Hamiltonian paths and cycles | Disproved conjectures | Planar graphs
In mathematics, Tait's conjecture states that "Every 3-connected planar cubic graph has a Hamiltonian cycle (along the edges) through all its vertices". It was proposed by P. G. Tait and disproved by W. T. Tutte, who constructed a counterexample with 25 faces, 69 edges and 46 vertices. Several smaller counterexamples, with 21 faces, 57 edges and 38 vertices, were later proved minimal by .The condition that the graph be 3-regular is necessary due to polyhedra such as the rhombic dodecahedron, which forms a bipartite graph with six degree-four vertices on one side and eight degree-three vertices on the other side; because any Hamiltonian cycle would have to alternate between the two sides of the bipartition, but they have unequal numbers of vertices, the rhombic dodecahedron is not Hamiltonian. The conjecture was significant, because if true, it would have implied the four color theorem: as Tait described, the four-color problem is equivalent to the problem of finding 3-edge-colorings of bridgeless cubic planar graphs. In a Hamiltonian cubic planar graph, such an edge coloring is easy to find: use two colors alternately on the cycle, and a third color for all remaining edges. Alternatively, a 4-coloring of the faces of a Hamiltonian cubic planar graph may be constructed directly, using two colors for the faces inside the cycle and two more colors for the faces outside. (Wikipedia).
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
A (compelling?) reason for the Riemann Hypothesis to be true #SOME2
A visual walkthrough of the Riemann Zeta function and a claim of a good reason for the truth of the Riemann Hypothesis. This is not a formal proof but I believe the line of argument could lead to a formal proof.
From playlist Summer of Math Exposition 2 videos
The Field With One Element and The Riemann Hypothesis (Full Video)
A crash course of Deninger's program to prove the Riemann Hypothesis using a cohomological interpretation of the Riemann Zeta Function. You can Deninger talk about this in more detail here: http://swc.math.arizona.edu/dls/ Leave some comments!
From playlist Riemann Hypothesis
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
Knots, three-manifolds and instantons – Peter Kronheimer & Tomasz Mrowka – ICM2018
Plenary Lecture 11 Knots, three-manifolds and instantons Peter Kronheimer & Tomasz Mrowka Abstract: Over the past four decades, input from geometry and analysis has been central to progress in the field of low-dimensional topology. This talk will focus on one aspect of these developments
From playlist Plenary Lectures
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
The Four-Color Theorem and an Instanton Invariant for Spatial Graphs I - Peter Kronheimer
Peter Kronheimer Harvard University October 13, 2015 http://www.math.ias.edu/seminars/abstract?event=83214 Given a trivalent graph embedded in 3-space, we associate to it an instanton homology group, which is a finite-dimensional Z/2 vector space. The main result about the instanton hom
From playlist Geometric Structures on 3-manifolds
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
A Beautiful Proof of Ptolemy's Theorem.
Ptolemy's Theorem seems more esoteric than the Pythagorean Theorem, but it's just as cool. In fact, the Pythagorean Theorem follows directly from it. Ptolemy used this theorem in his astronomical work. Google for the historical details. Thanks to this video for the idea of this visual
From playlist Mathy Videos
Lars Hesselholt: Around topological Hochschild homology (Lecture 2)
The lecture was held within the framework of the (Junior) Hausdorff Trimester Program Topology: "Workshop: Hermitian K-theory and trace methods" Introduced by Bökstedt in the late eighties, topological Hochschild homology is a manifestation of the dual visions of Connes and Waldhausen to
From playlist HIM Lectures: Junior Trimester Program "Topology"
Marc Levine - "The Motivic Fundamental Group"
Research lecture at the Worldwide Center of Mathematics.
From playlist Center of Math Research: the Worldwide Lecture Seminar Series
Perfectoid spaces (Lecture 3) by Kiran Kedlaya
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
Ptolemy's theorem and generalizations | Rational Geometry Math Foundations 131 | NJ Wildberger
The other famous classical theorem about cyclic quadrilaterals is due to the great Greek astronomer and mathematician, Claudius Ptolemy. Adopting a rational point of view, we need to rethink this theorem to state it in a purely algebraic way, without resort to `distances' and the correspon
From playlist Math Foundations
Justin Noel: Galois descent and redshift in algebraic K theory
The lecture was held within the framework of the Hausdorff Trimester Program: K-Theory and Related Fields. Justin Noel: Galois descent and redshift in algebraic K-theory Abstract: One of the fundamental results of Thomason states that the algebraic K-theory of discrete commutative rings
From playlist HIM Lectures: Trimester Program "K-Theory and Related Fields"
Andrew Wiles | Twenty Years of Number Theory | 1998
Notes for this talk: https://drive.google.com/file/d/1eJXPwL772Z00egvLjO3VHmvCv7mwcv6Q/view?usp=sharing Twenty Years of Number Theory Andrew Wiles Princeton University ICM Berlin 19.08.1998 https://www.mathunion.org/icm/icm-videos/icm-1998-videos-berlin-germany/icm-berlin-videos-2708
From playlist Number Theory
The Four-Color Theorem and an Instanton Invariant for Spatial Graphs II - Tomasz Mrowka
Tomasz Mrowka Massachusetts Institute of Technology October 13, 2015 http://www.math.ias.edu/seminars/abstract?event=83214 Given a trivalent graph embedded in 3-space, we associate to it an instanton homology group, which is a finite-dimensional Z/2 vector space. The main result about t
From playlist Geometric Structures on 3-manifolds
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
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
How to develop a proper theory of infinitesimals I | Famous Math Problems 22a | N J Wildberger
Infinitesimals have been contentious ingredients in quadrature and calculus for thousands of years. Our definition of the term starts with the Wikipedia entry, modified a bit to reduce the dependence on "real numbers", which is actually quite unnecessary--- but as a logical definition it i
From playlist Famous Math Problems