POLYMATH Noga Alon Birthday Conference Gil Kalai Einstein

POLYMATH! Noga Alon Birthday Conference Gil Kalai Einstein Institute of Mathematics Hebrew University of Jerusalem.

Noga’s formulas 1) Seperators for Minor-free graphs (Bhargav Narayanan) 2) Aaaalmost Stanley-Wilf conj. (Ian Alevy) 3) All pairs shortest path. (Josh Alman) 4) A bound on the linear arboricity la(G) of a graph G of maximum degree d from a paper published in 1988. (Ross Kang) 5) Maximum number of directed Hamiltonian path in a tournament (Shakhar Smorodinsky, Louis De. Biasio) 6) Shannon capacity of a union (Jean Cardinal, Daniel Soltész) 7) (corrected) Turan number for very asymmetric bipartite graphs (Adam Sheffer x 1. 5) (Sebi Cioaba, Ferdinand Ihringer) 8) Largest number of edges in a bipartite subgraph (Sebi Cioaba)

Noga’s formulas 2 1) 2) Alon-Boppana (Lior Silberman) x 1. 5 Ramsey number R(C 4, Kn ) (Sushant Sachdeva &David Conlon) 3) The discrete Cheeger-Buser inequality. (Emmanuel Kowalski x 2) 4) Monotone circuits for cliques (Gautam Kamath) 5) Maximum number of copies of a graph H in a graph with l edges. (Yufei Zhao x 2) 6) 123 -theorem of Alon-Yuster (Nicolas Rivera) 7) Lower bound an restricted sum of A and B. (Shachar Lovett)

Noga and I: academic twins Ph. D. Students of Micha A. Perles Altshuler (**) Katchalski (74) Koren (7? ) Kallay (79) Linial (80) Shemer (82) Alon (83) Kalai (83) Kupitz (85) Adin(*) (93) Smilanski Deutch Pinchasi (*) Magazanik (07) Sigron Nitzan (*) Kammal Keller

Polymath A polymath (Greek: πολυμαθής, polymathēs, having learned much") is a person whose expertise spans a significant number of different subject areas; such a person is known to draw on complex bodies of knowledge to solve specific problem the idea of the polymath was expressed by Leon Battista Alberti (1404– 1472), in the statement, most suitable to Noga "a man (who) can do all things if he will".

3 -regular subgraphs Conjecture (Berge): Every simple 4 -regular graph contains a 3 -regular subgraph. Theorem (Alon, Friedland Kalai, 1984): Every 4 regular graph plus an edge contains a 3 -regular graph. Theorem (AFK, 84) A graph with n vertices and 2 n+1 edges contains a nonempty subgraph with all degrees 0(mod 3).

Collaboration • Gil’s top coauthors 1) Kahn 8 2) Alon 7 3) Meshulam 5 4 -6) Bourgain, Friedgut, Linial 4 7 -9) Bjorner, Keller, Mossel 3 10 -24) Bar-Hillel, Bar-Nathan, Benjamini (*), Friedland, Kleinschmidt, Matouskek (*), Mc. Kay, Meisinger, Nisan, Nevo(*), Novik(*), Richlin, Schramm(*), Stockmayer, Tennenholtz(*) 2 • Noga’s top coauthors (incomplete) 1) Krivelevich 32 (AKS 9) 2) Sudakov 29 3) Yuster 20 4) Azar 17 5) Tennenholz 14 6 -7) Kleitman, Shapira 13 7 -9) Lubetzky, Rodl , Spencer 11 10) Caro 9 11 -12) Feldman, Seymour 8 13 -16) Fischer, Furedi , Kostochka, Stav 7 17 -23) Asodi, Bollobas, Erdos, Kalai, Linial, M. Naor , M. Szegedi 6

The polynomial method

3 -regular subgraphs (2) Taskinov proved Berge’s conjecture using direct graph theoretical arguments. Rodl and Szemeredi proved that a graph with n vertices and Cnlogn edges always contains a 3 -regular graph, but constructed examples of graphs with C’nloglogn edges without a 3 -regular subgraph.

Regular subgraphs: problems Does every 4 -regular graph plus an edge (or 5 regular graph, or just 4 -regular graph) have a type I (3 -edge colorable) 3 -regular subgraph? Is there a polynomial-time algorithm to find a subgraph with all degrees divisible by 3 in a graph with n vertices and 2 n+1 edges. How many edges guarantee a 3 -regular subgraph? What about subgraphs with vertex degrees divisible by m, when m is not a prime-power? How many vectors in (Z/m. Z)n guarantees a subset summing up to 0?

Shellability and upper bound theorems

Some of the places where our roads crossed again Few polytopes; many spheres: Goodman Pollack (85), Alon(87), Kalai (88) Transversal theorems: (Alon & Kalai (95) , Alon, Kalai, Matousek, Mashulam (02)) (Following Alon. Kleitman’s solution of Hadwiger-Debrunner’s conjecture. ) Spectral methods, Collective coin flipping, mobile user tracking and distributed job scheduling, social choice, hereditary discrepancy of HAP.

JERUSALEM COMBINATORICS 1993

Polymath (modern use) Polymath projects are a form of open Internet collaboration aimed towards a major mathematical goal, usually to settle a major mathematical problem. This is a concept introduced in 2009 by Tim Gowers who described it as “a large collaboration in which no single person has to work all that hard. ” It is in line with other forms of open Internet research activity which include Math. Overflow and are sometimes referred to as Science 2. 0. “Sometimes it is not market forces that achieve efficiencies, but cooperatives. ” (Nature, editorial 2014)

Polymath projects (*) Polymath 1 (Density Hales Jewett); (TG) Polymath 3 (Polynomial Hirsch Conjecture) (GK) Polymath 4 (Deterministically finding primes)(TT) (*) Polymath 5 (Erdos discrepancy problem) (TG) Polymath 7 (Hot spots) (TT) (*) Polymath 8 a and 8 b (Gaps in primes) (TT) Polymath 10 (Sunflower conjecture) (GK) A few dozens polymath proposals.

The IAS polymath debate (2010) The polymath format could be a better way of doing mathematics “If the polymath model were to grow to dominate mathematics, younger mathematicians would be driven in the direction of online collaboration at the cost of traditional mathematics, which has produced the likes of Alexander Grothendieck, whose individual contributions to the field have been revolutionary. ” “There is no danger and no hope that online collaboration will supersede lone endeavors” (but I like it) Interesting, but runs the danger of making us “managers” rather than researchers

Polymath 10: The Erdos-Rado Sunflower Conjecture A collection of sets is a sunflower if every element which is included in two sets is included in all of them. The sets are called petals; their intersection is called the head. Early heroes: Abott, Hanson and Sauer, Spencer; Kostochka

The conjecture Let f(k, r) denotes the maximum size of a family of k-sets without a subflower of size r. Erdős-Rado's sunflower conjecture: There is a constant C=Cr such that f(k, r) < Crk.

Balanced families A family of k-sets is balanced if we can color the ground set with k colors such that every set is colorful. Let fb (k, r) denotes the maximum size of a family of k-sets without a sunflower of size r. Lemma: f(k, r) < ek fb (k, r). Problem: exact lower bounds for balanced subhypergraphs of hypergraphs with e edges like Noga’s formula 8.

A refinement Let f(k, r, n, m) be the maximum number of ksubsets from [n] without a sunflower with r petals whose head has less than m elements. For r=2 and arbitrary m, and for m=1 and arbitrary r we get well-known questions of Erdos-Ko-Rado type.

Homology Let F be a family of sets, let K be the simplicial complex spanned by F and let Hi (K) be the ith (reduced) homology group of K.

Homology chains

Weighted homology

Cycles for multiple boundaries

Homology for multiple boundaries

Conjectures Some partial results Are available. Analogous results are available (when you replace [2, 1] by [1, 1] and [2 k, k] by [k, k]) , but The conjecture is largely a wishful thinking, so far.

More from polymath 10 • Phillip Gibbs found a better connection between the balanced and general cases. If the sunflower conjecture is true then f(k, r) < (1+o(1))k fb (k, r). • Dömötör Pálvölgyi shot down various working conjectures, and raised interesting questions. • Some computer experimentation was made by several participants. • Tim Gowers proposed a “pseudorandom” approach • Shachar Lovett commented on a special case related to matrix multiplication • Ferdinand Ihringer found inequalities for the refined version. • Eran Nevo showed that for balanced families, intersecting implies acyclic. Extensions of this result are crucial for partial results towards Conjecture A. • Karim Adiprasito discussed the Cohen-Macaulay case • Some new extremal results for balanced families were found.

Conjectures (late update) Some partial results Are available. e ls a F Analogous results are available, but The conjecture is largely a wishful thinking, so far.

Are polymath projects, and, more generally, “Science 2. 0” viable ideas? 1. Putting aside questions about incentives, can it lead to a better way of doing science (at least in some cases)? or to other major fruits? 2. What about incentives?

Social choice Both Noga and me did some work on social choice: question about aggregating information and preferences of many individuals.

Large committees require large budget! Given a committee of k members (k an odd integer) that need to give m followships. (The number of applications is unlimited). Is there a rule that exclude the possibility of a candidate not getting the fellowship in spite having the majority of the committee preferring her to all those who received the fellowship. • Condorcet: No: for k=3, m=1. • Alon, Brightwell, Kierstead, Kostochka and Winkler: NO for m< Ck/ log k; YES for m>C k log k. Question: what about general voting rules instead of majority? Condorcet 1743 -1794

Happy birthday dear Noga