Oberseminare und sonstige Vorträge

Abstract: Given a 4-dimensional Einstein orbifold that cannot be desingularized by smooth Einstein metrics, we investigate the existence of an ancient solution to the Ricci flow coming out of such a singular space. In this talk, we will focus on singularities modeled on a cone over RP3 that are desingularized by gluing Eguchi-Hanson metrics to get a first approximation of the flow. We show that a parabolic version of the corresponding obstructed gluing problem has a smooth solution: the bubbles are shown to grow exponentially in time, a phenomenon that is intimately connected to the instability of such orbifolds.

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.

We study stochastic differential equations with simple additive noise under the impact of a shear force. We fully characterize the behaviour of these systems, depending on the shear function, in terms of global and local properties of the stochastic flow.
In more detail, we show criteria for weak and strong completeness and the existence of random set and point attractors. We also demonstrate the possibility of negative and positive Lyapunov exponents, in combination with all other behaviours, indicating synchronization or chaos as characterizations of the random attractors if they exist.

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.

Let X be a finite connected graph and let G be a vertex-transitive group of automorphisms of X. The pair (X, G) is locally-L if the group induced by the action of the stabiliser Gv on the neighbourhood of a vertex v is permutation isomorphic to L. Using this language, a classical theorem of Tutte states that for locally-A3 and locally-S3 pairs, |G| grows linearly with |V(X)|. More generally, given a transitive permutation group L, we are interested in determining the growth of |G| as a function of |V(X)| for locally-L pairs (X,G). We present new results on this topic and highlight an exciting connection with the study of eigenspaces of graphs over finite fields.

Please see attachment

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.