Theorems in topology | Diffeomorphisms | 3-manifolds | Conjectures
In mathematics, the Smith conjecture states that if f is a diffeomorphism of the 3-sphere of finite order, then the fixed point set of f cannot be a nontrivial knot. Paul A. Smith showed that a non-trivial orientation-preserving diffeomorphism of finite order with fixed points must have a fixed point set equal to a circle, and asked in if the fixed point set could be knotted. Friedhelm Waldhausen proved the Smith conjecture for the special case of diffeomorphisms of order 2 (and hence any even order). The proof of the general case was described by John Morgan and Hyman Bass and depended on several major advances in 3-manifold theory, In particular the work of William Thurston on hyperbolic structures on 3-manifolds, and results by William Meeks and Shing-Tung Yau on minimal surfaces in 3-manifolds, with some additional help from Bass, Cameron Gordon, Peter Shalen, and Rick Litherland. Deane Montgomery and Leo Zippin gave an example of a continuous involution of the 3-sphere whose fixed point set is a wildly embedded circle, so the Smith conjecture is false in the topological (rather than the smooth or PL) category. Charles Giffen showed that the analogue of the Smith conjecture in higher dimensions is false: the fixed point set of a periodic diffeomorphism of a sphere of dimension at least 4 can be a knotted sphere of codimension 2. (Wikipedia).
Number theory Full Course [A to Z]
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure #mathematics devoted primarily to the study of the integers and integer-valued functions. Number theorists study prime numbers as well as the properties of objects made out of integers (for example, ratio
From playlist Number Theory
Theory of numbers: Congruences: Euler's theorem
This lecture is part of an online undergraduate course on the theory of numbers. We prove Euler's theorem, a generalization of Fermat's theorem to non-prime moduli, by using Lagrange's theorem and group theory. As an application of Fermat's theorem we show there are infinitely many prim
From playlist Theory of numbers
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
From playlist Theory of numbers
What is the Riemann Hypothesis?
This video provides a basic introduction to the Riemann Hypothesis based on the the superb book 'Prime Obsession' by John Derbyshire. Along the way I look at convergent and divergent series, Euler's famous solution to the Basel problem, and the Riemann-Zeta function. Analytic continuation
From playlist Mathematics
In this video, we explore the "pattern" to prime numbers. I go over the Euler product formula, the prime number theorem and the connection between the Riemann zeta function and primes. Here's a video on a similar topic by Numberphile if you're interested: https://youtu.be/uvMGZb0Suyc The
From playlist Other Math Videos
Shou-Wu Zhang: Congruent number problem and BSD conjecture
Abstract : A thousand years old problem is to determine when a square free integer n is a congruent number ,i,e, the areas of right angled triangles with sides of rational lengths. This problem has a some beautiful connection with the BSD conjecture for elliptic curves En:ny2=x3−x. In fact
From playlist Jean-Morlet Chair - Research Talks - Prasad/Heiermann
John Pardon: Totally disconnected groups (not) acting on three-manifolds
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 Geometry
Theory of numbers: Fermat's theorem
This lecture is part of an online undergraduate course on the theory of numbers. We prove Fermat's theorem a^p = a mod p. We then define the order of a number mod p and use Fermat's theorem to show the order of a divides p-1. We apply this to testing some Fermat and Mersenne numbers to se
From playlist Theory of numbers
The Hecke category action on the principal block via Smith theory - Geordie Williamson
Geometric and Modular Representation Theory Seminar Topic: The Hecke category action on the principal block via Smith theory Speaker: Geordie Williamson Affiliation: University of Sydney; Distinguished Visiting Professor, School of Mathematics Date: January 27, 2021 For more video please
From playlist Geordie Williamson external seminars
Number Theory | A very special case of Fermat's Last Theorem
We prove a very simple case of Fermat's Last Theorem. Interestingly, this case is fairly easy to prove which highlights the allure of the theorem as a whole -- especially given the fact that much of modern number theory was developed as part of the program that ended in the full proof. ht
From playlist Number Theory
Geordie Williamson: Miraculous Treumann-Smith theory and geometric Satake
Abstract: This talk will be about geometric approaches to the representation theory of reductive algebraic groups in positive characteristic p. A cornerstone of the geometric theory is the geometric Satake equivalence, which gives an incarnation of the category of representations as a cate
From playlist Geordie Williamson: Representation theory and the Geometric Satake
Mateusz Michalek: "Algebraic methods to construct tensors"
Tensor Methods and Emerging Applications to the Physical and Data Sciences 2021 Workshop III: Mathematical Foundations and Algorithms for Tensor Computations "Algebraic methods to construct tensors" Mateusz Michalek - Universität Konstanz, Institute of Mathematics Abstract: We will prese
From playlist Tensor Methods and Emerging Applications to the Physical and Data Sciences 2021
Mertens Conjecture Disproof and the Riemann Hypothesis | MegaFavNumbers
#MegaFavNumbers The Mertens conjecture is a conjecture is a conjecture about the distribution of the prime numbers. It can be seen as a stronger version of the Riemann hypothesis. It says that the Mertens function is bounded by sqrt(n). The Riemann hypothesis on the other hand only require
From playlist MegaFavNumbers
Geordie Williamson: Parity sheaves and modular representations II
This is a talk of Gordie Williamson given at the Harvard CDM Conference of November 23, 2019.
From playlist Geordie Williamson: Parity sheaves and modular representations
New Age Linkage - Daniel Juteau
Geometric and Modular Representation Theory Seminar Topic: New Age Linkage Speaker: Daniel Juteau Affiliation: CNRS, Université Paris Diderot; Member, School of Mathematics Date: January 20, 2021 For more video please visit http://video.ias.edu
From playlist Seminar on Geometric and Modular Representation Theory
Smith theory and Langlands functoriality - Tony Feng
Workshop on Representation Theory and Geometry Topic: Smith theory and Langlands functoriality Speaker: Tony Feng Affiliation: Member, School of Mathematics Date: March 29, 2021 For more video please visit http://video.ias.edu
From playlist Mathematics
Xin Zhou - Recent developments in constant mean curvature hypersurfaces I
We will survey some recent existence theory of closed constant mean curvature hypersurfaces using the min-max method. We hope to discuss some old and new open problems on this topic as well. Xin Zhou (Cornell)
From playlist Not Only Scalar Curvature Seminar
2^k-Selmer groups and Goldfeld's conjecture. - Smith - Workshop 2 - CEB T2 2019
Alexander Smith (Harvard University) / 25.06.2019 2^k-Selmer groups and Goldfeld's conjecture. Take E to be an elliptic curve over a number field whose four torsion obeys certain technical conditions. In this talk, we will outline a proof that 100% of the quadratic twists of E have rank
From playlist 2019 - T2 - Reinventing rational points
The Prime Number Theorem, an introduction ← Number Theory
An introduction to the meaning and history of the prime number theorem - a fundamental result from analytic number theory. Narrated by Cissy Jones Artwork by Kim Parkhurst, Katrina de Dios and Olga Reukova Written & Produced by Michael Harrison & Kimberly Hatch Harrison ♦♦♦♦♦♦♦♦♦♦ Ways t
From playlist Number Theory