My exams are over – I start my summer holidays. First spot is Topics in Geometric Group Thoery School/Conference. I thought I’m not really into GGT, but while preparing for this event I changed my mind a little bit. However, I came here mostly because people who came here are great if one wants to talk about mathematics (of any sort).

Today we just came to Bedlewo Math Conference Center, all the real stuff is starting tomorrow (it’s after midnight here, so actually today), but we had some time to drink few beers and talk about mathematics. I learned about at least two very cool things.

As a bonus, read about my very very weird mathematical dream :-).

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This one wins all the prizes for today; it was a question on pre-admission test for phd studies in Wroclaw (one of the biggest math centers in Poland). Hardcore version: prove or disprove whether it’s possible that an interesection of parabola and circle on a plane can consist of exactly two different point of which one is a point of tangency and the other is not a point of tangency.

The real version (softened by examining committee) was actually to prove that it’s possible :-).

For me it was a real surprise! Quite counterintuitive, very interesting. I won’t give you any hints, ’cause it’s best to ponder it alone for a moment.

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The second one is about Mapping Class Group. MCG is fairly simple concept. One takes a fixed manifold X and defines MCG(X) as a quotient of all homeomorphisms X –> X modulo these homeomorphisms which are homotopic to identity. It is discrete group. The soft version of what we talked about is to find a surjective morphism from MCG(S³ x S³) to SL(2,Z). This is fairly easy and uses only a fact that S³ is a group.

The hardcore version, to which solution I don’t understand, is to find a kernel of this morhpism. It has something to do with Milnor’s exotic spheres (!).

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And here’s my dream: Jean-Louis Loday is apparently giving some kind of a lecture. He writes some stupid philosophical question on a blackboard, which all the audience, including me, treats very seriously. He asks for some comments and I raise my hand. He chooses me to speak but just before I begin to speak he starts to laught at me, because I don’t like writing style of Jean Pierre Serre (He explicetely points to “Course in Arithmetics” by JPS). Then all the audience laguhs at me :-).

There were also some non-mathematical parts of this dream which I’ll mercifully omit. :-)

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I hope to write some notes from tomorrow’s lectures.

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