## April 19

Thursday at *16:15*, in Room 102

*Prof. Domokos Szász*, Budapest University of Technology and Economics
: Deterministic Chaos

*Abstract:* a bit more technical (mathematically involved) to be displayed in HTML this time, you might view it here

## April 12

Thursday at *16:15*, in the Main Lecture Hall of the Renyi Institute

*Prof. Gábor Domokos and Péter Várkonyi*, Budapest University of Technology and Economics
: Mono-monostatic bodies: the story of Arnold and the tortoise (the "Gömböc")

*Abstract:* V.I. Arnold conjectured that convex, homogeneous bodies with less
than four equilibria (also called mono-monostatic) may exist. Not only
did his conjecture turn out to be true, the newly discovered objects show
various interesting features. Our goal is to give an overview of these
findings as well as to present some new results. We will point out
that mono-monostatic bodies are neither flat, nor thin, they are not
similar to typical objects with more equilibria and they are hard to
approximate by polyhedra. Despite these "negative" traits, there seems
to be strong indication that these forms appear in Nature due to their
special mechanical properties.

## March 22

Thursday at *16:15*, in Room 102

*Prof. László Babai*, University of Chicago (USA)
: Modulo 2 geometry, linear vs. statistical independence,
guessing your hat, and why computer scientists and
economists should care

*Abstract:* Statistical independence (of events or of random variables) and
linear independence (of vectors) are concepts belonging to
two separate areas of mathematics (probability theory and linear
algebra). The two concepts seem to have little in common other
than their names. Pure coincidence. Or is it?

While the hat puzzle in question is for sheer mathematical delight (for now, at least), for a challenge, here is another hat puzzle, arising from electronic commerce, a recently popular branch of microeconomics with applications to such multibillion dollar businesses as Google ad auctions.

Suppose members of an athletic team, numbered 1 through n (on their shirts), are in a room, each wearing a red or a blue hat. Each can see everybody else's hat but not their own; they also see everybody's number, including their own. They are not allowed to communicate. Each gets a ballot sheet; privately and simultanously, they have to vote either "democrat" or "republican." When the results are revealed, both the red-hats and the blue-hats must be as evenly divided among the two parties as possible, i.e., if there is an even number of blue hats, say, then exactly half of the blue hats has to vote for each party; if there is an odd number, then "half" must be rounded either down or up. If they succeed, the team is rewarded. The athletes have no political views; their single aim is to get the reward.

Prior to this test, the athletes are permitted to have a long session (bare-headed) to agree on a strategy. The hats are subsequently distributed in a dark room by an adversary who eavesdropped on the strategy session.

Help the team devise a strategy that will win no matter how the hats are distributed.

The same problem with more than two parties is open; a solution would have further applications to electronic commerce.

## February 23

Friday at *14:30*, in Room *TBA*

*Prof. Béla Bollobás*, Trinity College, University of Cambridge (Cambridge, UK) and University of Memphis (USA)
: Some Coffee Time Problems

*Abstract:* Many mathematical problems can be formulated as
amusing puzzles to be discussed over a cup of coffee, although they
do illustrate beautiful mathematical principles. In the talk I shall
present some of these `coffee time problems' and their solutions.

## February 22

Feedback Session at *16:15*, in Room *102*

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. Next day, Friday, is the deadline for registration, if you are uncertain what to
keep and what to drop, the 'Feedback' will help to solve your clue. We finally have to form the classes. Having any
problems in organizing your life in Budapest? We all come together on Thursday to help each other.

## February 15

Thursday 16.30 pm:*a movie about Paul Erdős.*Please note that the movie will be shown at the Renyi Institute, which you can find according to this map.

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