RTG Lecture Abstracts
Wilderich Tuschmann: Moduli spaces of Riemannian metrics.
Consider a smooth manifold with a Riemannian metric satisfying some sort of curvature or other geometric constraint like, for example, positive scalar curvature, non-negative Ricci or negative sectional curvature, being Einstein, Kähler, Sasaki, etc. A natural question to ponder is then what the space of all such metrics does look like. Moreover, one can also pose this question for corresponding moduli spaces of metrics, i.e., quotients of the former by (suitable subgroups of) the diffeomorphism group of the manifold, acting by pulling back metrics.
These spaces are customarily equipped with the topology of smooth convergence on compact subsets and the quotient topology, respectively, and their topological properties then provide the right means to measure 'how many' different metrics and geometries the given manifold actually does exhibit; but one can topologize and view those also in very different manners.
The study of spaces of metrics and their moduli has been a topic of interest for differential geometers, global and geometric analysts and topologists alike, and in the course I will introduce to and survey main results and open questions in the field with a focus on non-negative Ricci or sectional curvature as well as other lower curvature bounds on closed and open manifolds, and, in particular, also discuss broader non-traditional approaches from metric geometry and analysis to these objects and topics.
Ana Chavez Caliz: A glimpse to Projective Geometry
In his book "Projective Geometry", Harold Coxeter introduces projective geometry by stating:
"Is it possible to develop a geometry having no circles, no distances, no angles, no intermediacy (or 'betweenness'), and no parallelism? Surprisingly, the answer is Yes; what remains is projective geometry: a beautiful and intricate system of propositions, simpler than Euclid's, but not too simple to be interesting."
And, in words of Arthur Cayley,
"Metrical geometry is thus part of descriptive geometry, and descriptive geometry is all geometry."
Thus, working with geometrical objects in the frame of Projective geometry allows us to see beyond angles and distances and explore the most fundamental geometric structures and incidence relations.
We plan to go over some basic definitions on projective geometry, and show how affine geometry fits in this geometrical model. We will talk about Configuration theorems, and some geometric spaces (like spaces of lines, polygons, to mention a few) that have a natural projective structure.