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MATH-328: Algebraic geometry I - Curves

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Lectures in this course (36)

Projective Geometry: Tangent Lines Log in to Mediaspace to watch this video

Discusses the correction of tangent lines and projective plane curves, exploring topological applications and curve structure.

Tangent Lines and Equality Log in to Mediaspace to watch this video

Explores the intersection multiplicity of curves and the absence of common tangent lines.

Projective Plane Curves Log in to Mediaspace to watch this video

Introduces projective plane curves, degrees, components, multiplicities, intersection numbers, tangents, and multiple points, culminating in the statement of Bézout's theorem and its consequences.

Algebraic Varieties in Linear Algebra Log in to Mediaspace to watch this video

Explores algebraic varieties in linear algebra, focusing on their nature, determinants, irreducibility, prime properties, and geometric representation theory.

Elliptic Curves: Theory and Applications Log in to Mediaspace to watch this video

Covers the theory and applications of elliptic curves in cryptography and number theory.

Bézout's Theorem and Cayley-Bacharach Log in to Mediaspace to watch this video

Explores Bézout's theorem and Cayley-Bacharach, discussing intersection multiplicities and linear combinations in projective geometry.

Incidence Geometry & Elliptic Curves Log in to Mediaspace to watch this video

Explores the Cayley-Bacharach theorem in incidence geometry and introduces elliptic curves with a commutative law.

Lenstra's Algorithm: Integer Factorization Log in to Mediaspace to watch this video

Covers Lenstra's Algorithm for integer factorization, which efficiently computes prime factors of an integer.

Elliptic Curves: Group Structure and Isomorphism Log in to Mediaspace to watch this video

Explores the group structure and isomorphism of elliptic curves, including inverses, associativity, and compactification to the torus.

Algebraic Subsets of A^1 Log in to Mediaspace to watch this video

Covers algebraic subsets of A^1 and ideals with a finite set of points.

Irreducible Algebraic Sets Log in to Mediaspace to watch this video

Explores irreducible algebraic sets and their unique decomposition into components, with a focus on prime ideals and plane subsets classification.

Affine Algebraic Varieties: Zariski Topology Log in to Mediaspace to watch this video

Explores affine algebraic varieties, emphasizing the Zariski Topology and regular functions.

Variety Defined as the Closure of VCA Log in to Mediaspace to watch this video

Explores the concept of variety defined as the closure of VCA and its applications in algebraic geometry.

Morphisms between Affine Varieties Log in to Mediaspace to watch this video

Covers defining morphisms between affine algebraic varieties and constructing morphisms using algebraic homomorphisms.

Regular Functions on Quasi-Affine Varieties Log in to Mediaspace to watch this video

Explores regular functions on quasi-affine varieties, defining morphisms, local rings, and rational functions.

Projective Varieties: An Algebraic Study Log in to Mediaspace to watch this video

Covers the study of projective varieties and their relation to compact manifolds.

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