective Varieties | Noether Normalization | Affine Morphisms | Finite Morphisms | Sheaves | ?s Theorem | Kahler Differentials | Canonical Bundles | Riemann-Hurwitz Formula

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.405J Advanced Complexity Theory (MIT)

Description

This graduate-level course focuses on current research topics in computational complexity theory. Topics include: Nondeterministic, alternating, probabilistic, and parallel computation models; Boolean circuits; Complexity classes and complete sets; The polynomial-time hierarchy; Interactive proof systems; Relativization; Definitions of randomness; Pseudo-randomness and derandomizations;Interactive proof systems and probabilistically checkable proofs.

Subjects

18.405 | 6.841 | Polynomial hierarchy | time-space lower bounds | approximate counting | ?s Theorem | Relativization | Baker-Gill-Solovay | switching lemma | Razborov-Smolensky | NEXP vs. ACC0 | Communication complexity | PCP theorem | Hadamard code | Gap amplification | Natural proofs

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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