Conjectures | Algebraic number theory
Kronecker's Jugendtraum or Hilbert's twelfth problem, of the 23 mathematical Hilbert problems, is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. That is, it asks for analogues of the roots of unity, as complex numbers that are particular values of the exponential function; the requirement is that such numbers should generate a whole family of further number fields that are analogues of the cyclotomic fields and their subfields. The classical theory of complex multiplication, now often known as the Kronecker Jugendtraum, does this for the case of any imaginary quadratic field, by using modular functions and elliptic functions chosen with a particular period lattice related to the field in question. Goro Shimura extended this to CM fields. In the special case of totally real fields, a solution was given by Dasgupta and Kakde. This provides an effective method to construct the maximal abelian extension of any totally real field. The method rests on p-adic integration and the solution it provides for totally real fields is different in nature from what Hilbert had in mind in his original formulation. A solution in the more special case of totally real quadratic fields, also resting on p-adic methods, was given by Darmon, Pozzi and Vonk. The general case of Hilbert's 12th Problem is still open as of 2022. Leopold Kronecker described the complex multiplication issue as his liebster Jugendtraum or “dearest dream of his youth”. (Wikipedia).
Hilbert's 10th Problem: Decision Problem on Solvability of Diophantine Equations
#shorts #mathonshorts
From playlist Elementary Number Theory
M. Kisin - Hilbert's thirteenth problem and the moduli space of abelian varieties
The (multi-valued) solution of a general polynomial of degree n is a priori a function of n-1 variables. Hilbert's thirteenth problem and its variants ask when such functions can be written as a composite of functions in a smaller number of variables. I will explain some progress on this q
From playlist Arithmetic and Algebraic Geometry: A conference in honor of Ofer Gabber on the occasion of his 60th birthday
Turing Machines & The Halting Problem (Part 1)
In the year 1900, David Hilbert gave a list of 23 mathematics problems for the mathematicians of the new generation. His tenth problem proved to be an enigma for many years until Alan Turing solved it while simultaneously creating the modern computer. Watch the video to see how Alan Turi
From playlist Math
Space-Filling Curves (2 of 4: Hilbert Curve)
More resources available at www.misterwootube.com
From playlist Exploring Mathematics: Fractals
Yuri Matiyasevich - On Hilbert's 10th Problem [2000]
On Hilbert's 10th Problem - Part 1 of 4 Speaker: Yuri Matiyasevich Date: Wed, Mar 1, 2000 Location: PIMS, University of Calgary Abstract: A Diophantine equation is an equation of the form $ D(x_1,...,x_m) $ = 0, where D is a polynomial with integer coefficients. These equations were n
From playlist Number Theory
The Geometry of Hilbert's 13th problem - Jesse Wolfson
Special Seminar on Hilbert's 13th Problem I Topic: The Geometry of Hilbert's 13th problem Speaker: Jesse Wolfson Affiliation: University of California, Irvine Date: December 5, 2019 For more video please visit http://video.ias.edu
From playlist Mathematics
Lecture 24. Hilbert basis theorem
From playlist Abstract Algebra 2
This shows a 3d print of a mathematical sculpture I produced using shapeways.com. This model is available at http://shpws.me/2toQ.
From playlist 3D printing
Explicit formulae for Stark Units and Hilbert's 12th problem - Samit Dasgupta
Joint IAS/Princeton University Number Theory Seminar Topic: Explicit formulae for Stark Units and Hilbert's 12th problem Speaker: Samit Dasgupta Affiliation: Duke University Date: October 11, 2018 For more video please visit http://video.ias.edu
From playlist Mathematics
MATH1081 Discrete Maths: Chapter 4 Question 26
Here we find the number of solutions to x1+x2+...+xr less than or equal n with xi be positive integer or zero by using the basic fact that the number of solutions to x1+x2+...+xm = n with xi be positive integer or zero is binomial coefficients m+n-1 choose m-1. Presented by Peter Brown of
From playlist MATH1081 Discrete Mathematics
The Riemann Hypothesis is one of the Millennium Prize Problems and has something to do with primes. What's that all about? Rather than another hand-wavy explanation, I've tried to put in some details here. Some grown-up maths follows. More information: http://www.claymath.org/publications
From playlist My Maths Videos
Eyal Markman: Hyperholomorphic sheaves and generalized deformations of K3 surfaces
This talk will elaborate on the role hyperholomorphic sheaves play in generalized deformations of K3 surfaces, described in the talk of Sukhendu Mehrotra. The lecture was held within the framework of the Junior Hausdorff Trimester Program Algebraic Geometry. (12.2.2014)
From playlist HIM Lectures: Junior Trimester Program "Algebraic Geometry"
Explicit formulae for Gross-Stark units and Hilbert’s 12th problem by Mahesh Kakde
PERFECTOID SPACES ORGANIZERS: Debargha Banerjee, Denis Benois, Chitrabhanu Chaudhuri, and Narasimha Kumar Cheraku DATE & TIME: 09 September 2019 to 20 September 2019 VENUE: Madhava Lecture Hall, ICTS, Bangalore Scientific committee: Jacques Tilouine (University of Paris, France) Eknath
From playlist Perfectoid Spaces 2019
Non-equilibrium and periodically driven quantum systems-3 by Arnab Sen
PROGRAM : BANGALORE SCHOOL ON STATISTICAL PHYSICS - XII (ONLINE) ORGANIZERS : Abhishek Dhar (ICTS-TIFR, Bengaluru) and Sanjib Sabhapandit (RRI, Bengaluru) DATE : 28 June 2021 to 09 July 2021 VENUE : Online Due to the ongoing COVID-19 pandemic, the school will be conducted through online
From playlist Bangalore School on Statistical Physics - XII (ONLINE) 2021
Spenta Wadia - Fermion-Boson Duality in 2+1 dim. large N Gauge Theories (1)
PROGRAM: THE 8TH ASIAN WINTER SCHOOL ON STRINGS, PARTICLES AND COSMOLOGY DATES: Thursday 09 Jan, 2014 - Saturday 18 Jan, 2014 VENUE: Blue Lily Hotel, Puri PROGRAM LINK: http://www.icts.res.in/program/asian8 The 8th Asian Winter School on Strings, Particles and Cosmology is part of a seri
From playlist The 8th Asian Winter School on Strings, Particles and Cosmology
Eureka Math Grade 4 Module 5 Lesson 33 (updated)
EngageNY/Eureka Math Grade 4 Module 5 Lesson 33 For more Eureka Math (EngageNY) videos and other resources, please visit http://EMBARC.online PLEASE leave a message if a video has a technical difficulty (audio separating from the video, writing not showing up, etc). Occasionally, Explain
From playlist Eureka Math Grade 4 Module 5
Introduction to Outlier Detection Methods (Part 2) - Wolfram Livecoding Session
Andreas Lauschke, a senior mathematical programmer, live-demos key Wolfram Language features useful in data science. In this seventh session, the introduction to outlier detection methods continues, and the basics of continuous probability theory are recapped. Then learn about the built-in
From playlist Data Science with Andreas Lauschke
EngageNY Grade 5 Module 3 Lesson 12
EngageNY/Eureka Math Grade 5 Module 3 Lesson 12 For more Eureka Math (EngageNY) videos and other resources, please visit http://EMBARC.online PLEASE leave a message if a video has a technical difficulty (audio separating from the video, writing not showing up, etc). Occasionally, Explain
From playlist Eureka Math Grade 5 Module 3
The Most Difficult Math Problem You've Never Heard Of - Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a millennium prize problem, one of the famed seven placed by the Clay Mathematical Institute in the year 2000. As the only number-theoretic problem in the list apart from the Riemann Hypothesis, the BSD Conjecture has been haunting mathematicians
From playlist Math
Comedy in Shakespeare's Twelfth Night
In this lecture, Dr Sophie Duncan (University of Oxford) thinks about comedy in Shakespeare's Twelfth Night, focusing in particular on the idea of genre in early modern England and whether we should trust the First Folio’s categorisation of Twelfth Night as a comedy. This lecture is part
From playlist English Literature