Commutative algebra 59: Krull's principal ideal theorem
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 give some applications of the theorems we proved about the dimension of local rings. We first show that the dimension of a
From playlist Commutative algebra
Abstract Algebra | Principal Ideals of a Ring
We define the notion of a principal ideal of a ring and give some examples. We also prove that all ideals of the integers are principal ideals, that is, the integers form a principal ideal domain (PID). http://www.michael-penn.net https://www.researchgate.net/profile/Michael_Penn5 http://
From playlist Abstract Algebra
Commutative algebra 58: System of parameters versus Krull
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 show that the smallest size of a system of parameters of a Noetherian local ring is at most the Krull dimension. The proof
From playlist Commutative algebra
P. Scholze - p-adic K-theory of p-adic rings
The original proof of Grothendieck's purity conjecture in étale cohomology (the Thomason-Gabber theorem) relies on results on l-adic K-theory and its relation to étale cohomology when l is invertible. Using recent advances of Clausen-Mathew-Morrow and joint work with Bhatt and Morrow, our
From playlist Arithmetic and Algebraic Geometry: A conference in honor of Ofer Gabber on the occasion of his 60th birthday
Introduction to additive combinatorics lecture 10.8 --- A weak form of Freiman's theorem
In this short video I explain how the proof of Freiman's theorem for subsets of Z differs from the proof given earlier for subsets of F_p^N. The answer is not very much: the main differences are due to the fact that cyclic groups of prime order do not have lots of subgroups, so one has to
From playlist Introduction to Additive Combinatorics (Cambridge Part III course)
Abstract Algebra | k[x] is a PID
We prove that every ideal of the polynomial ring k[x], where k is a field, is principal. That is, k[x] is a principal ideal domain. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Personal Website: http://www.michael-penn.net Randolph College Math: http://www
From playlist Abstract Algebra
Commutative algebra 55: Dimension of local rings
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 give 4 definitions of the dimension of a Noetherian local ring: Brouwer-Menger-Urysohn dimension, Krull dimension, degree o
From playlist Commutative algebra
Arthur Bartels: K-theory of group rings (Lecture 1)
The lecture was held within the framework of the Hausdorff Trimester Program: K-Theory and Related Fields. Arthur Bartels: K-theory of group rings The Farrell-Jones Conjecture predicts that the K-theory of group rings RG can be computed in terms of K-theory of group rings RV where V vari
From playlist HIM Lectures: Trimester Program "K-Theory and Related Fields"
Commutative algebra 57: Krull versus Hilbert
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 continue the previous video by showing that the Krull dimension of a Noetherian local ring is at most the dimension defined
From playlist Commutative algebra
Origin and Development of Valuation Theory by Sudesh Khanduja
DATES: Monday 29 Aug, 2016 - Tuesday 30 Aug, 2016 VENUE: Madhava Lecture Hall, ICTS Bangalore Emmy Noether (1882-1935) is well known for her famous contributions to abstract algebra and theoretical physics. Noether’s mathematical work has been divided into three ”epochs”. In the first (
From playlist The Legacy of Emmy Noether
Emmy Noether in Erlangen and Göttingen by Ravi Rao
DATES: Monday 29 Aug, 2016 - Tuesday 30 Aug, 2016 VENUE: Madhava Lecture Hall, ICTS Bangalore Emmy Noether (1882-1935) is well known for her famous contributions to abstract algebra and theoretical physics. Noether’s mathematical work has been divided into three ”epochs”. In the first (
From playlist The Legacy of Emmy Noether
Alfred Geroldinger: A characterization of class groups via sets of lengths
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 Combinatorics
Introduction to additive combinatorics lecture 1.8 --- Plünnecke's theorem
In this video I present a proof of Plünnecke's theorem due to George Petridis, which also uses some arguments of Imre Ruzsa. Plünnecke's theorem is a very useful tool in additive combinatorics, which implies that if A is a set of integers such that |A+A| is at most C|A|, then for any pair
From playlist Introduction to Additive Combinatorics (Cambridge Part III course)
Introduction to additive combinatorics lecture 5.8 --- Freiman homomorphisms and isomorphisms.
The notion of a Freiman homomorphism and the closely related notion of a Freiman isomorphism are fundamental concepts in additive combinatorics. Here I explain what they are and prove a lemma that states that a subset A of F_p^N such that kA - kA is not too large is "k-isomorphic" to a sub
From playlist Introduction to Additive Combinatorics (Cambridge Part III course)
Rings and modules 4 Unique factorization
This lecture is part of an online course on rings and modules. We discuss unique factorization in rings, showing the implications (Integers) implies (Euclidean domain) implies (Principal ideal domain) implies (Unique factorization domain). We give a few examples to illustrate these implic
From playlist Rings and modules
Seminar on Applied Geometry and Algebra (SIAM SAGA): Sonja Petrović
Date: Tuesday, January 12, 2021 at 11:00am EST (5:00pm CET) Speaker: Sonja Petrović, Illinois Institute of Technology Title: Random Monomial Ideals Abstract: Joint work with Jesus A. De Loera, Lily Silverstein, Despina Stasi, Dane Wilburne. Inspired by the study of random graphs and sim
From playlist Seminar on Applied Geometry and Algebra (SIAM SAGA)
Notes by Keith Conrad to follow along: https://kconrad.math.uconn.edu/blurbs/linmultialg/modulesoverPID.pdf
From playlist Abstract Algebra 2
The Fundamental Theorem of Calculus | Algebraic Calculus One | Wild Egg
In this video we lay out the Fundamental Theorem of Calculus --from the point of view of the Algebraic Calculus. This key result, presented here for the very first time (!), shows how to generalize the Fundamental Formula of the Calculus which we presented a few videos ago, incorporating t
From playlist Algebraic Calculus One