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 In And Around Combinatorics: The 18th Midrasha Mathematicae. Jerusalem, JANUARY 1831
 Mathematical Gymnastics
 Media Item from “Haaretz” Today: “For the first time ever…”
 Jim Geelen, Bert Gerards, and Geoﬀ Whittle Solved Rota’s Conjecture on Matroids
 Media items on David, Amnon, and Nathan
 Next Week in Jerusalem: Special Day on Quantum PCP, Quantum Codes, Simplicial Complexes and Locally Testable Codes
 Happy Birthday Ervin, János, Péter, and Zoli!
 My Mathematical Dialogue with Jürgen Eckhoff
 Test Your Intuition (23): How Many Women?
Top Posts & Pages
 In And Around Combinatorics: The 18th Midrasha Mathematicae. Jerusalem, JANUARY 1831
 Believing that the Earth is Round When it Matters
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 Answer: Lord Kelvin, The Age of the Earth, and the Age of the Sun
 In how many ways you can chose a committee of three students from a class of ten students?
 Why Quantum Computers Cannot Work: The Movie!
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Category Archives: Convex polytopes
The Polynomial Hirsch Conjecture – How to Improve the Upper Bounds.
I can see three main avenues toward making progress on the Polynomial Hirsch conjecture. One direction is trying to improve the upper bounds, for example, by looking at the current proof and trying to see if it is wasteful and if so where … Continue reading
Posted in Convex polytopes, Open discussion, Open problems
Tagged Discussion, Hirsch conjecture
14 Comments
The Polynomial Hirsch Conjecture, a Proposal for Polymath3 (Cont.)
The Abstract Polynomial Hirsch Conjecture A convex polytope is the convex hull of a finite set of points in a real vector space. A polytope can be described as the intersection of a finite number of closed halfspaces. Polytopes have … Continue reading
Posted in Open problems, Convex polytopes, Open discussion
Tagged Hirsch conjecture, Polymath proposals
5 Comments
The Polynomial Hirsch Conjecture: A proposal for Polymath3
This post is continued here. Eddie Kim and Francisco Santos have just uploaded a survey article on the Hirsch Conjecture. The Hirsch conjecture: The graph of a dpolytope with n vertices facets has diameter at most nd. We devoted several … Continue reading
(Eran Nevo) The gConjecture II: The Commutative Algebra Connection
Richard Stanley This post is authored by Eran Nevo. (It is the second in a series of five posts.) The gconjecture: the commutative algebra connection Let be a triangulation of a dimensional sphere. Stanley’s idea was to associate with a ring … Continue reading
How the gConjecture Came About
This post complements Eran Nevo’s first post on the conjecture 1) Euler’s theorem Euler Euler’s famous formula for the numbers of vertices, edges and faces of a polytope in space is the starting point of many mathematical stories. (Descartes came close … Continue reading
(Eran Nevo) The gConjecture I
This post is authored by Eran Nevo. (It is the first in a series of five posts.) Peter McMullen The gconjecture What are the possible face numbers of triangulations of spheres? There is only one zerodimensional sphere and it consists … Continue reading
Posted in Combinatorics, Convex polytopes, Guest blogger, Open problems
Tagged face rings, gconjecture, Polytopes
5 Comments
Ziegler´s Lecture on the Associahedron
The associahedron in 3 dimension, and James Stasheff. This picture is taken from Bill Casselman’s article on the associahedron. The article is entitled “Strange Associations” and starts with “There are many other polytopes that can be described in purely combinatorial terms. Among the … Continue reading
Posted in Convex polytopes
Tagged Associahedron, Cyclohedron, Permutahedron, Permutoassociahedron
7 Comments
Telling a Simple Polytope From its Graph
Peter Mani (a photograph by Emo Welzl) Simple polytopes, puzzles Micha A. Perles conjectured in the ’70s that the graph of a simple polytope determines the entire combinatorial structure of the polytope. This conjecture was proved in 1987 by Blind … Continue reading
Posted in Convex polytopes, Open problems
Tagged Eric Friedman, Peter Mani, Roswitta Blind
4 Comments
A Diameter problem (7): The Best Known Bound
Our Diameter problem for families of sets Consider a family of subsets of size d of the set N={1,2,…,n}. Associate to a graph as follows: The vertices of are simply the sets in . Two vertices and are adjacent if . … Continue reading
A Diameter Problem (6): Abstract Objective Functions
George Dantzig and Leonid Khachyan In this part we will not progress on the diameter problem that we discussed in the earlier posts but will rather describe a closely related problem for directed graphs associated with ordered families of sets. The role models for … Continue reading
Posted in Combinatorics, Convex polytopes, Open problems
Tagged Hirsch conjecture, Linear programming
7 Comments