abstracts singularities-2024

The classical umkehr map of Hopf assigns to a map of oriented manifolds, f:M \to N, `wrong-way’ homomorphisms in homology f_!: H_*(N) \to H_*(M) and in cohomology f^!:H^*(M) \to H^*(N), the latter a version of `integration over the fibers’. Similar wrong-way maps, sometimes known as transfer maps or Gysin maps, are defined for other generalized (co)homology theories as long as the manifolds are suitably oriented and have had many applications. While these maps are defined only for manifolds there has long been interest in extending them to singular spaces. I’ll discuss joint work with Markus Banagl and Paolo Piazza in which we capitalize on recent work on the index theory of signature operators to give analytic definitions of transfer maps in K-homology for stratified spaces and relate them to topological orientations.

Geometric (pre)quantization can be performed only for integral symplectic manifolds. In 1989 Alan Weinstein conjectured that using the notion of Diffeology, as well as Noncommutative Geometry methods, one might obtain the representations required to “quantise the unquantisable” from a torus bundle rather than a line bundle. In joint work with P. Antonini, we showed that the obstruction to integrality can be lifted by adding extra dimensions and passing to the diffeological category. In fact, the added dimensions force the C*-algebra associated with this construction to be nothing else than the crossed product algebra associated with a torus action.

In this talk I’ll discuss recent and ongoing work concerning two geometric flows. The first result concerns “convergence stability” for normalized Ricci flow near rotationally symmetric asymptotically hyperbolic metrics. The second result concerns a stability result for a higher-order geometric flow modeled on the Bach tensor of conformal geometry. Our analytic methods exploit semigroup techniques, and I’ll discuss a criterion for a geometric elliptic operator to generate an analytic semigroup on weighted Hölder spaces on manifolds of bounded geometry. This talk is based on joint work with Guenther, Isenberg and Mazzeo.

This is a joint work with Ilaria Mondello and David Tewodrose. In 1985, Bakry and Emery have introduced a curvature condition for Weighted Riemannian manifold generalizing the lower bound on the Ricci curvature. This Bakry-Emery curvature have been extensively studied and have lead to the notion of synthetic lower bound on Ricci curvature for metric measure space by J.Lott-C.Villani et K-T. Sturm.
The Kato condition for the Ricci curvature is a weakening of the lower Ricci bound condition. With Ilaria Mondello and David Tewodrose, we have investigated limit of Riemannian manifold with such a Kato condition. Recently using a change times in Brownian motion, we have highlighted a relationship between the Kato condition and the Bakry-Emery condition. I will explain these conditions and this relationship and its potential usefulness.

We consider nonlinear elliptic equations with a gradient-dependent nonlinearity and a Hardy-type potential in a bounded domain punctured at zero. We present new results on the existence and classification of the positive solutions near zero. In the case of a ball, using a dynamical systems approach, we prove the existence of positive radial solutions, manifesting near zero each the profiles we ascertain from our classification results. This proves not only the optimality of our classification results in a variety of ranges for our parameters but also gives further insight into the refined asymptotic behaviour of the solutions near zero. By a modified Kelvin transform, our results also render the