A variety V over a finite field with q elements has a finite number of rational points, as well as points over every finite field with qk elements containing that field. The generating function has coefficients derived from the numbers Nk of points over the (essentially unique) field with qk elements.
In algebraic geometry, an étale morphism (French:[eˈtal]) is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology.
The word étale is a French adjective, which means "slack", as in "slack tide", or, figuratively, calm, immobile, something left to settle.
Definition
Let be a ring homomorphism. This makes an -algebra. Choose a monic polynomial in and a polynomial in such that the derivative of is a unit in . We say that is standard étale if and can be chosen so that is isomorphic as an -algebra to and is the canonical map.
short introduction for The Weil Conjectures
#mathematics
#The Weil Conjectures
If you would like to know more about it , I recommend following books
https://amzn.to/46t6fmn
https://amzn.to/46ptttD
published: 29 Jun 2023
What are the Weil conjectures about?
The story of the Weil conjectures forms one of the most interesting chapters in the history of 20th century mathematics. The proof, following the conjectural proof Weil gave led to the Fields medals of Grothendieck and Deligne. This is the first video in a playlist concerning these conjectures where we give a gentle introduction into what they are about.
published: 31 Jul 2024
Timothy Gowers: The Weil conjectures explained
Sir William Timothy Gowers is a British mathematician and a Royal Society Research Professor at the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge.
This video is a clip from the Abel Prize Announcement in 2013.
published: 23 Dec 2019
Weil conjectures 1 Introduction
This talk is the first of a series of talks on the Weil conejctures.
We recall properties of the Riemann zeta function, and describe how Artin used these to motivate the definition of the zeta function of a curve over a finite field. We then describe Weil's generalization of this to varieties of
higher dimension over finite fields, and conclude by stating the Weil conjectures about these zeta functions, including the analog of the Riemann hypothesis.
published: 08 Oct 2020
The Weil Conjectures, from Abel to Deligne - Sophie Morel
The Weil Conjectures, from Abel to Deligne
Sophie Morel
Princeton University
October 14, 2013
published: 06 Jul 2017
The Weil conjectures - Lothar Goettsche - 2016
BASIC NOTIONS SEMINAR SERIES 2016
Thursday, 19 May, at 16:00 hrs.
Lothar Goettsche (ICTP)
Title: The Weil conjectures
Abstract: The Weil conjectures is a deep and famous theorem (conjectured by Andre Weil and proven after work of others by Pierre Deligne) in algebraic geometry, which relates the homology (thus the topology) of complex projective algebraic varieties to counting of points of algebraic varieties over finite fields.
The aim of this lecture is to give an idea of what the theorem says and give some elementary examples.
published: 20 May 2016
The Biggest Project in Modern Mathematics
In a 1967 letter to the number theorist André Weil, a 30-year-old mathematician named Robert Langlands outlined striking conjectures that predicted a correspondence between two objects from completely different fields of math. The Langlands program was born. Today, it's one of the most ambitious mathematical feats ever attempted. Its symmetries imply deep, powerful and beautiful connections between the most important branches of mathematics. Many mathematicians agree that it has the potential to solve some of math's most intractable problems, in time, becoming a kind of “grand unified theory of mathematics," as the mathematician Edward Frenkel has described it. In a new video explainer, Rutgers University mathematician Alex Kontorovich takes us on a journey through the continents of mathem...
published: 01 Jun 2022
Weil conjectures for beginners
A lecture series for advanced undergrads and grad students in mathematics.
published: 08 Mar 2023
Dan Boneh on André Weil's advice that every mathematician spend some time in jail
This always cracks me up :)
Original video here (at 00:35:05): https://www.youtube.com/watch?v=8WDOpzxpnTE
published: 06 May 2020
Etale Cohomology and the Weil Conjectures
This was prepared for Matt Satriano's PMATH 965 course on algebraic stacks at the University of Waterloo in the Winter 2022 term. We give a quick introduction to etale and l-adic cohomology groups of a scheme, and discuss how they were used to prove the Weil conjectures by Grothendieck, M. Artin, Verdier, Dwork, and Deligne.
short introduction for The Weil Conjectures
#mathematics
#The Weil Conjectures
If you would like to know more about it , I recommend following books
https:/...
short introduction for The Weil Conjectures
#mathematics
#The Weil Conjectures
If you would like to know more about it , I recommend following books
https://amzn.to/46t6fmn
https://amzn.to/46ptttD
short introduction for The Weil Conjectures
#mathematics
#The Weil Conjectures
If you would like to know more about it , I recommend following books
https://amzn.to/46t6fmn
https://amzn.to/46ptttD
The story of the Weil conjectures forms one of the most interesting chapters in the history of 20th century mathematics. The proof, following the conjectural pr...
The story of the Weil conjectures forms one of the most interesting chapters in the history of 20th century mathematics. The proof, following the conjectural proof Weil gave led to the Fields medals of Grothendieck and Deligne. This is the first video in a playlist concerning these conjectures where we give a gentle introduction into what they are about.
The story of the Weil conjectures forms one of the most interesting chapters in the history of 20th century mathematics. The proof, following the conjectural proof Weil gave led to the Fields medals of Grothendieck and Deligne. This is the first video in a playlist concerning these conjectures where we give a gentle introduction into what they are about.
Sir William Timothy Gowers is a British mathematician and a Royal Society Research Professor at the Department of Pure Mathematics and Mathematical Statistics a...
Sir William Timothy Gowers is a British mathematician and a Royal Society Research Professor at the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge.
This video is a clip from the Abel Prize Announcement in 2013.
Sir William Timothy Gowers is a British mathematician and a Royal Society Research Professor at the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge.
This video is a clip from the Abel Prize Announcement in 2013.
This talk is the first of a series of talks on the Weil conejctures.
We recall properties of the Riemann zeta function, and describe how Artin used these to mo...
This talk is the first of a series of talks on the Weil conejctures.
We recall properties of the Riemann zeta function, and describe how Artin used these to motivate the definition of the zeta function of a curve over a finite field. We then describe Weil's generalization of this to varieties of
higher dimension over finite fields, and conclude by stating the Weil conjectures about these zeta functions, including the analog of the Riemann hypothesis.
This talk is the first of a series of talks on the Weil conejctures.
We recall properties of the Riemann zeta function, and describe how Artin used these to motivate the definition of the zeta function of a curve over a finite field. We then describe Weil's generalization of this to varieties of
higher dimension over finite fields, and conclude by stating the Weil conjectures about these zeta functions, including the analog of the Riemann hypothesis.
BASIC NOTIONS SEMINAR SERIES 2016
Thursday, 19 May, at 16:00 hrs.
Lothar Goettsche (ICTP)
Title: The Weil conjectures
Abstract: The Weil conjectures is a...
BASIC NOTIONS SEMINAR SERIES 2016
Thursday, 19 May, at 16:00 hrs.
Lothar Goettsche (ICTP)
Title: The Weil conjectures
Abstract: The Weil conjectures is a deep and famous theorem (conjectured by Andre Weil and proven after work of others by Pierre Deligne) in algebraic geometry, which relates the homology (thus the topology) of complex projective algebraic varieties to counting of points of algebraic varieties over finite fields.
The aim of this lecture is to give an idea of what the theorem says and give some elementary examples.
BASIC NOTIONS SEMINAR SERIES 2016
Thursday, 19 May, at 16:00 hrs.
Lothar Goettsche (ICTP)
Title: The Weil conjectures
Abstract: The Weil conjectures is a deep and famous theorem (conjectured by Andre Weil and proven after work of others by Pierre Deligne) in algebraic geometry, which relates the homology (thus the topology) of complex projective algebraic varieties to counting of points of algebraic varieties over finite fields.
The aim of this lecture is to give an idea of what the theorem says and give some elementary examples.
In a 1967 letter to the number theorist André Weil, a 30-year-old mathematician named Robert Langlands outlined striking conjectures that predicted a correspond...
In a 1967 letter to the number theorist André Weil, a 30-year-old mathematician named Robert Langlands outlined striking conjectures that predicted a correspondence between two objects from completely different fields of math. The Langlands program was born. Today, it's one of the most ambitious mathematical feats ever attempted. Its symmetries imply deep, powerful and beautiful connections between the most important branches of mathematics. Many mathematicians agree that it has the potential to solve some of math's most intractable problems, in time, becoming a kind of “grand unified theory of mathematics," as the mathematician Edward Frenkel has described it. In a new video explainer, Rutgers University mathematician Alex Kontorovich takes us on a journey through the continents of mathematics to learn about the awe-inspiring symmetries at the heart of the Langlands program, including how Andrew Wiles solved Fermat's Last Theorem.
Read more at Quanta Magazine: https://www.quantamagazine.org/what-is-the-langlands-program-20220531/
00:00 A map of the mathematical world
00:25 The land of Number Theory"
00:39 The continent of Harmonic Analysis
01:20 A bridge: the Langlands Program
01:46 Robert Langlands' conjectures link the two worlds
02:40 Ramanujan Discriminant Function
03:00 Modular Forms
04:36 Pierre Deligne's proof of Ramanujan's conjecture
04:47 Functoriality
05:03 Pierre De Fermat's Last Theorem
06:13 Andrew Wiles builds a bridge
06:30 Elliptic curves
07:07 Modular arithmetic
08:56 Infinite power series
09:20 Taniyama - Shimura - Weil conjecture
10:40 Frey's counterexample to Frey's last theorem
11:30 Wiles' proof of Fermat's Last Theorem
- VISIT our Website: https://www.quantamagazine.org
- LIKE us on Facebook: https://www.facebook.com/QuantaNews
- FOLLOW us Twitter: https://twitter.com/QuantaMagazine
Quanta Magazine is an editorially independent publication supported by the Simons Foundation https://www.simonsfoundation.org/
In a 1967 letter to the number theorist André Weil, a 30-year-old mathematician named Robert Langlands outlined striking conjectures that predicted a correspondence between two objects from completely different fields of math. The Langlands program was born. Today, it's one of the most ambitious mathematical feats ever attempted. Its symmetries imply deep, powerful and beautiful connections between the most important branches of mathematics. Many mathematicians agree that it has the potential to solve some of math's most intractable problems, in time, becoming a kind of “grand unified theory of mathematics," as the mathematician Edward Frenkel has described it. In a new video explainer, Rutgers University mathematician Alex Kontorovich takes us on a journey through the continents of mathematics to learn about the awe-inspiring symmetries at the heart of the Langlands program, including how Andrew Wiles solved Fermat's Last Theorem.
Read more at Quanta Magazine: https://www.quantamagazine.org/what-is-the-langlands-program-20220531/
00:00 A map of the mathematical world
00:25 The land of Number Theory"
00:39 The continent of Harmonic Analysis
01:20 A bridge: the Langlands Program
01:46 Robert Langlands' conjectures link the two worlds
02:40 Ramanujan Discriminant Function
03:00 Modular Forms
04:36 Pierre Deligne's proof of Ramanujan's conjecture
04:47 Functoriality
05:03 Pierre De Fermat's Last Theorem
06:13 Andrew Wiles builds a bridge
06:30 Elliptic curves
07:07 Modular arithmetic
08:56 Infinite power series
09:20 Taniyama - Shimura - Weil conjecture
10:40 Frey's counterexample to Frey's last theorem
11:30 Wiles' proof of Fermat's Last Theorem
- VISIT our Website: https://www.quantamagazine.org
- LIKE us on Facebook: https://www.facebook.com/QuantaNews
- FOLLOW us Twitter: https://twitter.com/QuantaMagazine
Quanta Magazine is an editorially independent publication supported by the Simons Foundation https://www.simonsfoundation.org/
This was prepared for Matt Satriano's PMATH 965 course on algebraic stacks at the University of Waterloo in the Winter 2022 term. We give a quick introduction t...
This was prepared for Matt Satriano's PMATH 965 course on algebraic stacks at the University of Waterloo in the Winter 2022 term. We give a quick introduction to etale and l-adic cohomology groups of a scheme, and discuss how they were used to prove the Weil conjectures by Grothendieck, M. Artin, Verdier, Dwork, and Deligne.
This was prepared for Matt Satriano's PMATH 965 course on algebraic stacks at the University of Waterloo in the Winter 2022 term. We give a quick introduction to etale and l-adic cohomology groups of a scheme, and discuss how they were used to prove the Weil conjectures by Grothendieck, M. Artin, Verdier, Dwork, and Deligne.
short introduction for The Weil Conjectures
#mathematics
#The Weil Conjectures
If you would like to know more about it , I recommend following books
https://amzn.to/46t6fmn
https://amzn.to/46ptttD
The story of the Weil conjectures forms one of the most interesting chapters in the history of 20th century mathematics. The proof, following the conjectural proof Weil gave led to the Fields medals of Grothendieck and Deligne. This is the first video in a playlist concerning these conjectures where we give a gentle introduction into what they are about.
Sir William Timothy Gowers is a British mathematician and a Royal Society Research Professor at the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge.
This video is a clip from the Abel Prize Announcement in 2013.
This talk is the first of a series of talks on the Weil conejctures.
We recall properties of the Riemann zeta function, and describe how Artin used these to motivate the definition of the zeta function of a curve over a finite field. We then describe Weil's generalization of this to varieties of
higher dimension over finite fields, and conclude by stating the Weil conjectures about these zeta functions, including the analog of the Riemann hypothesis.
BASIC NOTIONS SEMINAR SERIES 2016
Thursday, 19 May, at 16:00 hrs.
Lothar Goettsche (ICTP)
Title: The Weil conjectures
Abstract: The Weil conjectures is a deep and famous theorem (conjectured by Andre Weil and proven after work of others by Pierre Deligne) in algebraic geometry, which relates the homology (thus the topology) of complex projective algebraic varieties to counting of points of algebraic varieties over finite fields.
The aim of this lecture is to give an idea of what the theorem says and give some elementary examples.
In a 1967 letter to the number theorist André Weil, a 30-year-old mathematician named Robert Langlands outlined striking conjectures that predicted a correspondence between two objects from completely different fields of math. The Langlands program was born. Today, it's one of the most ambitious mathematical feats ever attempted. Its symmetries imply deep, powerful and beautiful connections between the most important branches of mathematics. Many mathematicians agree that it has the potential to solve some of math's most intractable problems, in time, becoming a kind of “grand unified theory of mathematics," as the mathematician Edward Frenkel has described it. In a new video explainer, Rutgers University mathematician Alex Kontorovich takes us on a journey through the continents of mathematics to learn about the awe-inspiring symmetries at the heart of the Langlands program, including how Andrew Wiles solved Fermat's Last Theorem.
Read more at Quanta Magazine: https://www.quantamagazine.org/what-is-the-langlands-program-20220531/
00:00 A map of the mathematical world
00:25 The land of Number Theory"
00:39 The continent of Harmonic Analysis
01:20 A bridge: the Langlands Program
01:46 Robert Langlands' conjectures link the two worlds
02:40 Ramanujan Discriminant Function
03:00 Modular Forms
04:36 Pierre Deligne's proof of Ramanujan's conjecture
04:47 Functoriality
05:03 Pierre De Fermat's Last Theorem
06:13 Andrew Wiles builds a bridge
06:30 Elliptic curves
07:07 Modular arithmetic
08:56 Infinite power series
09:20 Taniyama - Shimura - Weil conjecture
10:40 Frey's counterexample to Frey's last theorem
11:30 Wiles' proof of Fermat's Last Theorem
- VISIT our Website: https://www.quantamagazine.org
- LIKE us on Facebook: https://www.facebook.com/QuantaNews
- FOLLOW us Twitter: https://twitter.com/QuantaMagazine
Quanta Magazine is an editorially independent publication supported by the Simons Foundation https://www.simonsfoundation.org/
This was prepared for Matt Satriano's PMATH 965 course on algebraic stacks at the University of Waterloo in the Winter 2022 term. We give a quick introduction to etale and l-adic cohomology groups of a scheme, and discuss how they were used to prove the Weil conjectures by Grothendieck, M. Artin, Verdier, Dwork, and Deligne.
A variety V over a finite field with q elements has a finite number of rational points, as well as points over every finite field with qk elements containing that field. The generating function has coefficients derived from the numbers Nk of points over the (essentially unique) field with qk elements.
I don't love, I don't give, I don't care' cause I've already been there I don't want, I don't need, I don't take, 'cause I had it all I don't guess, I just know, what this is, is a place to die from I am not what you see, you are real but not for real I'm not scared, I'm not high, I'm a man and I'm all right We are here, we are gone, and the rest are left to wonder What is life, in the hand of illusion, what is life to a man in confusion What is right, for a life that is losin', what is time, what is, what is, what is? Take look, open up, look inside, what is there In your heart, in your mind, in your soul there's the answer Why you want, why you need, the side effects of what you see What is life, in the hand of illusion What is life, to a man in confusion What is right for a life that is losin', what is time, what is, what is, what is? I don't pay, I just know what this is,