## May 9

Thursday at *16:15*, In room 102, Bethlen ter

** BSM EUR Conference **

The Elective Undergraduate Research Groups will present their findings

## April 27

Saturday * from 10:00 am to 15:00 pm*, In room 105, Bethlen ter

**BSM local mathematics competition **

The local math competition * comes with cash prizes * as well as serves as a selection for the team representing BSM at the
International Mathematics Competition for University Students 2013.
Anyone is very welcome to compete and the participation of good problem solvers is highly
encouraged!

## April 25

Thursday at *16:15*, at BSM, in Room 102, Bethlen tér

*Prof. Kimberly ROTH*, Juniata College:
**Variations on a fractal theme: The Mandelbrot Set and Multibrot Sets**
We will look at three types of fractals: the Mandelbrot set discovered by Benoit Mandelbrot in 1975, the Multibrot sets that are a generalization of the Mandelbrot set, and the Julia sets related to them. We will learn about how the sets are mathematically generated, relationships between them, and some known and conjectured properties. No prior knowledge of fractals or complex analysis is needed.

*Abstract:* We will look at three types of fractals: the Mandelbrot set discovered by Benoit Mandelbrot in 1975, the Multibrot sets that are a generalization of the Mandelbrot set, and the Julia sets related to them. We will learn about how the sets are mathematically generated, relationships between them, and some known and conjectured properties. No prior knowledge of fractals or complex analysis is needed.

*REMARK:*Prof. Roth is a BSM alumna form the "good old days" visiting BSM and AIT in Budapest. She will also be available for a chat after the talk.

## April 19

Friday at *16:45 (apprx.)*, at Szeged University (room M8 = 201, Mars ter 20, Szeged, Hungary)

*Prof. Gábor CZÉDLI: *, Bolyai Institute, University of Szeged:
**On the mathematics of simple juggling patterns**

*Abstract: * Simple juggling patterns are described by certain finite sequences of nonnegative integers, the so-called juggling sequences. We characterize juggling sequences mathematically. Also, we count these sequences, that is, the simple juggling patterns, for any given number of balls and given maximal height of throws.

The first theorems in the Mathematics of Juggling are due to two very famous mathematicians, Claude E. Shannon (also a computer scientist, 1916-2001) and
Ronald L. Graham (president both of the American Mathematical Society in 1993, and the International Juggler's Association, 1972; 28 joint papers with Paul Erdős). This lecture is based on Burkard Polster's book "The Mathematics of Juggling", 2003, but only some mathematically easy questions will be discussed.

Sometimes, we would like to, or have to, convince outsiders that mathematics is useful, interesting and beautiful. The talk offers a possible way to do so, because it tries to follow the following scheme:

- start with a topic outside mathematics that is sufficiently interesting to many non-mathematicians,

- disregard details of little importance,

- set up a mathematical model that is at least partially clear for outsiders,

- find and prove mathematical theorems in the model that are more or less clear for outsiders,

- draw non-mathematical conclusions for the original topic that are very clear to anyone.

In our case, the non-mathematical conclusion was the discovery of a simple juggling patterns, not known by professional jugglers before their juggling sequences were found. (Perhaps, one of these juggling patterns will be shown.)

*REMARK:*This will be an exceptional, out of the town colloquium lecture at an exceptional time, Friday afternoon. This Friday the last classes (from 1pm) might be canceled (and made up later) and the group will leave from Nyugati railway station at 13:53 with TÖMÖRKÉNY intercity train. You will be met at the Szeged railway station at the arrival time, at quarter past 4 and be ushered to the site of the lecture, stopping by at the dormitory on the route. 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 21

Thursday at *17:00*, at Eotvos University, (Pazmany Peter setany
1/A - north building, "Északi tömb" - room 1.71; see remark below)

*Prof. András Szűcs*, Eötvös University: **Soups and hedgehogs**

*Abstract:* “It is a mystery why mathematics is such an efficient tool in the hands of scientists
in understanding the world” - wrote a Hungarian-American Nobel prize winner physicist,
Eugene Wigner. Probably the clue to this mystery is that mathematical abstraction helps
us to discover the same essence behind many, seemingly different phenomena. The talk
will show examples demonstrating this statement:

1. “The soup can not be mixed properly”, i.e. there always will be a particle which
returns to its original position.

2. ‘”A hedgehog can not be combed”. In mathematical terms: there is no nowhere
vanishing, continuous, tangent vector field on the 2-dimensional sphere.
3. “The Fundamental Theorem of Algebra”, i.e any polynomial (of degree at least 1
over the complex field) has a root.

It turns out that all these statements (and many more) can be proved using the same
elementary notion, the so called

*Before the talk there will be chance to learn about the English language MSc program of the Institute of Mathematics of Eotvos University.*

*REMARK:*Another exceptional, out of the building colloquium lecture. The venue will be at the Eötvös University, Faculty of Sciences campus, north building (Északi tömb, 1/A on this map).

You find the Eotvos University campus by taking trams 4 or 6 southbound from Blaha Lujza, go with them across the Danube and leave the trams immediately on the Buda side. Walk along the Danube southbound until you reach the first of the two huge buildings of the Faculty of Sciences (North building)

A more detailed map of the immediate neighborhood .

The timing will allow all of you to reach the place.

The room is on the first floor. On this plan of the building the red cross shows the location of the room. You need to find an elevator (X) or staircase to go one floor up.

## March 7

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

*Prof. László Csirmaz*, Central European University and Rényi Institute:
**Gentle introduction to elliptic curves**

*Abstract:*Elliptic curves are playing an important role in modern cryptography.
Their most important property is to admit an operation -- addition --
on their points, which is then used to hide information, send secret
messages between parties. During the lecture we'll look at those
interesting curves. Study their connections with Pappos theorem in
projective geometry, with Bezout theorem in algebraic geometry, and
find out how the above operation is defined, and why it forms a group.
If time permits, we show some interesting cryptographic applications.

*REMARK:*This will be the first 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 and map of the neighborhood.

This will be an excellent opportunity to meet the students at CEU - some of them BSM alumni - and chat with them about the program.

## February 28

Thursday at *16:15*, at BSM, in Room 102, Bethlen tér

*Prof. Tibor Beke*, University of Massachusetts, Lowell:
**The sign pattern theorem**

*Abstract:*This talk is about an unexpected occurrence of algebraic topology in combinatorics or, if you like, combinatorics in algebraic topology. The good thing about it is that one does not need a lot of mathematical machinery to appreciate it! Much of it is very visual.

On the topological end, there is the Brouwer fixed point theorem: any continuous map of the n-dimensional ball into itself will have a fixed point, i.e. a point not moved by the mapping. This statement is surprising even for the 2-dimensional "ball", that is, the disk.

On the combinatorial end: if a string of + and - symbols begins with `+' and ends with `-', then it contains adjacent + and - symbols somewhere. The sign pattern theorem is the analogue of this trivial observation for matrices whose entries are sign-pairs "++", "+-", "-+" or "--": if the matrix satisfies certain boundary conditions then it contains a 2x2 submatrix with a specific sign pattern. (Come to the talk to hear what it is!) There is an n-dimensional version, concerning n-dimensional matrices whose entries are strings of + and - symbols of length n.

Via an easy application of compactness, the sign pattern theorem implies a vector-valued form of the intermediate value theorem, which in turn implies the Brouwer fixed point theorem. Though the statement of the sign pattern theorem is completely combinatorial, all the proofs I know secretly use ideas from algebraic topology. Curiously, while the Brouwer fixed point theorem is classical and well understood, there are open questions about the combinatorics of sign patterns, for example, the computational complexity of finding them.

## February 21

Thursday at*16:15*, in Room

*102*

*Feedback Session*

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, the morning of this day 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*:

*"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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