MathFest - Gil Kalai @ 60
The Hebrew University of Jerusalem, June 15-16, 2015
Maison De France, Edmond J. Safra Campus, Givat Ram
Abstracts
Nati Linial - Q-hypertrees and beyond: On the combinatorics of simplicial complexes
Günter M. Ziegler - Polytopes for the Book of Examples
Gil has suggested that besides the BOOK of proofs, there should be a BOOK of Examples. For a start, in this lecture I want to describe some remarkable polytopes and their properties ... and some intriguing results ond conjectures they relate to. Naturally, some of the nicest of these conjectures due to Gil.Roy Meshulam - Gil and the combinatorics of d-Leray complexes
One of Gil's earliest breakthroughs was his proof of the Katchalski-Perles conjecture on families of convex sets. Gil's theorem is in fact more general and establishes an upper bound theorem for d-Leray complexes, i.e. simplicial complexes all of whose links have vanishing homology in dimensions d and above. In this talk I'll discuss this and further Helly type results and problems concerning d-Leray complexes.Anders Björner -Shifting and Speculating with Gil
One of Gil’s most surprising contributions to mathematics, perhaps the most original of them all, is the theory of algebraic shifting. Positioned at the confluence of set-theoretic extremal combinatorics, algebraic combinatorics and commutative algebra, this theory provides a powerful tool for the study of f-vectors. Having had the opportunity to collaborate with Gil on applications of algebraic shifting at an early stage, I will in this talk recall some of the results and speculations from that time. I plan to review what algebraic shifting is and does, give a couple of examples of its uses, and recall the pleasures of collaboration with Gil.Rom Pinchasi - Every finite family of pseudodiscs must contain a small pseudodisc
We show that there exists an absolute constant $c < 200$ such that in every finite family $\F$ of pseduodiscs one can find one disc $D$ with the following property. Among the members of $\F$ that intersect with $D$ there are at most $c$ pairwise disjoint pseduodiscs. This result has some nice applications in two and three dimensions.Imre Bárány - Tensors, colours, octahedra
Several classical results in convexity, like the theorems of Caratheodory, Helly, and Tverberg, have colourful versions. In this talk I plan to explain how two methods, the octahedral construction and Sarkaria's tensor trick, can be used to prove further extensions and generalizations of such colourful theorems.Ron Adin - SYT Enumeration - Then and Now
More than a hundred years ago, Frobenius and Young based the emerging representation theory of the symmetric group on the combinatorial objects now called Standard Young Tableaux (SYT). Many important features of these classical objects have since been discovered, including some surprising interpretations and the celebrated hook length formula for their number. In recent years, SYT of non-classical shapes have come up in research and were shown to have, in many cases, surprisingly nice enumeration formulas. The talk will present some gems from the study of SYT over the years, including some exciting recent progress. It is partially based on a survey chapter, joint with Yuval Roichman, in the recent CRC Handbook of Combinatorial Enumeration.Eran Nevo - Gil and Rigidity
In his 1987 Invent. Math. paper "Rigidity and the Lower Bound Theorem 1" Gil showed how framework rigidity can be used to prove and generalize Barnette's LBT on face numbers of simplicial polytopes. Since then, framework rigidity had become an important tool to produce further LBTs, for other simplicial or polyhedral complexes. I will recall Gil's proof and discuss further developments and open problems, including some new results, in this area.Micha Perles - Crossing matchings and circuits are of maximal length
Speaker: Let V be a finite set of points in the Euclidean plane. A (geometric) graph G on V is a graph whose edges are nondegenerate closed line segments with endpoints in V . The length ℓ(G) of such a graph is the sum of lengths of its edges. A matching M on V (|V|= 2m even) is a 1-regular graph on V , i.e., a set of m pairwise vertex-disjoint edges. M is a crossing matching (cm) if every two edges of M cross. (Two segments cross if their relative interiors share a unique point.) One can easily show that V admits at most one cm.Theorem 1. If M is a cm on V, then M is longer than any other matching on V.
Key Lemma. Suppose |V| = 2m. If M is a cm on V, and {{a_1,b_1}, {a_2,b_2},..., {a_m,b_m}} is an arbitrary pairing of V , then there exist polygonal paths P_1,...,P_m, such that
(a): P_i connects a_i to b_i (i = 1, 2,...,m);
(b): P_i is contained in the union of segments in M, for i = 1,2,...,m;
(c): for 1 <= i < j <= m, P_i and P_j meet in a finite number of points.
A Hamiltonian circuit C on V is an asterisk (or a crossing circuit) if every two non-adjacent edges of C cross. This can happen only when |V| = 2m+1 is odd. Using Theorem 1, one can prove:
Theorem 2. If A is an asterisk on V , then A is longer than any other 2-regular (multi)graph on V. This implies that an asterisk on V , if it exists, is unique. Joint work with Yaakov S. Kupitz and Hagit Last
