## 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).

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