Author Archives: Gil Kalai

Analysis of Boolean Functions – week 1

Home page of the course. In the first lecture I defined the discrete n-dimensional cube and  Boolean functions. Then I moved to discuss five problems in extremal combinatorics dealing with intersecting families of sets. 1) The largest possible intersecting family … Continue reading

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Open Collaborative Mathematics over the Internet – Three Examples

After much hesitation, I decided to share with you the videos of my lecture: Open collaborative mathematics over the internet – three examples, that I gave last January in Doron Zeilberger’s seminar at Rutgers on experimental mathematics. Parts of the 47-minutes … Continue reading

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Poznań: Random Structures and Algorithms 2013

   Michal Karonski (left) who built Poland’s probabilistic combinatorics group at Poznań, and a sculpture honoring the Polish mathematicians who first broke the Enigma machine (right, with David Conlon, picture taken by Jacob Fox). I am visiting now Poznań for the 16th … Continue reading

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BosonSampling and (BKS) Noise Sensitivity

Following are some preliminary observations connecting BosonSampling, an interesting  computational task that quantum computers can perform (that we discussed in this post), and noise-sensitivity in the sense of Benjamini, Schramm, and myself (that we discussed here and here.) BosonSampling and computational-complexity hierarchy-collapse Suppose that … Continue reading

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Lawler-Kozdron-Richards-Stroock’s combined Proof for the Matrix-Tree theorem and Wilson’s Theorem

   David Wilson and a cover of Shlomo’s recent book “Curvature in mathematics and physics” A few weeks ago, in David Kazhdan’s basic notion seminar, Shlomo Sternberg gave a lovely presentation Kirchho ff and Wilson via Kozdron and Stroock. The lecture is based on … Continue reading

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

Auction-based Tic Tac Toe: Solution

Reshef, Moshe and Sam The question: (based on discussions with Reshef Meir, Moshe Tennenholtz, and Sam Payne) Tic Tac Toe is played since anciant times. For the common version, where the two players X and O take turns in marking … Continue reading

Posted in Games, Test your intuition | Tagged | 6 Comments

Some old and new problems in combinatorics and geometry

Paul Erdős in Jerusalem, 1933  1993 I just came back from a great Erdős Centennial conference in wonderful Budapest. I gave a lecture on old and new problems (mainly) in combinatorics and geometry (here are the slides), where I presented twenty … Continue reading

Posted in Combinatorics, Geometry, Open problems | Tagged | 4 Comments

The Kadison-Singer Conjecture has beed Proved by Adam Marcus, Dan Spielman, and Nikhil Srivastava

…while we keep discussing why mathematics is possible… The news Adam Marcus, Dan Spielman, and Nikhil Srivastava posted a paper entitled “Interlacing Families II: Mixed Characteristic Polynomials and the Kadison-Singer Problem,” where they prove the 1959 Kadison-Singer conjecture. (We discussed part … Continue reading

Posted in Analysis, Computer Science and Optimization, Physics, Updates | Tagged , , , , , | 29 Comments

Why is Mathematics Possible: Tim Gowers’s Take on the Matter

In a previous post I mentioned the question of why is mathematics possible. Among the interesting comments to the post, here is a comment by Tim Gowers: “Maybe the following would be a way of rephrasing your question. We know … Continue reading

Posted in Open discussion, Philosophy, What is Mathematics | Tagged , , , | 23 Comments

Polymath8: Bounded Gaps Between Primes

Yitang Zhang’s very recent shocking paper demonstrated that bounded gaps between primes occur infinitely often, with the explicit upper bound of 70,000,000 given for this gap. Polymath8 was launched for the dual purpose of learning Zhang’s proof and improving the upper bound … Continue reading

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