This is the webpage of the seminar of the Research Cluster in Geometry at the Mathematics Department of the University of Luxembourg.
Unless otherwise specified, the seminar will be on Monday, 2-3pm and will take place in Room S-6A (MNO).
Here is a link to the seminars of former years.
Organizers: Clara Aldana, Clément Guérin, Vincent Pecastaing.
The simplicial volume is a homotopy invariant of manifolds
defined by Gromov in 1982. Roughly speaking, the simplicial volume of M
measures the minimal size of real fundamental cycles for M. It is
related to many topological and geometric invariants (most notably, to
the Riemannian volume), and it is usually surprisingly difficult to
compute.
We define a variation of the simplicial volume which still enjoys many
features of its classical counterpart, while being easier to compute.
Our new invariant is defined for manifolds with boundary, and differs
from the ordinary simplicial volume because ideal simplices are now
allowed to appear in fundamental cycles. We discuss some applications of
the ideal simplicial volume to the study of maps between hyperbolic
3-manifolds with geodesic boundary.
Thurston asked whether a mapping torus admits a fibration of closed minimal surfaces and he speculated that the answer is no for mapping tori with short curves. We confirm this with a quantified theorem for C^2 minimal fibrations. This is a computer assisted proof: we prove necessary conditions, then use programs SNAPPY and TWISTER to find examples of mapping tori which satisfy these necessary conditions. This is joint work with B. Wang.
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We will explain what we think is a good definition of a constant scalar curvature Kähler metric with conic singularities. A construction of such special metrics over certain ruled manifolds will be given. To do so, we present a Fredholm alternative result for the Lichnerowicz operator. This also requires to study Hermitian-Einstein metrics on parabolic vector bundles with good regularity properties. Eventually, we will explain how these results fit in the logarithmic version of Yau-Tian-Donaldson conjecture that relates differential geometry to algebraic geometry. This is a joint work with Kai Zheng (Univ. of Warwick).
A famous inverse problem posed by Kac in the 1960s is to determine the shape of a drum from the set of resonant frequencies at which it vibrates. In this talk, I will discuss recent results in this direction pertaining to elliptical drumheads. In particular, I will outline a proof of “spectral rigidity” using a new parametrix for the wave propagator, which makes an interesting connection to billiards and Birkhoff s conjecture.