A fake projective plane is a smooth complex projective algebraic surface whose Betti numbers are same as those of the complex projective plane but which is not the complex projective plane. The first fake projective plane was constructed by David Mumford in 1978 using p-adic uniformization. This construction is so indirect that it is hard to obtain geometric properties of the fake projective plane. A major problem in the theory of complex algebraic surfaces was to find all fake projective planes in a way which allows us to discover their geometric properties.
Universitat Poliecnica de Catalunya, Barcelona, Spain
February 15, 2012
We study particular solutions of the "inner equation" associated to the splitting of separatrices on "generalized standard maps". An exponentially small complete expression for their difference is obtained. We also provide numerical evidence that the inner equation provides quantitative information of the splitting of separatrices even in the case when the limit flow does not predict correctly its asymptotic behavior.
Zaremba's 1971 conjecture predicts that every integer appears as the denominator of a finite continued fraction whose partial quotients are bounded by an absolute constant. We confirm this conjecture for a set of density one.
In this talk I will present a colored version of Tverberg's theorem about partitioning finite point sets in R^d into rainbow groups whose convex hulls intersect. This settles the famous Bárány-Larman conjecture (1992) for primes minus one, and asymptotically in general. It implies some complexity estimates in computational geometry. I will also give some generalizations that connect to mass partition and center point theorems. The proofs use equivariant topology, which I will try to keep as elementary as possible. This is joint work with Pavle V.M.
We study the setting in which the bits of an unknown infinite binary sequence x are revealed sequentially to an observer. We show that very limited assumptions about x allow one to make successful predictions about unseen bits of x . Our main focus is the problem of successfully predicting a single 0 from among the bits of x . In our model we get just one chance to make a prediction, at a time of our choosing. This models a variety of situations in which we need to perform an action of fixed duration, and must predict a "safe" time-interval to perform it.