Theorems in algebraic geometry | Lemmas in algebra | Theorems in abstract algebra | Algebraic number theory

In mathematics, Abhyankar's lemma (named after Shreeram Shankar Abhyankar) allows one to kill tame ramification by taking an extension of a base field. More precisely, Abhyankar's lemma states that if A, B, C are local fields such that A and B are finite extensions of C, with ramification indices a and b, and B is tamely ramified over C and b divides a, then the compositumAB is an unramified extension of A. (Wikipedia).

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Yves André - Direct summand conjecture and perfectoid Abhyankar lemma: an overview

Séminaire Paris Pékin Tokyo / Mercredi 2 novembre 2016 According to Hochster's direct summand conjecture (1973), a regular ring R is a direct summand, as an R-module, of every finite extension ring. We shall outline our recent proof which relies on perfectoid techniques. Similar arguments

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Zero dimensional valuations on equicharacteristic (...) - B. Teissier - Workshop 2 - CEB T1 2018

Bernard Teissier (IMJ-PRG) / 06.03.2018 Zero dimensional valuations on equicharacteristic noetherian local domains. A study of those valuations based, in the case where the domain is complete, on the relations between the elements of a minimal system of generators of the value semigroup o

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Frédéric Mangolte: Algebraic models of the line in the real affine plane

Abstract: We study the following real version of the famous Abhyankar-Moh Theorem: Which real rational map from the affine line to the affine plane, whose real part is a non-singular real closed embedding of ℝ into ℝ^2, is equivalent, up to a birational diffeomorphism of the plane, to the

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Regularity methods in combinatorics, number theory, and computer science - Jacob Fox

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9. Szemerédi's graph regularity lemma IV: induced removal lemma

MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019 Instructor: Yufei Zhao View the complete course: https://ocw.mit.edu/18-217F19 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP62qauV_CpT1zKaGG_Vj5igX Prof. Zhao explains a strengthening of the graph regulari

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6. Szemerédi's graph regularity lemma I: statement and proof

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Theory of numbers: Gauss's lemma

This lecture is part of an online undergraduate course on the theory of numbers. We describe Gauss's lemma which gives a useful criterion for whether a number n is a quadratic residue of a prime p. We work it out explicitly for n = -1, 2 and 3, and as an application prove some cases of Di

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A stable arithmetic regularity lemma in finite (...) - C. Terry - Workshop 1 - CEB T1 2018

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7. Szemerédi's graph regularity lemma II: triangle removal lemma

MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019 Instructor: Yufei Zhao View the complete course: https://ocw.mit.edu/18-217F19 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP62qauV_CpT1zKaGG_Vj5igX Continuing the discussion of Szemerédi's graph regularity

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László Lovász: The many facets of the Regularity Lemma

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