May 7

Thursday at 16:15, in Room 102

Prof. Gyula Karolyi, Eotvos University: What are polynomials good for?

Abstract: What is the minimum number of planes that contain every point of an n by n by n grid in 3-space? Does every square submatrix of the Cayley addition table of a cyclic group of odd order have a Latin transversal? What is the minimum number of points such that every line in the affine plane over a finite field is incident to at least one of them?
The common in these apparently unrelated problems is that their solution depend on a simple fact on the zeroes of appropriate multivariate polynomials.
The polynomial method based on the so-called Combinatorial Nullstellensatz formulated by Noga Alon ten years ago became very successful in extremal combinatorics. It should be in the tool-kit of every working mathematician.

REMARK: Please note that this colloquium lecture was originally planned at the Eotvos University but finally, due to different reasons, is offered at Bethlen ter.

April 27

MONDAY at 16:15, in Room 105

Prof. Laszlo Babai , University of Chicago, BSM founder: Playing Rubik's Cube with matrices

Abstract: Suppose we pull Rubik's Cube apart and then randomly reassemble the pieces. How long does it take to find out whether or not the configuration obtained is feasible, i.e., whether it can be solved by a sequence of legal moves?
A moment's reflection shows that the question is about subgroup membership: given a group G, a list of generators of a subgroup H, and an element g in G, determine whether or not g belongs to H.
The problem becomes especially intriguing when G is the group of invertible n by n matrices mod p. This problem has connections to number theory, elementary combinatorics, discrete probability, random walks, elementary group theory, and quite profound group theory.
I will illustrate some of these connections, leading up to a recent breakthrough (STOC 2009) that combines the work of several research groups, spanning a quarter century. My coauthors on the 2009 paper are Bob Beals (BSM Spring 1987) and Akos Seress.
No prior knowlegde of group theory is required.

REMARK: Please note that this colloquium lecture is exceptionally scheduled on MONDAY afternoon.

April 23

Thursday at 16:30, at the Central European University (Zrinyi utca 14. 3. floor 310/A , Budapest, Hungary)

Prof. Pal HEGEDUS , Central European University: Electric networks, wanderings of a particle

Abstract: This talk will focus on graphs, especially trees and so called Cayley graphs, finite or infinite. We will consider the graph a grid of electric wires and some batteries hence there will be a resulting electric current. It turns out that the current has strong resemblance to a random walk of a hypothetical positively charged particle. We shall also see an application for the number of spanning trees of a graph.

REMARK: This will be the second lecture in the sequence introducing fine Hungarian higher education mathematics centers. CEU is an international graduate level university in the truest sense: Its students come from more than 80 countries; its faculty, from more than 30 countries--with the mix of nationalities increasing every year. The language of instruction and communication is English. It has it's own mathematics department and MSc and PhD math programs, which will also be shortly introduced. The venue of the lecture is the headquarters of the Department of Mathematics and it's Application at Zrinyi utca, downtown Budapest. You are assumed to find the location by yourself, with the following help: general directions general directions and map of the neighborhood.

April 17

Friday at 16:15 (apprx.), at Bolyai Institute (Aradi vertanuk tere 1, Szeged, Hungary)

Prof. Laszlo HATVANI, Bolyai Institute, University of Szeged: Can the mathematical pendulum be chaotic?

Abstract: It turns out that the damped and periodically forced mathematical pendulum has infinitely many chaotic motions. Roughly speaking, by chaos we mean that nothing can be predicted about these motions. Computer simulations and an outlined proof will be presented.

REMARK: This will be an exceptional, out of town colloquium lecture at an exceptional time, Friday afternoon. This Friday the last classes (from noon) will be canceled (and made up later) and the group will leave from Nyugati railway station at 12:53 with Mora intercity train. You will be met at the Szeged railway station and ushered to the site of the lecture (which is, in case you need to find it, here). Anna takes care of the train ticket (to Szeged) and the dormitory type accommodation, see her in case you intend to attend the lecture and visit Szeged.

March 26

Thursday at 16:15, at Bethlen ter, location TBA

Prof. Karoly SIMON, Renyi Institute and Budapest University of Technology and Economics : Deterministic and random fractals

Abstract: This will be an introductory talk into the beautiful world of deterministic and random fractals. I will show a number of examples and the notion of fractal dimension will be discussed.

March 19

Thursday at 16:15, in Room 102

Prof. Horst Martini, Technical University of Chemnitz, Germany : Some results and open problems from geometry

Abstract: In this talk some results and open problems from different parts of geometry will be presented.
More precisely, the fields convex geometry, elementary geometry, and geometry of finite dimensional real Banach spaces will be presented in this way. In particular we will take care for the "geometric kernel" of proof ideas and problem presentations. Due to this "descriptive approach", the lecture might be stimulating even for students to attack themselves problems of this type.

March 12

Thursday at 16:15, in Room 102

Prof. István Juhász, Rényi Institute : Axioms, consistency and independence

Abstract: Gödel's incompleteness results show that mathematics is inherently incomplete but for a long time it was believed that all "really relevant" mathematical statements are decidable by the standard axiom system of set theory. The aim of this talk is to describe several examples which should convince you that this belief is not justified. Undoubtedly, the most well-known such independent statement is the continuum hypothesis which states that any infinite subset of the real line R either has the same size as the set N of natural numbers or as R itself. This problem was the first on Hilbert's famous list that he presented to the International Congress of Mathematicians held in Paris in 1900. After explaining the axioms of set theory I intend to give you a few more, less well-known examples: projective determinacy that implies regularity properties (e.g. Lebesgue measurability) of all projective subsets of Euclidean spaces or, if time permits, results on S and L spaces that come up in set-theoretic topology, my own field of research.

February 26

Feedback Session
Thursday at 16:15, in Room 102

Having any problems in organizing your life in Budapest? We all come together on Thursday to help each other.
This is the perfect opportunity to discuss your first impression about the courses, instructors, and the BSM program. Your opinion can be valuable to us, as well as to others in making the big decision.
Also, this late afternoon is the deadline for registration. If you are uncertain what to keep and what to drop, the 'Feedback' will help to solve this clue. In any case, we finally have to form the classes, decide the fate of ones with low/high audience.

February 19

Thursday 16.15 pm in Room 102

"N is a number", a movie about Paul Erdős. Please note that the time is changed to 16:15 and the show is changed from the Main Lecture Hall of the Renyi Institute to Room 102 at Bethlen ter (due to the ongoing Turan Memorial Lectures, unfortunately the Lecture Hall at Renyi is not available at this time).