Polyhedral combinatorics | Disproved conjectures | Linear programming | Conjectures
In mathematical programming and polyhedral combinatorics, the Hirsch conjecture is the statement that the edge-vertex graph of an n-facet polytope in d-dimensional Euclidean space has diameter no more than n − d. That is, any two vertices of the polytope must be connected to each other by a path of length at most n − d. The conjecture was first put forth in a letter by to George B. Dantzig in 1957 and was motivated by the analysis of the simplex method in linear programming, as the diameter of a polytope provides a lower bound on the number of steps needed by the simplex method. The conjecture is now known to be false in general. The Hirsch conjecture was proven for d < 4 and for various special cases, while the best known upper bounds on the diameter are only sub-exponential in n and d. After more than fifty years, a counter-example was announced in May 2010 by Francisco Santos Leal, from the University of Cantabria. The result was presented at the conference 100 Years in Seattle: the mathematics of Klee and Grünbaum and appeared in Annals of Mathematics. Specifically, the paper presented a 43-dimensional polytope of 86 facets with a diameter of more than 43. The counterexample has no direct consequences for the analysis of the simplex method, as it does not rule out the possibility of a larger but still linear or polynomial number of steps. Various equivalent formulations of the problem had been given, such as the d-step conjecture, which states that the diameter of any 2d-facet polytope in d-dimensional Euclidean space is no more than d; Santos Leal's counterexample also disproves this conjecture. (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
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
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 hyperbolic Ax-Lindemann conjecture - Emmanuel Ullmo
Emmanuel Ullmo Université Paris-Sud February 7, 2014 The hyperbolic Ax Lindemann conjecture is a functional transcendental statement which describes the closure of "algebraic flows" on Shimura varieties. We will describe the proof of this conjecture and its consequences for the André-Oort
From playlist Mathematics
Weil conjectures 4 Fermat hypersurfaces
This talk is part of a series on the Weil conjectures. We give a summary of Weil's paper where he introduced the Weil conjectures by calculating the zeta function of a Fermat hypersurface. We give an overview of how Weil expressed the number of points of a variety in terms of Gauss sums. T
From playlist Algebraic geometry: extra topics
“Gauss sums and the Weil Conjectures,” by Bin Zhao (Part 3 of 8)
“Gauss sums and the Weil Conjectures,” by Bin Zhao. The topics include will Gauss sums, Jacobi sums, and Weil’s original argument for diagonal hypersurfaces when he raised his conjectures. Further developments towards the Langlands program and the modularity theorem will be mentioned at th
From playlist CTNT 2016 - ``Gauss sums and the Weil Conjectures" by Bin Zhao
The Quasi-Polynomial Freiman-Ruzsa Theorem of Sanders - Shachar Lovett
Shachar Lovett Institute for Advanced Study March 20, 2012 The polynomial Freiman-Ruzsa conjecture is one of the important open problems in additive combinatorics. In computer science, it already has several diverse applications: explicit constructions of two-source extractors; improved bo
From playlist Mathematics
When do fractional differential equations have maximal solutions?
When do fractional differential equations have maximal solutions? This video discusses this question in the following way. Firstly, a comparison theorem is formulated that involves fractional differential inequalities. Secondly, a sequence of approximative problems involving polynomials
From playlist Research in Mathematics
Are Prime Numbers Made Up? | Infinite Series | PBS Digital Studios
Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: https://to.pbs.org/donateinfi Is math real or simply something made up by mathematicians? You can’t physically touch a number yet using numbers we’re able to build skyscrapers and launch rockets int
From playlist An Infinite Playlist
On characterization of monomial irreducible representations by Pooja Singla
DATE & TIME 05 November 2016 to 14 November 2016 VENUE Ramanujan Lecture Hall, ICTS Bangalore Computational techniques are of great help in dealing with substantial, otherwise intractable examples, possibly leading to further structural insights and the detection of patterns in many abstra
From playlist Group Theory and Computational Methods
Alexander Black: Modifications of the Shadow Vertex Pivot Rule
The shadow vertex pivot rule is a fundamental tool for the probabilistic analysis of the Simplex method initiated by Borgwardt in the 1980s. More recently, the smoothed analysis of the Simplex method first done by Spielman and improved upon by Dadush and Huiberts relied on the shadow verte
From playlist Workshop: Tropical geometry and the geometry of linear programming
Jim Lawrence: The concatenation operation for uniform oriented matroids and simplicial...
Abstract: Some problems connected with the concatenation operation will be described. Recording during the meeting "Combinatorial Geometries: Matroids, Oriented Matroids and Applications" the September 24, 2018 at the Centre International de Rencontres Mathématiques (Marseille, France) F
From playlist Combinatorics
Caustics of fronts and the arborealization conjecture - Daniel Alvarez-Gavela
Short talks by postdoctoral members Topic: Caustics of fronts and the arborealization conjecture Speaker: Daniel Alvarez-Gavela Affiliation: Member, School of Mathematics Date: September 25, 2018 For more video please visit http://video.ias.edu
From playlist Mathematics
Galois theory II | Math History | NJ Wildberger
We continue our historical introduction to the ideas of Galois and others on the fundamental problem of how to solve polynomial equations. In this video we focus on Galois' insights into how extending our field of coefficients, typically by introducing some radicals, the symmetries of the
From playlist MathHistory: A course in the History of Mathematics
Are Geniuses Born or Made? A Conversation with Dr. Joy Hirsch | BEST OF 2015 | Big Think
Are Geniuses Born or Made? Watch the newest video from Big Think: https://bigth.ink/NewVideo Join Big Think Edge for exclusive videos: https://bigth.ink/Edge ---------------------------------------------------------------------------------- Is there a way to bring out the genius within a
From playlist Best Videos | Big Think
Genius is a Continuum, with Dr. Joy Hirsch | Big Think.
Genius is a Continuum, with Dr. Joy Hirsch Watch the newest video from Big Think: https://bigth.ink/NewVideo Join Big Think Edge for exclusive videos: https://bigth.ink/Edge ---------------------------------------------------------------------------------- Can you find "genius" in the bra
From playlist Best Videos | Big Think
What is Linear Programming (LP)? (in 2 minutes)
Overview of Linear Programming in 2 minutes. ---------------------- Additional Information on the distinction between "Polynomial" vs "Strongly Polynomial" algorithms: An algorithm for solving LPs of the form max c^t x s.t. Ax \le b runs in polynomial time if its running time can be boun
From playlist Under X-Minutes Visual Explanations
Are Geniuses Born or Made? A Conversation with Dr. Joy Hirsch | Big Think.
Are Geniuses Born or Made? Watch the newest video from Big Think: https://bigth.ink/NewVideo Join Big Think Edge for exclusive videos: https://bigth.ink/Edge ---------------------------------------------------------------------------------- Is there a way to bring out the genius within al
From playlist Best Videos | Big Think
D. Yogeshwaran: The Poisson-OSSS inequality and an application to Confetti percolation
I will present a version of the OSSS inequality (proved by O’Donnell, Saks, Schramm and Servedio (2005)) to functionals of general Poisson point processes. This inequality can significantly simplify the proofs of sharp phase-transition in continuum percolation models. We shall illustrate t
From playlist Workshop: High dimensional spatial random systems
👉 Learn about the remainder theorem and the factor theorem. The remainder theorem states that when a polynomial is divided by a linear expression of the form (x - k), the remainder from the division is equivalent to f(k). Similarly, when a polynomial is divided by a linear expression of th
From playlist Remainder and Factor Theorem | Learn About