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.

## 4 comments

July 23, 2007 at 10:46 am

guestCan you provide a hint for this phd-exam question? It’s very counter-intutive unless I don’t understand something…

August 19, 2007 at 11:54 pm

sirixTake a prabola and a point on it that doesn’t lie on parabola’s axis of symmetry. Now take a circle that is strictly tangent to a parabola in a chosen point – this circle actually cuts a parabola in three pieces.

August 20, 2007 at 2:10 am

John ArmstrongNow that the answer’s been given, I can say that I’m surprised that it counts as a Ph.D. level problem. In general position, the intersection of two conics is in four points (with multiplicities). Three can be used together to give a point of tangency and the fourth is left elsewhere.

August 20, 2007 at 11:44 am

sirixFor a record: as I wrote, this was an admission question, so it hardly counts as a phd level question – I would call it “master level” question.

Still, I guess you’re right that such a question wouldn’t be present on an admission exam in Princeton.