## December 3

Thursday at *16:15*, in Room 102

*Brett FRANKEL*, The Johns Hopkins University and * BSM *: Quadratic Forms and Topographs

*Abstract: * Binary quadratic forms, expressions of the form ax^2+bxy+cy^2, are among the most well-studied objects in number theory. In particular, Gauss's Disquisitiones Arithmeticae gives a very comprehensive treatment of the subject. It is therefore surprising that a powerful-yet-elementary approach to working with quadratic forms was developed as recently as 1991. In this talk I will present topographs, a visual approach to quadratic forms due to Conway.

## November 19

Thursday at *16:15*, in Room 102

*Prof. Ervin GYŐRI*, Rényi Institute: Funny numbers, functions and graphs

*Abstract: * The lecture finds surprising connection among the following unrelated (?) subjects:

1. Representation of rational numbers

2. Monotone increasing real functions with discontinuity at each rational number

3. The maximum number of triangles sharing a common edge in graphs with large minimum degree

The talk is self-contained, no prerequisites are needed.

## November 12

Thursday at *16:15*, in Room 102

*Prof. Ian BIRINGER*, Yale University: Combinatorics and Geometry on Surfaces!

*Abstract: * Let us consider the following game. Pick a point on the circle. Rotate it by some angle, say 'x'. Rotate the result again by angle x, and continue the process, say, 418 times. We then obtain a set of points on the circle; this set divides the circle into a bunch of arcs. The 3-Gap Theorem, a (partially Hungarian) result in combinatorics from the 1960's, says that although there may be many arcs, they have only 3 distinct lengths!

In this talk, we'll discuss this theorem and how to play a similar game on 2-dimensional surfaces instead of the circle. The flavor will be mostly geometric, with a hint of combinatorics.

## November 5

Thursday at *16:15*, in Room 102

*Prof. Pál HEGEDŰS*, Central European University: Transcending
boundaries

*Abstract: * A Hungarian thinker, Tabor, once explained that miracle is something
that can be understood only when one "grows by one dimension". In this
talk I will review many proofs and concepts that are miraculous in this
sense. They illuminate phenomena from an outside source that explains
away the complexity of the problem. The difficulty ranges from high
school mathematics to graduate level.

## October 29

Thursday at *16:15*, in Room 102

*Prof. Miklós ABÉRT*, Rényi Institute and University of Chicago: Why is group theory cool?

*Abstract: * I will tell you about my angle on groups: why are they beautiful and why is it necessary to study them. I will illustrate my point of view on a specific direction of research that is related to graph theory and probability theory.

## October 8

Thursday at *16:15*, in Room 102

*Prof. Richard Rimányi*, University of North Carolina / Rényi Institute
: Intersections (or, the cohomology ring of moduli spaces)

*Abstract:* We will discuss enumerative geometry problems similar to the following: given four
straight lines in space, how many straight lines intersect all four of them? Our approach
will be based on understanding the intersections of various subsets of the set
parameterizing straight lines of the space.

## October 1

Thursday at *17:00*, at Eotvos University, (Pazmany Peter setany 1/C - south building - room 0-805; see remark below)

*Prof. Gyula Károlyi*, Eötvös University: Incidence geometry in combinatorial arithmetic

*Abstract:* What has throwing dice, drawing graphs and
counting incidences between points and lines to do
with value sets of certain polynomials?

An entirely
elementary introduction into one of the most rapidly
developing areas of 21st century mathematics.

*REMARK:*Another exceptional, out of the building colloquium lecture. The venue will be at the Eötvös University, Faculty of Sciences campus, south building (Deli tomb, 1/C on this map). A more detailed map, and the sketch of the interior of the building . If you enter the building at the entrance to the west (nyugati bejarat), turn immediately to the left and you will find room 0-805, the venue of the lecture in 10 seconds.

The time is 5pm, allowing all of you to reach the place.

You will have a chance to meet some of the faculty and math students of the Institute of Mathematics over there.

## September 24

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.

## September 18

Friday at *16:45 (apprx.)*, at Bolyai Institute (Aradi vertanuk tere 1, Szeged, Hungary)

*Prof. Tibor KRISZTIN*, Bolyai Institute, University of Szeged: Differential equations with delay

*Abstract: * Differential equations with delay appear in several applications, e.g.,
in models for the growth of a single species.
Some basic properties of equations with delay will be compared to those
of ordinary and partial differential equations.
A simple nonlinear equation - modeling a basic feedback mechanism -
will be considered to show how oscillation, periodicity and more
complicated behavior arise naturally.

*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 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. 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.

## September 17

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.

## September 10

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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