Vorträge des SFB 1442

I will discuss the construction of integral models of Hecke correspondences and the corresponding extension of Hecke correspondences on automorphic vector bundles.

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.

1st talk:
A classical strategy to study the existence question of positive scalar curvature (PSC) metrics is to (successively) reduce the dimension by one to simplify the problem. This goes back to the 1979 proof of the positive mass theorem by Schoen?Yau via minimal hypersurfaces. More recently, an augmentation of minimal hypersurfaces known as "mu-bubbles? has greatly extended the power of this strategy and lead to advances such as the non-existence of PSC metrics on aspherical 4- and 5-manifolds due to Chodosh?Li and Gromov. In this talk, in addition to a gentle introduction and a survey of recent results, I will discuss a conjecture due to Rosenberg and Stolz which predicts that a closed manifold $M$ (of dimension $\neq 4$) admits a positive scalar curvature (PSC) metric if and only if $M \times \mathbb{R}$ admits a complete PSC metric. I will explain a proof of this for orientable manifolds up to ambient dimension 7 by combining mu-bubble techniques with positive scalar curvature cobordism theory. This result was obtained in a joint work together with Cecchini and Räde.

2nd talk:
I will explain the role of cobordism theory in the study of positive scalar curvature on high--dimensional manifolds. This starts with the Gromov--Lawson surgery theorem and unfolds its consequences, some of which are classical and some of which have been discovered by the speaker and his collaborators in the last decade. \begin{enumerate} \tem The surgery theorem reduces the existence question for metrics of positive scalar curvature (in the high--dimensional regime) to a problem in cobordism theory which can in some cases be completely solved by means of stable homotopy theory. \item The homotopy type of the space $\mathcal{R}^+ (M)$ of metrics of positive scalar curvature only depends on the cobordism class of $M$, when the notion of cobordism is interpreted correctly. \item In some cases, that homotopy type does not even depend on the cobordism class of $M$. \item The natural action of the diffeomorphism group of $M$ on $\mathcal{R}^+ (M)$ has a remarkable rigidity property; saying that often the diffeomorphism group acts trivially in a suitable sense. \item This rigidity property, combined with high--dimensional Madsen--Weiss theory, leads to the construction of a great wealth of elements in the homotopy groups in $\mathcal{R}^+ (M)$. \end{enumerate} The results we explain are either classical (mostly due to Gromov--Lawson and Stolz) or were obtained in joint works with Randal--Williams, Botvinnik and Wiemeler; or proven in the theses by Frenck, Bantje and Mantione.