## May 4

Friday at *15:45 (apprx.)*, at Bolyai Institute (Aradi vertanuk tere 1, Szeged, Hungary)

*Prof. Viktor Vigh: *, Bolyai Institute, University of Szeged:
**Spindle convexity - an introduction**

*Abstract: * In the first part of the talk, we
introduce the notion of spindle convexity. The idea is that the role of
closed half-spaces is played by closed unit balls, so, roughly speaking,
spindle convex sets are the (not necessarily finite) intersections of unit
balls. We present the basic properties of spindle convex sets, and compare
them with the properties of "normal" (linear) convex sets.

In the
second part (if time allows) we show some recent results concerning
spindle convexity. These theorems easily imply older, well-known results
from linear convexity, which proves the strength of this new concept.

*REMARK:*This will be an exceptional, out of the 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 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. Mariann 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.

## May 3

Thursday at *16:15*, in Room 102

*Prof. Sarah Kitchen*, Albert-Ludwigs Universit¨at Freiburg (BSM alumna):
** What is a D-module?**

*Abstract:* D-modules are modules for rings of differential operators. They give
us a way of studying differential equations algebraically. In this
talk, I will introduce rings of differential operators and modules for
such rings. We will take a careful look at the Weyl algebra in one
variable, which is the ring of polynomial differential operators for
polynomials in one complex variable. We'll look at the close
relationship between the D-module associated to a differential
equation and the solutions to that differential equation in the form
of the Riemann-Hilbert correspondence.

*REMARK:*Prof. Sarah Kitchen was a BSM student Fall 2003 and graduated later with a PhD from the University of Utah. Currently, she is a research assistant (postdoc fellow) at the Albert-Ludwigs Universität Freiburg. After her talk the audience is invited to a free conversation, question-and-answer session, about life at and after BSM including all kinds of info on graduate life.

## April 26

Thursday at *16:15*, in Room 102

*Adam Hesterberg*, Rényi Institute and Princeton University:
** Logic Puzzles in Linguistics**

*Abstract:* We'll solve and discuss some linguistics-themed logic puzzles of the sort used in the International Linguistics Olympiad and the just-started Hungarian Linguistics Olympiad. For instance, in the following easy sample puzzle, the 6 English sentences correspond to six Kurdish sentences, in random order. Figure out which is which:

1. Ez h'irç'e^ dibînim

A. You see Bear.

B. You see me.

C. Bear runs.

D. You run.

E. I see Bear.

F. I run.

2. Tu dir'evî

3. Tu min dibînî

4. H'irç' dir'eve

5. Ez dir'evim

6. Tu h'irç'e^ dibînî

## April 19

Thursday at *16:15*, in Room 102

*Prof. Antal Balog*, Rényi Institute and BSM:
** What shall we do for an ABEL?**

*Abstract:* We give a short account about the work of Endre Szemeredi, the 2012 Abel Prize laureate. Although some proofs are included, no special expertise beyond the general undergraduate knowledge is expected.

## April 12

Thursday at *16:15*, in Room 102

*Prof. János Pach*, Rényi Institute:
** Szemerédi strikes back**

*Abstract:*
By an argument reminiscent of Furstenberg's original ergodic
theoretic proof for Szemeredi's Theorem on arithmetic
progressions, Furstenberg and Weiss (2003) proved the
following result. For every k and l, there exists an integer
n(k,l) such that no matter how we color the vertices of a
complete binary tree of depth n>n(k,l) with k colors, it
always contains a monochromatic equispaced complete binary
subtree T' of depth l; that is, a complete binary subtree T'
of depth l which has the property that all of its vertices
are of the same color and every vertex at level i of T' lies
at level j+id in T. (Here j and d are suitable integers and
0 <= i <= l.) Moreover, the two children of any vertex
v of T' are descendants of different children of v in T.
Furstenberg and Weiss also established several density
versions of the above results, generalizing Szemeredi's
Theorem.

We show that all of these results can be obtained by
elementary combinatorial arguments, using Szemeredi's
classical theorem itself. Joint work with J. Solymosi and G.
Tardos.

## March 22

Thursday at *16:15*, in Room 102

*Prof. József Pelikán*, Eötvös University: **Partitions and formal power series**

*Abstract: * The partition function p(n) is defined as the number of essentially
different ways of writing n as a sum of positive integers. For
instance p(4)=5, because of the five representations 4=3+1=2+2=2+1+1=
1+1+1+1. This function p(n) occurs in several places in mathematics
and has many mysterious properties discovered by Euler, Ramanujan,...
Many questions about them are still unsolved.

Formal power series are a powerful technique for handling partitions,
but they find application in other areas of mathematics as well.

We plan to present some of the interesting results in this area of
mathematics.

## March 10

Sarturday * from 10:00 am to 15:00 pm*, in Room 220

**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 2012 (Blagoevgrad, Bulgaria, 26th July
- 1st August).
Anyone is very welcome to compete and the participation of good problem solvers is highly
encouraged!

## March 8

Thursday at *16:15*, in Room 102

*Prof. Miklós Abért*, Rényi Institute:
**Expander graphs, or how to save on the price of randomness**

*Abstract:* We will introduce the notion of expansion for graphs and
talk about its relevance for fields outside mathematics. In
particular, we describe how to find a sick sheep in the flock using as
few random coin-tosses as possible. We also outline the connection of
expansion to linear algebra and prove some nice starting results.

## February 23

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, 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.

followed by a lecture of *David Rudel*, BSM alumn

**Mathematical Modeling in the Real World (and Science Education)**

*Abstract:* : I will discuss the topic of applied mathematical modeling from the perspective of a professional system designer. This will include a cursory discussion of several problems I've worked on for clients in the education industry and a detailed investigation of the question "How much faster do gas molecules move on average than molecules in a solid?" The answer is sure to surprise you!
Remarks: David Rudel is a senior editor for ExploreLearning.com, where he has designed and written curriculum for many science Gizmos^{TM}. He is also a writer, having published seven books in the fields of theology, chess, and science education. His most recent book is the second in a series on common science myths and misconceptions perpetuated by standard science textbooks in grades 6-12.

## February 16

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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