Daniel Ballesteros, SIlesian University of Technology
On the prescribed curvature problem for starshaped hypersurfaces.
The problem of finding hypersurfaces with curvature given by a function defined in the Riemannian or Lorentzian ambient space, has attracted the attention of geometers for many years.
With the help of techniques like apriori estimates for fully non-linear elliptic partial differential equations, many of these problems have been solved.
In this talk we will discuss about the PDEs arising from these geometric questions, we will give an overview of some known results for the existence of starshaped hypersurfaces in Riemannian manifolds, and we will present our result for the existence of such hypersurfaces in de Sitter space.
Samuel Borza, Durham University
Warner's regularity conditions in sub-Riemannian geometry
In 1965, Warner described certain conditions of regularity and continuity that the Riemannian exponential map satisfies and that imply its non injectivity in a neighbourhood of a conjugate vector. We show how these conditions can be adapted to a sub-Riemannian context via the study of Jacobi fields along normal extremals. Finally, we apply these results to the Heisenberg group and the Grushin plane.
Diego Corro, UNAM
Singular Riemannian foliations and topology rigidity
In this talk we will cover the local theory of Singular Riemannian foliations with closed leaves, giving a list of invariants which determine locally the foliation. Then we will focus on how to relate this local invariants to global behavior of the foliation to deduce topological rigidity results: that is, we can state when two given manifolds, each with a singular Riemannian foliation, are diffeomorphic to each other.
In particular for low dimensions we can give an explicit classification of the diffeomorphism type of the manifold.
If time permits we will see that even for low dimensional foliations the local information can give us some global information about the topology of the manifold, such as the simplicial volume.
Karla García, UNAM
Spaces and moduli spaces of flat Riemannian metrics on closed manifolds.
We introduce a result by Wolf that allows us to compute the moduli space of flat metrics of a manifold, using the relation between flat metrics and Bieberbach groups. Then we will study the moduli space of flat metrics on low dimensional closed manifolds.
Katie Gittins, Durham University
Comparing Hodge spectra of manifolds and orbifolds
In the field of spectral geometry we explore the interplay between the eigenvalues of a differential operator and the geometry of the underlying object. A central question is: what can the eigenvalues tell us about the geometry?
In these talks, we first present some known results for the Laplacian acting on functions defined on smooth Riemannian manifolds and Riemannian manifolds with singularities (called orbifolds). We then consider the question: can closed orbifolds with singularities be spectrally distinguished from smooth manifolds?
We focus on the Hodge Laplacian acting on differential forms on closed Riemannian orbifolds. We apply the heat invariants for differential forms to obtain several positive results in this direction. For example, we obtain that the spectra of the Laplacian for functions and 1-forms together can detect the presence of singularities for orbifolds of dimension at most 3. Time-permitting, we may also discuss some negative results by presenting counterexamples. This is based on joint work with Carolyn Gordon, Magda Khalile, Ingrid Membrillo Solis, Mary Sandoval and Elizabeth Stanhope.
Phil Kamtue, Durham University
Introduction to Bakry-Emery and Ollivier Ricci curvature on graphs and networks.
In this talk, I will give an introduction to two different approaches to define the lower Ricci curvature bound on discrete spaces (e.g., graphs and networks). The first notion, known as Bakry-Emery curvature, is an analytic approach following the authors' celebrated work (1984) based on Bochner's formula in Riemannian geometry, and it was first studied on graphs by Schmuckenschlager (1996). The second notion, Ollivier Ricci curvature, is introduced by Ollivier (2009) based on the distance between two small balls (in the sense of Optimal transport).
To illustrate the behaviour of these two curvatures, I will show some examples of graphs where curvatures are explicitly calculated, with the help of Curvature Calculator invented by our colleagues, Cushing and Stagg. Lastly, I will discuss some relevant results discovered by our group and further applications of these curvatures.
Wilhelm Klingenberg, Durham University
Introduction to Sub-Riemannian Geometry
Sub-Riemannian manifolds form an interesting class of singular spaces. In this talk, we will review its fundamentals while emphasising its link with optimal control theory. In particular, the study of length minimizers reveals the existence of geodesics which can be normal, i.e. solving Hamilton's equation, or abnormal. Some problems, such as Sard theorem for the endpoint map, are still open. We will introduce the topic via Dido's problem.
Norbert Peyerimhoff, Durham University
Talk 1: More about Bakry-Emery curvature on graphs
Following the survey of Supanat Kamtue, I will present a proof of the Theorem of Bonnet-Myers, namely, that a connected graph satisfying the curvature-dimension condition $infinCD(K,\infty)$ of bounded vertex degree is finite and has a specific
upper diameter bound which is assumed for the hypercube. This is joint work with Shiping Liu and Florentin Muench. I will also discuss more recent results where we reformulate the computation of Bakry-Emery curvature as a problem of finding the smallest eigenvalue of a specific matrix called the curvature matrix. This reformulation has various interesting consequences, which I may also sketch if time permits. This second part of my talk is joint work with David Cushing, Supanat Kamtue and Shiping Liu.
Talk 2: More about Ollivier Ricci curvature on graphs
Following the survey of Supanat Kamtue, I will present a proof of Lichnerowicz Theorem, namely, that for a finite connected graph with positive Ollivier Ricci curvature the smallest positive Laplace eigenvalue is bounded below by this curvature. This classical result goes back to Ollivier himself and can also be found in a paper by Lin/Lu/Yau from 2011. I will also introduce the notion of Bonnet-Myers sharpness for Ollivier Ricci curvature and show a kind of suspension property for Bonnet-Myers sharp graphs, namely the fact that, for any pair of antipodal vertices x and y, all other vertices are contained in the geodesics between x and y. This latter result is part of a classification result of Bonnet-Myers sharp graphs obtained in collaboration with David Cushing, Supanat Kamtue, Jack Koolen, Shiping Liu and Florentin Muench.