Lecture

Elliptic Curves: Theory and Applications

Related lectures (32)

Covers the topology of Riemann surfaces, focusing on orientation and orientability.

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

Algebraic Curves: Normalization

Covers the normalization process of plane algebraic curves, focusing on irreducible polynomials and affine curves.

Directed Networks & Hypergraphs

Explores directed networks with asymmetric relationships and hypergraphs that generalize graphs by allowing edges to connect any subset of nodes.

Irreducible Polynomials: Degree and Roots

Explores irreducible polynomials, focusing on their degree and roots in different fields.

Differential Forms Integration

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.

Functional Equation of Zeta

Covers the functional equation of zeta function and Jensen's formula in complex analysis.

Projective Plane Curves

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

Equations: Degree and Solutions

Explores polynomial equations, their degree, solutions, and domain definition.

Introduction to Elliptic Curves

Covers the basics of elliptic curves, their significance in cryptography, and their applications in public key cryptography.

Proper Actions and Quotients

Covers proper actions of groups on Riemann surfaces and introduces algebraic curves via square roots.

Stein Algorithm: Polynomial Identity Testing

Explores the Stein algorithm for polynomial identity testing and the minimization of a cut problem.

Homotopy: Degree

Covers the concept of degree in homotopy theory and the structure of complexes with two simplices.

Fixed points of the Ruelle-Thurston operator

Covers the fixed points of the Ruelle-Thurston operator and its applications in complex analysis.

Polynomials on a Field: Basics and Operations

Introduces the basics of polynomials on a field, focusing on definitions, operations, and properties.

Galois Fields and Elliptic Curves

Introduces Galois fields, elliptic curves, factorization algorithms, and the discrete logarithm problem in cryptography.

Isogeny Graphs: Eigenvalues and Cryptography

Explores isogeny graphs of supersingular elliptic curves, showing optimal mixing times for random walks and applications to cryptography.

Local Degrees: Calculating Maps

Delves into local degrees of continuous maps from the sphere, showcasing their role in calculating map degrees.

Discrete Log Problem: Index Calculus

Explores the discrete log problem and index calculus method in cryptography.

The Degree of a Covering Space

Shows the equivalence between the index of a covering space's fundamental group and the degree of the covering space.