Fields of mathematics | Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations. Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique. In the 20th century, algebraic geometry split into several subareas. * The mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more generally to the points with coordinates in an algebraically closed field. * Real algebraic geometry is the study of the real points of an algebraic variety. * Diophantine geometry and, more generally, arithmetic geometry is the study of the points of an algebraic variety with coordinates in fields that are not algebraically closed and occur in algebraic number theory, such as the field of rational numbers, number fields, finite fields, function fields, and p-adic fields. * A large part of singularity theory is devoted to the singularities of algebraic varieties. * is an area that has emerged at the intersection of algebraic geometry and computer algebra, with the rise of computers. It consists mainly of algorithm design and software development for the study of properties of explicitly given algebraic varieties. Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology, differential and complex geometry. One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way which is very similar to its use in the study of differential and analytic manifolds. This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of classical algebraic geometry, mainly concerned with complex points, and of algebraic number theory. Wiles' proof of the longstanding conjecture called Fermat's Last Theorem is an example of the power of this approach. (Wikipedia).
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
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 topology: Introduction
This lecture is part of an online course on algebraic topology. This is an introductory lecture, where we give a quick overview of some of the invariants of algebraic topology (homotopy groups, homology groups, K theory, and cobordism). The book "algebraic topology" by Allen Hatcher men
From playlist Algebraic topology
algebraic geometry 14 Dimension
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the dimension of a topological space, algebraic set, or ring.
From playlist Algebraic geometry I: Varieties
An interesting homotopy (in fact, an ambient isotopy) of two surfaces.
From playlist Algebraic Topology
AlgTopReview: An informal introduction to abstract algebra
This is a review lecture on some aspects of abstract algebra useful for algebraic topology. It provides some background on fields, rings and vector spaces for those of you who have not studied these objects before, and perhaps gives an overview for those of you who have. Our treatment is
From playlist Algebraic Topology
algebraic geometry 15 Projective space
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It introduces projective space and describes the synthetic and analytic approaches to projective geometry
From playlist Algebraic geometry I: Varieties
algebraic geometry 29 Automorphisms of space
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It describes the automorphisms of affine and projective space, and gives a brief discussion of the Jacobian conjecture.
From playlist Algebraic geometry I: Varieties
Homeschool Geometry Before Algebra 2
TabletClass Math: https://tabletclass.com/ This video explains why you should homeschool geometry before algebra 2.
From playlist Homeschool Math
Homeschool Geometry - What Every Homeschool Parent Needs to Know
TabletClass Math Homeschool: https://tabletclass.com/ How to homeschool Geometry successfully. Need help with homeschooling Pre-Algebra, Algebra 1, Geometry, Algebra 2 and Pre-Calculus? Check out TabletClass Math for all your homeschooling needs: https://tabletclass.com/ .
From playlist Homeschool Geometry
Algebraic Structures: Groups, Rings, and Fields
This video covers the definitions for some basic algebraic structures, including groups and rings. I give examples of each and discuss how to verify the properties for each type of structure.
From playlist Abstract Algebra
High School Math – What to Expect and How to Pass
TabletClass Math: https://tcmathacademy.com/ A overview of high school math courses and how to pass. For more math help to include math lessons, practice problems and math tutorials check out my full math help program at https://tcmathacademy.com/ Math Notes: Pre-Algebra Note
From playlist GED Prep Videos
Holomorphic Cartan geometries on simply connected manifolds by Sorin Dumitrescu
Discussion Meeting Complex Algebraic Geometry ORGANIZERS: Indranil Biswas, Mahan Mj and A. J. Parameswaran DATE:01 October 2018 to 06 October 2018 VENUE: Madhava Lecture Hall, ICTS, Bangalore The discussion meeting on Complex Algebraic Geometry will be centered around the "Infosys-ICT
From playlist Complex Algebraic Geometry 2018
Lie groups: Bianchi classification
This lecture is part of an online graduate course on Lie groups. We give a sketch of the Bianchi classification of the Lie algebras and groups of dimension at most 3. We mention that this is related to the Thurston geometries of 3-manifolds. For the other lectures in the course see ht
From playlist Lie groups
Math Major Guide | Warning: Nonstandard advice.
A guide for how to navigate the math major and how to learn the main subjects. Recommendations for courses and books. Comment below to tell me what you think. And check out my channel for conversation videos with guests on math and other topics: https://www.youtube.com/channel/UCYLOc-m8Wu
From playlist Math
Pseudo-finite dimensions, modularity, and generalisations (...) - M. Bays - Workshop 1 - CEB T1 2018
Martin Bays (Münster) / 29.01.2018 Pseudo-finite dimensions, modularity, and generalisations of Elekes–Szab´o. Given a system of polynomial equations in m complex variables with solution set V of dimension d, if we take finite subsets Xi of C each of size N, then the number of solutions
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
A brief history of geometry II: The European epoch | Sociology and Pure Mathematics | N J Wildberger
Let's have a quick overview of some of the developments in the European story of geometry -- at least up to the 19th century. We'll discuss Cartesian geometry, Projective geometry, Descriptive geometry, Algebraic geometry and Differential geometry. This is meant for people from outside m
From playlist Sociology and Pure Mathematics
Geometry of tropical varieties with a view toward applications (Lecture 1) by Omid Amini
PROGRAM COMBINATORIAL ALGEBRAIC GEOMETRY: TROPICAL AND REAL (HYBRID) ORGANIZERS Arvind Ayyer (IISc, India), Madhusudan Manjunath (IITB, India) and Pranav Pandit (ICTS-TIFR, India) DATE & TIME: 27 June 2022 to 08 July 2022 VENUE: Madhava Lecture Hall and Online Algebraic geometry is
From playlist Combinatorial Algebraic Geometry: Tropical and Real (HYBRID)
Homeschool Algebra 2 - What Every Homeschool Parent Needs to Know
TabletClass Math Homeschool: https://tabletclass.com/ How to homeschool Algebra 2 successfully. Need help with homeschooling Pre-Algebra, Algebra 1, Geometry, Algebra 2 and Pre-Calculus? Check out TabletClass Math for all your homeschooling needs: https://tabletclass.com/ .
From playlist Homeschool Math
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