Veranstaltungsübersicht -- Sommersemester 2023

In a remarkable work, Duistermaat and Singer in 1976 studied the algebras of all classical pseudodifferential operators on smooth (boundaryless) manifolds. They gave a description of order preserving algebra isomorphism between the algebras of classical pseudodifferential operators of two manifolds. The subject of this talk is the generalisation of their results to manifolds with boundary. The role of the algebra of pseudodifferential operators that we are interested in is the Boutet de Monvel algebra.

The main fact of life about manifold with boundary is that vector fields do not define global flows and the "boundary conditions" are a way of dealing with this problem. The Boutet de Monvel algebra corresponds to the choice of local boundary conditions and is, effectively, a non-commutative completion of the manifold. One can think of it as a parametrised version of the classical Toeplitz algebra as a completion of the half-space.

What appears in the study of automorphisms are Fourier integral operators and we will try to explain their appearance - both in boundaryless and boundary case. as it turns out, the non-trivial boundary case introduces both some complications but also some simplifications of the analysis involved, Once this is done, the analysis that we need reduces to a high degree to relatively classical results about automorphisms and homology of the Toeplitz algebra and some basic facts from K-theory.

This is a joint work in progress with Elmar Schrohe.

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.

We describe a generalization of Riemannian metrics motivated from Kähler geometry of singular complex varieties. These generalizations are semipositive symmetric 2-tensors, but degenerate in such way, that e.g. they still induce a metric space structure on the underlying manifold.

In this talk, we will mostly use instructive examples to sketch how far Riemannian Geometry can (hopefully) be pursued for these ODD metrics.

In talk I will evaluate shifted convolution sums of divisor functions of the form $\displaystyle\sum_{n_1,n_2\in\mathbb{Z}\setminus{0}, n_1+n_2=n} Q_{d}^{(r_1,r_2)}\Big(\frac{n_2-n_1}{n_1+n_2}\Big)\sigma_{r_1}(n_1)\sigma_{r_2}(n_2)$ where $\sigma_{r}(n) = \sum_{d \mid n} d^ r$ 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 $\sum_{n_1 \in \mathbb{Z}} \sigma_{r_1}(n_1) \sigma_{r_2}(n_1) n_1^s$.

Key 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.

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.

A 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 $X$.

A 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.

In 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 $N \times \mathbb{R}$ 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.

Positively graded symplectic Q-manifolds encompass a lot of well-known mathematical structures, such as Poisson manifolds, Courant algebroids, etc. Lagrangian submanifolds of them are of special interest, since they simultaniously generalize coisotropic submanifolds, Dirac-structures and many more. In this talk we set up their deformation theory inside a symplectic Q-manifold via strong homotopy Lie algebras.

In this talk I will explain the Bakry-Emery Ricci tensor and the metric gluing construction between two (weighted) Riemannian manifolds along isometric parts of their boundary. When the (weighted) Riemannian manifolds admit a lower bound for the (Bakry-Emery) Ricci curvature, I will present a necessary and sufficient condition such that the metric glued space has synthetic Ricci curvature bounded from below.

Classical theorems of Conner-Floyd and Hopkins-Hovey say that complex $K$-theory is completely determined by unitary bordism and $\mathrm{Spin}^c$ bordism respectively. The isomorphisms appearing in these theorems are induced by the maps that send a bordism class to its orientation-class in complex $K$-theory. Despite this geometric description, the proofs that they are indeed isomorphism are rather abstract and homotpy-theoretical.

Motivated by theoretical physics, Baum, Joachim, Khorami and Schick extend Hopkins and Hovey’s result in a forthcoming paper to twisted $\mathrm{Spin}^c$ bordism and twisted $K$-theory. Here, the twists are given by (representatives of) elements in third integral cohomology.

Since every almost complex structure induces a $\mathrm{Spin}^c$ structure and since the classical Conner-Floyd orientation factors through the Hopkins-Hovey orientation, one may wonder whether there is a twisted unitary bordism theory and a twisted Conner-Floyd orientation that extends the result of Baum, Joachim, Khorami and Schick ‘to the left’. In this talk, I answer this question in the negative.

In der Modulraumtheorie gibt es eine tiefe, weitgehend unverstandene Dualität zwischen den Instanton-Zusammenhängen auf Hauptfaserbündeln über einer Mannigfaltigkeit und den kalibrierten Untermannigfaltigkeiten, welche als Modelle für „singuläre“ Instantonen auftreten. Während des Vortrags werde ich mithilfe von Dirac-Operatoren und Spinoren eine entsprechende Dualität (im adiabatischen Limes) der linearisierten Deformationstheorien dieser Modulräume herstellen. Das Hauptergebnis hat Anwendungen auf die Konstruktion von Orientierungsdaten in der Donaldson-Thomas Theorie.