Category Archives: Geometry

F ≤ 4E

1. E ≤ 3V Let G be a simple planar graph with V vertices and E edges. It follows from Euler’s theorem that E ≤ 3V In fact, we have (when V is at least 3,) that E ≤ 3V – 6. … Continue reading

Posted in Combinatorics, Convex polytopes, Geometry, Open problems | Tagged | 12 Comments

Symplectic Geometry, Quantization, and Quantum Noise

Over the last two meetings of our HU quantum computation seminar we heard two talks about symplectic geometry and its relations to quantum mechanics and quantum noise. Yael Karshon: Manifolds, symplectic manifolds, Newtonian mechanics, quantization, and the non squeezing theorem. … Continue reading

Posted in Computer Science and Optimization, Geometry, Physics | Tagged , , , , , , | 6 Comments

Greg Kuperberg: It is in NP to Tell if a Knot is Knotted! (under GRH!)

Wolfgang Haken found an algorithm to tell if a knot is trivial, and, more generally with Hemion, if two knots are equivalent. Joel Hass, Jeff Lagarias and Nick Pippinger proved in 1999 that telling that a knot is unknotted is … Continue reading

Posted in Computer Science and Optimization, Geometry | Tagged | 9 Comments

Exciting News on Three Dimensional Manifolds

The Virtually Haken Conjecture A Haken 3-manifold is a compact 3-dimensional manifold M which is irreducible (in a certain strong sense) but contains an incompressible surface S. (An embedded surface S is incompressible if the embedding indices an injection of its … Continue reading

Posted in Geometry, Updates | Tagged , , | 3 Comments

Fractional Sylvester-Gallai

Avi Wigderson was in town and gave a beautiful talk about an extension of Sylvester-Gallai theorem. Here is a link to the paper: Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes by Boaz Barak, Zeev … Continue reading

Posted in Combinatorics, Computer Science and Optimization, Geometry | Tagged , , , | 4 Comments

Course Announcement: High Dimensional Expanders

Alex Lubotzky and I  are running together a year long course at HU on High Dimensional Expanders. High dimensional expanders are simplical (and more general) cell complexes which generalize expander graphs. The course will take place in Room 110 of the mathematics building … Continue reading

Posted in Algebra and Number Theory, Combinatorics, Geometry, Teaching | 2 Comments

Polymath3 (PHC6): The Polynomial Hirsch Conjecture – A Topological Approach

This is a new polymath3 research thread. Our aim is to tackle the polynomial Hirsch conjecture which asserts that there is a polynomial upper bound for the diameter of graphs of -dimensional polytopes with facets. Our research so far was … Continue reading

Posted in Convex polytopes, Geometry, Polymath3 | Tagged , , | 37 Comments

János Pach: Guth and Katz’s Solution of Erdős’s Distinct Distances Problem

Click here for the most recent polymath3 research thread. Erdős and Pach celebrating another November day many years ago. The Wolf disguised as Little Red Riding Hood. Pach disguised as another Pach. This post is authored by János Pach A … Continue reading

Posted in Combinatorics, Geometry, Guest blogger, Open problems | Tagged , | 13 Comments

Benoît’s Fractals

Mandelbrot set Benoît Mandelbrot passed away a few dayes ago on October 14, 2010. Since 1987, Mandelbrot was a member of the Yale’s mathematics department. This chapterette from my book “Gina says: Adventures in the Blogosphere String War”   about fractals is brought here on this … Continue reading

Posted in Geometry, Obituary, Physics, Probability | 6 Comments

Answer to Test Your Intuition (3)

Question: Let be the -dimensional cube. Turn into a torus by identifying opposite facets. What is the minumum -dimensional volume of a subset of which intersects every non-trivial cycle in . Answer: Taking to be all points in the solid … Continue reading

Posted in Geometry, Test your intuition | Tagged | 2 Comments