Shifted Lagrangian structures in Poisson geometry
Tuesday, 2.5.23, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
It is well known that BG carries a 2-shifted symplectic structure. In this talk, I will study the shifted lagrangian groupoids of BG. I will show how many constructions on Poisson geometry unify using the language of shifted symplectic groupoids. This is work in progress with Daniel Alvarez and Henrique Bursztyn.
ODD Riemannian metrics
Monday, 8.5.23, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
We describe a generalization of Riemannian metrics motivated from\nKähler geometry of singular complex varieties. These generalizations\nare semipositive symmetric 2-tensors, but degenerate in such way, that\ne.g. they still induce a metric space structure on the underlying\nmanifold.\n\nIn this talk, we will mostly use instructive examples to sketch how far\nRiemannian Geometry can (hopefully) be pursued for these ODD metrics.\n
Shifted convolution sums
Monday, 15.5.23, 16:00-17:00, Raum 125, Ernst-Zermelo-Str. 1
In talk I will evaluate shifted convolution sums of divisor functions of the form \(\bdisplaystyle\bsum_{n_1,n_2\bin\bmathbb{Z}\bsetminus\b{0\b}, n_1+n_2=n}\nQ_{d}^{(r_1,r_2)}\bBig(\bfrac{n_2-n_1}{n_1+n_2}\bBig)\bsigma_{r_1}(n_1)\bsigma_{r_2}(n_2)\) where \(\bsigma_{r}(n) = \bsum_{d \bmid n} d^ r\) and \(Q_{d}^{(r_1,r_2)}(x)\) is the Jacobi function of the second kind. These sums can be considered as a shifted version of the Ramanujan sum \(\bsum_{n_1 \bin \bmathbb{Z}} \bsigma_{r_1}(n_1) \bsigma_{r_2}(n_1) n_1^s\). \n\nKey words that appear in the proof and the final result: non-holomorphic Eisenstein series, cusp forms, values of \(L\)-functions, Mellin transform and Whittaker's functions.
Metric inequalities with positive scalar curvature
Monday, 22.5.23, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
We will discuss various situations where a certain perturbation of the Dirac operator on spin manifolds can be used to obtain distance estimates from lower scalar curvature bounds. \n\nA first situation consists in an area non-decreasing map from a Riemannian spin manifold with boundary \(X\) into the round sphere under the condition that the map is locally constant near the boundary and has nonzero degree. Here a positive lower bound of the scalar curvature is quantitatively related to the distance from the support of the differential of f and the boundary of \(X\). \n\nA second situation consists in estimating the distance between the boundary components of Riemannian “bands” \(M×[−1,1]\) where \(M\) is a closed manifold that does not carry positive scalar curvature. Both situations originated from questions asked by Gromov. \n\nIn the final part, I will compare the Dirac method with the minimal hypersurface method and show that if \(N\) is a closed manifold such that the cylinder \(N \btimes \bmathbb{R}\) carries a complete metric of positive scalar curvature, then \(N\) also carries a metric of positive scalar curvature. This answers a question asked by Rosenberg and Stolz. Based on joint work with Daniel Raede and Rudolf Zeidler.