Programmdiskussion
Montag, 18.4.16, 16:15-17:15, Raum 404, Eckerstr. 1
Γ-structures and symmetric spaces
Montag, 25.4.16, 16:15-17:15, Raum 404, Eckerstr. 1
Γ-structures are weak forms of multiplications on closed oriented manifolds.\nAs shown by Hopf the rational cohomology algebras of manifolds admitting\nΓ-structures are free over odd degree generators. We prove that this condition\nis also sufficient for the existence of Γ-structures on manifolds which are\nnilpotent in the sense of homotopy theory. This includes homogeneous spaces\nwith connected isotropy groups.\n\nPassing to a more geometric perspective we show that on compact oriented\nRiemannian symmetric spaces with connected isotropy groups and free rational\ncohomology algebras the canonical products given by geodesic symmetries define\nΓ-structures. This extends work of Albers, Frauenfelder and Solomon on\nΓ-structures on Lagrangian Grassmannians.\n
t.b.a.
Montag, 2.5.16, 16:15-17:15, Raum 404, Eckerstr. 1
Adiabatic Limits of Eta Invariants
Montag, 9.5.16, 16:15-17:15, Raum 404, Eckerstr. 1
We will introduce eta invariants, which are spectral invariants of Dirac operators, and the notion of adiabatic limits. Then we present some known results by Bismut, Cheeger and Dai before we give a partial answer in a more general setting.
Surgery stability of the space of metrics with invertible Dirac operator
Montag, 23.5.16, 16:15-17:15, Raum 404, Eckerstr. 1
Surgery stability of the space of metrics with invertible Dirac operator
Montag, 23.5.16, 16:15-17:15, Raum 404, Eckerstr. 1
Ricci curvature of non-symmetric diffusion operators
Montag, 13.6.16, 16:15-17:15, Raum 414, Eckerstr. 1
Integral curvature and area of domains in surfaces
Montag, 20.6.16, 16:15-17:15, Raum 404, Eckerstr. 1
Minimal Geodesics on a K3
Montag, 27.6.16, 16:15-17:15, Raum 404, Eckerstr. 1
I will give an introduction to my PhD thesis topic. My emphasis will be on explaining the problem by stating known resutls of interest to a broader audience of differential geometers. Most of the talk should be understandable without a detailed knowledge of K3 surfaces.
Aufblasung affiner Varietäten
Montag, 4.7.16, 16:15-17:15, Raum 404, Eckerstr. 1
Higher homotopies
Montag, 11.7.16, 16:15-17:15, Raum 404, Eckerstr. 1
Starting with the relation of infinite loop spaces and generalized cohomology theories, we use some simple examples to illustrate some special homotopy invariant properties of infinite loop spaces. Then we go on and introduce various delooping machines. In the end of the talk, a description of infinite loop space by Gamma-space will be given.
TBA
Montag, 18.7.16, 16:15-17:15, Raum 404, Eckerstr. 1
A Local Index Formula for the Intersection Euler Characteristic of an Infinite Cone
Montag, 18.7.16, 16:15-17:15, Raum 404, Eckerstr. 1
The study of global analysis of spaces with (isolated) cone-like singularities has started with work of Cheeger in the 80s and has seen a rich development since. One important result is the generalisation of the Chern-Gauss-Bonnet theorem for these spaces, which is due to Cheeger. It establishes a relation between the \(L^2\)-Euler characteristic of the space, the integral over the Euler form and a local contribution \(\bgamma\) of the singularities. The ``Cheeger invariant'' \(\bgamma\) is a spectral invariant of the link manifold. \n\nThe aim of this talk is to establish a local index formula for the intersection Euler characteristic of a cone. This is done by studying local index techniques as well as the spectral properties of the model Witten Laplacian on the infinite cone. As a result we express the absolute and relative intersection Euler characteristic of the cone as a sum of two terms, one of which is Cheeger's invariant \(\bgamma\).
Smoothing theory
Montag, 25.7.16, 16:15-17:15, Raum 404, Eckerstr. 1
Since Milnor discovered exotic 7-spheres in 1956, it is known that a\ntopological manifold can have several non-diffeomorphic smooth\nstructures. The aim of smoothing theory is to calculate the set S(X) of\nsmoothings of a given topological manifold X in terms of a homotopy\ntheoretical property:\nS(X) turns out to be in bijection with the set of lifts of a certain\nclassifying map.\n\nIn my talk I will introduce all necessary concepts such as mircobundles\nand piecewise linear manifolds and try to illustrate their properties.\nThen the fundamental theorem will be stated and important parts of the\nproof will be sketched. In the end I hope to give some practical\nadvices on how to calculate structure sets.\n
Local Morse homology with finite-cyclic symmetry
Montag, 22.8.16, 16:15-17:15, Raum 404, Eckerstr. 1
Morse theory is concerned with the relationships between the\nstructure of the critical set of a function and the topology of the\nambient space where the function is defined. The applications of Morse\ntheory are ubiquitous in mathematics, since objects of interest\n(geodesics, minimal surfaces etc) are often critical points of a\nfunctional (length, area etc). In this talk I will review basic\nconcepts in Morse theory, and will focus on Hamiltonian dynamics where\nthe applications emerge from the fact that periodic solutions of\nHamilton's equations are critical points of the action functional. I\nwill explain how to define a local Morse homology of the action\nfunctional at an isolated periodic orbit which takes into account the\nsymmetries associated to time-reparametrization, and serves a\nwell-defined alternative to local contact homology. Then I will\nexplain dynamical applications. This is joint work with Doris Hein\n(Freiburg) and Leonardo Macarini (Rio de Janeiro).
Higson-Roe exact sequence and secondary \(\bell^2\)-invariants
Mittwoch, 28.9.16, 13:45-14:45, Raum 125, Eckerstr. 1
In this talk we give an overview of the Higson-Roe exact sequence for\ndiscrete groups, also known as the analytic surgery sequence, and explain its\nrelation with secondary invariants of type rho. Using the machinery of\nequivariant Roe-algebras we shall also outline a proof of some rigidity results\nof \(\bell^2\)-rho-invariants, generalizing earlier work of Higson and Roe. This\nis joint work with M.-T. Benameur.\n