In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation h(x, y, t) = 0 can be restricted to the affine algebraic plane curve of equation h(x, y, 1) = 0. These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. More generally, an algebraic curve is an algebraic variety of dimension one. Equivalently, an algebraic curve is an algebraic variety that is birationally equivalent to an algebraic plane curve. If the curve is contained in an affine space or a projective space, one can take a projection for such a birational equivalence. These birational equivalences reduce most of the study of algebraic curves to the study of algebraic plane curves. However, some properties are not kept under birational equivalence and must be studied on non-plane curves. This is, in particular, the case for the degree and smoothness. For example, there exist smooth curves of genus 0 and degree greater than two, but any plane projection of such curves has singular points (see Genus–degree formula). A non-plane curve is often called a space curve or a skew curve. (Wikipedia).
In this talk, we will define elliptic curves and, more importantly, we will try to motivate why they are central to modern number theory. Elliptic curves are ubiquitous not only in number theory, but also in algebraic geometry, complex analysis, cryptography, physics, and beyond. They were
From playlist An Introduction to the Arithmetic of Elliptic Curves
Algebraic geometry 44: Survey of curves
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It gives an informal survey of complex curves of small genus.
From playlist Algebraic geometry I: Varieties
Algebraic Expressions (Basics)
This video is about Algebraic Expressions
From playlist Algebraic Expressions and Properties
Maxima and Minima for Quadratic and Cubics | Algebraic Calculus One | Wild Egg
Tangents of algebraic curves are best defined purely algebraically, without recourse to limiting arguments! We apply our techniques for finding such tangents to derive some familiar results for quadratic and cubic polynomial functions and their maxima and minima. We compare also with the c
From playlist Algebraic Calculus One
An interesting homotopy (in fact, an ambient isotopy) of two surfaces.
From playlist Algebraic Topology
Complex numbers and curves | Math History | NJ Wildberger
In the 19th century, the study of algebraic curves entered a new era with the introduction of homogeneous coordinates and ideas from projective geometry, the use of complex numbers both on the curve and at infinity, and the discovery by the great German mathematician B. Riemann that topolo
From playlist MathHistory: A course in the History of Mathematics
Algebraic Calculus One ... and Two! | Algebraic Calculus Info | N J Wildberger
The online course Algebraic Calculus One at openlearning.com has had its first beta run at openlearning.com over the last two years. Overall it has been a very pleasant success. In this video we recount the main innovative aspects of this purely algebraic approach to a classical subject. T
From playlist Algebraic Calculus One Info
How do you graph an equation using the intercept method
👉 Learn about graphing linear equations. A linear equation is an equation whose highest exponent on its variable(s) is 1. i.e. linear equations has no exponents on their variables. The graph of a linear equation is a straight line. To graph a linear equation, we identify two values (x-valu
From playlist ⚡️Graph Linear Equations | Learn About
Introduction to Algebraic Equations (L5.1)
This video defines an algebraic equation. Then a solution of an equation is verified. Equivalent equations are also defined. Video content created by Jenifer Bohart, William Meacham, Judy Sutor, and Donna Guhse from SCC (CC-BY 4.0)
From playlist Introduction to Linear Equations in One Variable
Residual Intersections in Geometry and Algebra by David Eisenbud
DISTINGUISHED LECTURES RESIDUAL INTERSECTIONS IN GEOMETRY AND ALGEBRA SPEAKER: David Eisenbud (Director, Mathematical Sciences Research Institute, and Professor of Mathematics, UC Berkeley) DATE: 13 December 2019, 16:00 to 17:00 VENUE: Madhava Lecture Hall, ICTS-TIFR, Bengaluru In thi
From playlist DISTINGUISHED LECTURES
Differential Isomorphism and Equivalence of Algebraic Varieties Board at 49:35 Sum_i=1^N 2/(x-phi_i(y,t))^2
From playlist Fall 2017
Xevi Guitart : Endomorphism algebras of geometrically split abelian surfaces over Q
CONFERENCE Recording during the thematic meeting : "COUNT, COmputations and their Uses in Number Theory" the February 28, 2023 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Jean Petit Find this video and other talks given by worldwide mathematici
From playlist JEAN MORLET CHAIR
Kenneth Ascher: What is a moduli space?
Abstract: Moduli spaces are geometric spaces which parametrize equivalence classes of algebraic varieties. I will discuss the moduli space of algebraic curves equivalently Riemann surfaces) of genus g, and use this example to motivate some interesting questions in higher dimensions. Biogr
From playlist What is...? Seminars
Lazaro Recht: Metric geometry in homogeneous spaces of the unitary group of a C* -algebra. 2
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Algebraic and Complex Geometry
An Arithmetic Refinement of Homological Mirror Symmetry for the 2-Torus - Yanki Lekili
Yanki Lekili University of Cambridge November 9, 2012 We establish a derived equivalence of the Fukaya category of the 2-torus, relative to a basepoint, with the category of perfect complexes on the Tate curve over Z[[q]]. It specializes to an equivalence, over Z, of the Fukaya category o
From playlist Mathematics
Pierrick Bousseau - The Skein Algebra of the 4-punctured Sphere from Curve Counting
The Kauffman bracket skein algebra is a quantization of the algebra of regular functions on the SL_2 character of a topological surface. I will explain how to realize the skein algebra of the 4-punctured sphere as the output of a mirror symmetry construction based on higher genus Gromov-Wi
From playlist 2021 IHES Summer School - Enumerative Geometry, Physics and Representation Theory
Bi Polynumbers and Tangents to Algebraic Curves | Algebraic Calculus One | Wild Egg
We introduce the important technology of defining, and computing the tangent line to an algebraic curve at a point lying on it. We start with a discussion on bi polynumbers, which are two dimensional arrays that are equivalent to polynomials in two variables, but without us having to fret
From playlist Algebraic Calculus One from Wild Egg
Daniele Agostini - Curves and theta functions: algebra, geometry & physics
Riemann’s theta function is a central object throughout mathematics, from algebraic geometry to number theory, and from mathematical physics to statistics and cryptography. One of my long term projects is to develop a program to study and connect the various aspects - geometric, computatio
From playlist Research Spotlight
(1.8) Introduction to Solving Exact Differential Equations
This video introduces and explains how to solve an exact differential equation. https://mathispower4u.com
From playlist Differential Equations: Complete Set of Course Videos