This lecture is part of an online course on rings and modules. We continue the previous lecture on complete rings by discussing Hensel's lemma for finding roots of polynomials over p-adic rings or over power series rings. We sketch two proofs, by slowly improving a root one digit at a tim
From playlist Rings and modules
Commutative algebra 51: Hensel's lemma continued
This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. This lecture continues the discussion of Hensel's lemma. We first use it to find the structure of the group of units of the p-
From playlist Commutative algebra
NIP Henselian fields - F. Jahnke - Workshop 2 - CEB T1 2018
Franziska Jahnke (Münster) / 05.03.2018 NIP henselian fields We investigate the question which henselian valued fields are NIP. In equicharacteristic 0, this is well understood due to the work of Delon: an henselian valued field of equicharacteristic 0 is NIP (as a valued field) if and on
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
Ring Theory: We define rings and give many examples. Items under consideration include commutativity and multiplicative inverses. Example include modular integers, square matrices, polynomial rings, quaternions, and adjoins of algebraic and transcendental numbers.
From playlist Abstract Algebra
Borromean Onion Rings, the perfect way to top your Green Bean Matherole! Borromean onion rings were invented by special guest Marc ten Bosch (http://marctenbosch.com). Also shown are gelatinous cranberry cylinder, bread spheres and butter prism, mathed potatoes, apple pie, and pumpkin tau.
From playlist Doodling in Math and more | Math for fun and glory | Khan Academy
Commutative algebra 50: Hensel's lemma
This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. We describe Hensel's lemma for finding roots of polynomials over complete rings, and give some examples of using it to find wh
From playlist Commutative algebra
Ring Definition (expanded) - Abstract Algebra
A ring is a commutative group under addition that has a second operation: multiplication. These generalize a wide variety of mathematical objects like the integers, polynomials, matrices, modular arithmetic, and more. In this video we will take an in depth look at the definition of a rin
From playlist Abstract Algebra
Abstract Algebra: The definition of a Ring
Learn the definition of a ring, one of the central objects in abstract algebra. We give several examples to illustrate this concept including matrices and polynomials. Be sure to subscribe so you don't miss new lessons from Socratica: http://bit.ly/1ixuu9W ♦♦♦♦♦♦♦♦♦♦ We recommend th
From playlist Abstract Algebra
Kęstutis Česnavičius - Grothendieck–Serre in the quasi-split unramified case
Correction: The affiliation of Lei Fu is Tsinghua University. The Grothendieck–Serre conjecture predicts that every generically trivial torsor under a reductive group scheme G over a regular local ring R is trivial. We settle it in the case when G is quasi-split and R is unramified. To ov
From playlist Conférence « Géométrie arithmétique en l’honneur de Luc Illusie » - 5 mai 2021
Toward an imaginary Ax-Kochen-Ershov principle - S. Rideau - Workshop 2 - CEB T1 2018
Silvain Rideau (CNRS – Université Paris Diderot) / 09.03.2018 Toward an imaginary Ax-Kochen-Ershov principle. All imaginaries that have been classified in Henselian fields (possibly with operators) have been shown to be geometric in the sense of Haskell-HrushovskiMacpherson. In general,
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
A. Vasiu - On the classification of p-healthy regular schemes
A regular local ring R of dimension at least 2 and mixed characteristic (0,p) is called p-healthy if each p-divisible group over the the punctured spectrum of R extends to a p-divisible group over Spec R. In the book of Faltings and Chai, it has been claimed that, in the current language,
From playlist Arithmetic and Algebraic Geometry: A conference in honor of Ofer Gabber on the occasion of his 60th birthday
Brauer groups, Severi-Brauer schemes, Azumaya algebras, twisted sheaves
From playlist Étale cohomology and the Weil conjectures
This is part of a series of lectures on the Zermelo-Fraenkel axioms for set theory. We discuss the powerset axiom, the strongest of the ZF axioms, and explain why the notion of a powerset is so hard to pin down precisely. For the other lectures in the course see https://www.youtube.com
From playlist Zermelo Fraenkel axioms
The Mandelbrot set is a churning machine
Its job is to fling off the red pixels and hang onto the green ones. Audio by @Dorfmandesign
From playlist mandelstir
CTNT 2022 - Local Fields (Lecture 4) - by Christelle Vincent
This video is part of a mini-course on "Local Fields" that was taught during CTNT 2022, the Connecticut Summer School and Conference in Number Theory. More about CTNT: https://ctnt-summer.math.uconn.edu/
From playlist CTNT 2022 - Local Fields (by Christelle Vincent)
Gabriele Vezzosi - Applications of non-commutative algebraic geometry to arithmetic geometry
Abstract: We will briefly recall the general philosophy of non-commutative (and derived) algebraic geometry in order to establish a precise link between dg-derived category of singularities of Landau-Ginzburg models and vanishing cohomology, over an arbitrary henselian trait. We will then
From playlist Algebraic Analysis in honor of Masaki Kashiwara's 70th birthday
