## May 8

Thursday at *16:15*, in Room 102

*Prof. László Lovász*, Eötvös University: *Very large graphs*

*Abstract:* There are many areas of science where huge networks (graphs) are central
objects of study: the internet, the brain, various social networks, chip
layout, statistical physics, etc. To study these graphs, new paradigms are
needed: What are meaningful questions to ask? When are two huge graphs
"similar"? How to scale down these graphs without changing their
fundamental structure? How to generate random examples with the desired
properties?

## April 24

Thursday at *16:15*, in Room 102

*Prof. János Pintz*, Rényi Institute: *Landau's problems on primes*

*Abstract:* At the 1912 Cambridge International Congress Landau listed four basic problems about primes.
These problems were characterized in his speech as ``unattackable at the present state of science''.
The problems were the following:

(1) Are there infinitely many primes of the form n^2+1?

(2) The (Binary) Goldbach Conjecture, that every even number exceeding 2 can be written as the sum of two primes.

(3) The Twin Prime Conjecture.

(4) Does there exist always at least one prime between neighboring squares?

All these problems are still open. In the lecture a survey will be given about partial results in Problems (2)--(4), with special emphasis on the recent results of D. Goldston, C. Yildirim and the speaker on small gaps between primes.

## April 17

Thursday at *16:15*, in Room 102

*Prof. Péter Pál Pálfy *, Rényi Institute: *What are simple groups good for?*

*Abstract:* The Classification of Finite Simple Groups is a theorem that has by
far the longest proof in mathematics, its proof is scattered in
numerous papers that add up to more than 10,000 journal pages.
The statement is that all finite simple groups are known. Simple
groups are the building blocks of finite groups, and they are known
both in the technical sense that there is a list containing all
isomorphism types of finite simple groups, and in the informal sense
that if a general question about groups can be reduced to a problem
about simple groups then a solution often can be obtained just by
going through the list of finite simple groups.

In the talk I will sketch this list, that involves 18 infinite
series and 26 individual (so-called `sporadic') groups. I will
highlight some of these groups. As the 2008 Abel Prize has been
awarded to two groups theorists, John G. Thompson and Jacques Tits,
I should not miss mentioning the Thompson group (one of the sporadic
groups) and the Tits group (an exceptional group in one of the
infinite series).

I will briefly discuss the reliability of such an extremely complex
proof. Some consequences of the Classification Theorem will be
presented as well.

## April 11

*Friday at 14:15*, in Room *105*

*Prof. Béla Bollobás*, University of Cambridge and University of Memphis: *Gamblers, Prisoners and Mathematicians*

## April 10

Thursday at *16:15*, in Room 102

*Prof. Jerry Kazdan*, University of Pennsylvania: *Using symmetry*

*Abstract:* I give examples of using symmetry to problems in various
aspects of mathematics - including an example using the * same* group in
both a problem in number theory and in special relativity.

## April 3

Thursday at *16:15*, in Room 102

This day we will have visitors from the University of Cambridge, two Part III (kind of Master program) students, one of them a BSM alumnus.
They will speak about there life in Cambridge and about mathematics as well. Titles and abstracts are:

*Nathan Kaplan *: Coefficients of Cyclotomic Polynomials

*Abstract:* The *n*th cyclotomic polynomial, Phi_n(x) is the unique monic polynomial whose roots are the primitive *n*th roots of unity. It is not difficult to show that these polynomials have integer coefficients and are irreducible. In this talk we will discuss some problems related to the size of coefficients of cyclotomic polynomials.

*Carl Erickson*: Prime Numbers and the Riemann Hypothesis

*Abstract:* It is common knowledge that the zeros of the Riemann zeta function are connected with the distribution of the prime numbers, and that the Riemann hypothesis, which says that the non-trivial zeros all lie on a certain line, is a very beautiful statement about the primes. We will discuss the precise connection between the zeros and the primes in order to illustrate the meaning and importance of the Riemann hypothesis with respect to the prime numbers.

## March 27

Thursday at *16:15*, in Room 102

*Prof. Dezső Miklós*, BSM and Rényi Institute: On extremal sets of the vertices of the hypercube, subset sums of a finite sum and database security

*Abstract:* If we are given some real numbers with a positive sum, it is a natural question to ask how many sumbsums (maybe of a certain property, like with all of them having the same number of elements) should always be positive. A similar question is: how many subsums of * any * given n numbers (but they may not be equal to 0) can be equal to zero (at most).

Another - seemingly rather different - question is the following: how many sums from a given set of numbers can be disclosed without enabling the
audience to find out the single values.

Both of the above questions turn out to be connected to (and answered via) the following general combinatorial problem:
What is the maximum size of a subset M of the vertices of the n-dimensional
hypercube assuming that the span (convex span) of these vertices in M completely avoids (or does not contain) the hyperplane of the cube consisting of the vertices of a certain weight m (where the weight of a vertex is the number of 1 coordinates of it)?

## March 6

Thursday at *16:15*, in Room 102

*Prof. Miklós Simonovits*, Rényi Institute(Budapest): The evolution of random graphs.

*Abstract:* One of the very important and fast developing branches of Discrete Mathematics
is the theory of Random Structures. In the lecture I will give a
introduction into the theory and application of random graphs.

Random graphs were first used to prove the existence of some combinatorial
structures that were difficult (or sometimes seem even today to be impossible)
to construct using the classical methods.

From these methods developed (first in the works of Erdős and Rényi) the
theory of Random graphs.

Random graphs are used in algorithms, in existence theorem, in computer
science, and in many other fields.

## February 21

Feedback Session 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 14

Thursday 16.30 pm:*"N is a number", a movie about Paul Erdős.*Please note that the movie will be shown in the Main Lecture Hall of the Renyi Institute, which you can find according to this map.

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