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.

References:

Domokos, G., Ruina, A., Papadopoulos, J.: Static equilibria of rigid bodies: is there anything new? J. Elasticity, 36 (1994), 59-66.

Varkonyi, P. , Domokos G.: Static equilibria of rigid bodies: dice, pebbles and the Poincare-Hopf Theorem. J. Nonlinear Science 16 (2006), 255-281.

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?

We shall discuss surprising interactions between these two notions of independence, with applications ranging from derandomization of algorithms to error correcting codes (built into your CDs and cell phones) to the solution of a recently popular hat puzzle discussed on the New York Times.

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.

There are no prerequisites for the talk: it should be understandable by everybody with a little mathematical maturity.

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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