Combinatorics - Summer SchoolGulbenkian Foundation, LISBON — 23 to 27 JULY 2012
sketch for the panel Começar by José de Almada Negreiros - image reproduced with kind permission of the Gulbenkian Foundation ProgramThis school comprises three 5-lecture courses (in English) aimed at 1st- or 2nd-year undergraduate students of mathematics, complemented by problem sessions.CoursesSymmetric polynomials and Schubert polynomialsby Allen KNUTSON (Cornell University)Abstract: Consider the ring of polynomials in variables x1, x2,... and call a polynomial symmetric in xi and xj if switching them results in the same polynomial., e.g. x1 + x2 or x1x2. Highly symmetric polynomials tend to have a lot of terms, which suggests that writing them in terms of monomials (as usual) might not be the best way to study them. We'll introduce Schubert polynomials, which are a different basis of the ring of polynomials in infinitely many variables. While their definition will be clear from an algebraic point of view, it will turn out (quite nonobviously) that they have positive integer coefficients, so combinatorics enters. The Schubert polynomials in k variables and symmetric in all k are the much more classical "Schur polynomials" and we'll investigate the even richer theory of the multiplication on Schur polynomials. The only background for the course is familiarity with polynomials and permutations. Probabilistic Methodsby Benny SUDAKOV (University of California at Los Angeles)Abstract: The Probabilistic Method is a powerful tool in tackling many problems in Combinatorics. It belongs to those areas of mathematics which have experienced a most impressive growth in the past few decades. Roughly speaking, its basic idea can be described as follows. In order to prove existence of a combinatorial structure with certain properties, we construct an appropriate probability space, and show that a randomly chosen element of this space has the desired property, with positive probability. This course provides a gentle introduction to the Probabilistic Method, with emphasis on methodology. We will try to illustrate the main ideas by showing how to use probabilistic reasoning to solve various combinatorial problems. Most of the course will be self contained (we review the tools and results which we use) but some elementary knowledge of probability and graph theory would certainly be helpful. Combinatorics and Geometry - a match made in heavenby Nathan LINIAL (Hebrew University of Jerusalem)Abstract: I will present several topics which involve a mix of combinatorics, linear algebra and geometry. The girth of a graph G is defined as the length of the shortest cycle in G. It is not so simple to construct graphs of high girth, i.e., graphs without short cycles. Concretely, for given d and n we consider the question: What is the largest possible girth of a d-regular graph with n vertices? One specific topic on which I will concentrate in this area is the (almost) complete characterization of k-regular graphs on k^2+1 vertices of girth 5. The girth problem is closely related to the notion of linear error-correcting codes and I will present next some basic results from that area. If time permits I will go into somewhat more advanced topics. One particularly beautiful topic I hope to be able to present is the refutation of the Borsuk conjecture from geometry. Schedule
Friday, July 27, at 21:30 there will be a gathering with the lecturers at Hotel Príncipe Lisboa, Av. Duque de Ávila, 201 (right next to Hotel Alif Avenidas) where participants will be able to talk to the lecturers about their careers in a relaxed setting. Mini-conferenceOn Saturday, July 28, there will be student mini-conference at the Gulbenkian Foundation which will include talks by Profs. Knutson and Sudakov.All school participants are invited to attend this mini-conference. Schedule for mini-conference (28/July/2011) AssistantsJoão Guerreiro (guerreiro@math.columbia.edu) and João Gouveia (jgouveia@mat.uc.pt)VenueThe school takes place in the headquarters of the Gulbenkian Foundation (Avenida de Berna, Lisbon), with lectures and problem sessions in Sala 1. |
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