In mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes how an algebraic variety over one field determines a variety over a bigger field, or the pullback of a family of varieties, or a fiber of a family of varieties. Base change is a closely related notion. (Wikipedia).
This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne.. We define fibered products of schemes, sketch their construction, and give a few examples to illustrate their slightly odd behavior.
From playlist Algebraic geometry II: Schemes
Introduction to Fiber Bundles part 1: Definitions
We give the definition of a fiber bundle with fiber F, trivializations and transition maps. This is a really basic stuff that we use a lot. Here are the topics this sets up: *Associated Bundles/Principal Bundles *Reductions of Structure Groups *Steenrod's Theorem *Torsor structure on arith
From playlist Fiber bundles
A Concrete Introduction to Tensor Products
The tensor product of vector spaces (or modules over a ring) can be difficult to understand at first because it's not obvious how calculations can be done with the elements of a tensor product. In this video we give an explanation of an explicit construction of the tensor product and work
From playlist Tensor Products
This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne.. We use the fiber product define last lecture to define group schemes, and give a few non-classical examples of them.
From playlist Algebraic geometry II: Schemes
What is the dot product of two vectors? How is it useful? Free ebook https://bookboon.com/en/introduction-to-vectors-ebook (updated link) Test your understanding via a short quiz http://goo.gl/forms/2SGI5Kvpk9
From playlist Introduction to Vectors
Multivariable Calculus | The dot product.
We present the definition of the dot product as well as a geometric interpretation and some examples. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Vectors for Multivariable Calculus
What is the cross product of two vectors? How is it useful? Free ebook https://bookboon.com/en/introduction-to-vectors-ebook (updated link) Test your understanding via a short quiz http://goo.gl/forms/Ii3hPtwksX
From playlist Introduction to Vectors
Lecture 27. Properties of tensor products
0:00 Use properties of tensor products to effectively think about them! 0:50 Tensor product is symmetric 1:17 Tensor product is associative 1:42 Tensor product is additive 21:40 Corollaries 24:03 Generators in a tensor product 25:30 Tensor product of f.g. modules is itself f.g. 32:05 Tenso
From playlist Abstract Algebra 2
Dot products and duality | Chapter 9, Essence of linear algebra
Why the formula for dot products matches their geometric intuition. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Home page: https://www.3blue1brown.com/ Dot products are a nice geometric tool for
From playlist Essence of linear algebra
Ivan Mirkovic: Loop Grassmanians and local spaces
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
Tony Feng - 1/3 Derived Aspects of the Langlands Program
We discuss ways in which derived structures have recently emerged in connection with the Langlands correspondence, with an emphasis on derived Galois deformation rings and derived Hecke algebras. Michael Harris (Columbia Univ.) Tony Feng (MIT)
From playlist 2022 Summer School on the Langlands program
Clark Barwick - 3/3 Exodromy for ℓ-adic Sheaves
In joint work with Saul Glasman and Peter Haine, we proved that the derived ∞-category of constructible ℓ-adic sheaves ’is’ the ∞-category of continuous functors from an explicitly defined 1-category to the ∞-category of perfect complexes over ℚℓ. In this series of talks, I want to offer s
From playlist Summer School 2020: Motivic, Equivariant and Non-commutative Homotopy Theory
Dennis Gaitsgory - 1/4 Singular support of coherent sheaves
Singular support is an invariant that can be attached to a coherent sheaf on a derived scheme which is quasi-smooth (a.k.a. derived locally complete intersection). This invariant measures how far a given coherent sheaf is from being perfect. We will explain how the subtle difference betwee
From playlist Dennis Gaitsgory - Singular support of coherent sheaves
Some directions in derived geometry - Gabriele Vezzosi
Gabriele Vezzosi March 10, 2015 Workshop on Chow groups, motives and derived categories More videos on http://video.ias.edu
From playlist Mathematics
Cyril Demarche: Cohomological obstructions to local-global principles - lecture 4
Hasse proved that for quadrics the existence of rational points reduces to the existence of solutions over local fields. In many cases, cohomological constructions provide obstructions to such a local to global principle. The objective of these lectures is to give an introduction to these
From playlist Algebraic and Complex Geometry
Richard Thomas - Vafa-Witten Invariants of Projective Surfaces 4/5
This course has 4 sections split over 5 lectures. The first section will be the longest, and hopefully useful for the other courses. 1. Sheaves, moduli and virtual cycles 2. Vafa-Witten invariants: stable and semistable cases 3. Techniques for calculation --- virtual degeneracy loci, c
From playlist 2021 IHES Summer School - Enumerative Geometry, Physics and Representation Theory
Peter Scholze: Local acyclicity in p-adic geometry
Abstract: Motivated by applications to the geometric Satake equivalence and in particular the construction of the fusion product, we define a notion of universally locally acyclic for rigid spaces and diamonds, and prove that it has the expected properties. Recording during the meeting "p
From playlist Algebraic and Complex Geometry
Infinite Products of Projective Schemes Don't Exist
In this video we explain why infinite products of projective schemes don't exist as objects in the category of schemes.
From playlist Schemes
This is the third video of a series from the Worldwide Center of Mathematics explaining the basics of vectors. This video explains the precise definition of dot product (also known as scalar product) and shows some examples of calculated dot products. For more math videos, visit our channe
From playlist Basics: Vectors