Veranstaltungen am Mathematischen Institut

Abstract: Gromov proved the following ''Limit theorem'': Let g be a C^2 Riemannian metric on a smooth manifold M (without boundary). If a sequence of C^2 Riemannian metrics on M converges to g in the C^0 sense, and each scalar curvature is bounded from below by k. Then the scalar curvature of the limiting metric g is also bounded from below by k. In this talk, I'd like to talk about some total-scalar-curvature-version theorems of this limit theorem. I also consider the limit theorem for an weighted total scalar curvature and as a corollary, I give a definition of scalar curvature lower bound in a weak sense. To prove these, we use the Ricci flow. If I have time, I also would like to talk about limit theorems for the upper bound of the total scalar curvature. Compared to the above results, we use a different geometric flow: the Yamabe flow.

Abstract: In joint work with Rachael Boyd and Corey Bregman we study the classifying space B Diff(M) of the diffeomorphism group of a connected, compact, orientable 3-manifold M. By a theorem of Milnor every such M has a unique prime decomposition as a connected sum of prime 3-manifolds. The purpose of this talk is to explain how one can compute the moduli space B Diff(M) in terms of the moduli spaces of prime factors. We show that certain space of systems of reducing spheres is contractible. (This can be thought of as saying that the modular infinity-operad of 3-manifolds is freely generated by irreducible manifolds.) We use this to prove that if M has non-empty boundary, then B Diff(M rel boundary) has the homotopy type of a finite CW complex, as was conjectured by Kontsevich.

Abstract: Let (M,g) be a four-dimensional closed connected oriented (possibly non-spin) Riemannian manifold with scalar curvature bounded below by 12. We prove that, if f is a smooth distance non-increasing map of non-zero degree from (M, g) to the unit four-sphere, then f is an isometry. This removes the spin condition in Llarull's scalar curvature rigidity theorem of spheres in dimension four. We utilize mu-bubbles and a version with coefficients of the rigidity of the three-sphere to rule out the case where all the inequalities are strict. Our proof of rigidity exploits monotonicity results for the harmonic map heat flow coupled with the Ricci flow due to Lee and Tam This is joint work with J. Wang, Z. Xie and B. Zhu.

Abstract: For every n and a fixed prime p, we construct a new congruence for the orbifold Euler characteristic of a group which we call the chromatic congruence at the height n. Here the word ?chromatic? refers to the chromatic stable homotopy theory, though to understand this talk no background in stable homotopy theory is required. The p-adic limit of these congruences when n tends to infinity recovers the well-known Brown-Quillen congruence. We apply these results to mapping class groups and using Harer-Zagier we get an infinite family of congruences for Bernoulli numbers. At the end we will see that these congruences in particular recover classical congruences for Bernoulli numbers due to Kummer, Voronoi, Carlitz and Cohen.