Weyl formulae for some singular metrics with application to acoustic modes in gas giants
Montag, 11.11.24, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
We prove eigenvalue asymptotics of the Laplace-Beltrami operator for certain singular Riemannian metrics. This is motivated by the study of propagation of soundwaves in gas planets. This is joint work with Yves Colin de Verdière, Maarten de Hoop and Emmanuel Trélat.
Symplectic topology and rectangular peg problem
Montag, 18.11.24, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
The rectangular peg problem, an extension of the square peg problem which has a long history, is easy to outline but challenging to prove through elementary methods. I will report the recent progress on the existence and multiplicity results, utilizing advanced concepts from symplectic topology, e.g. J-holomorphic curves and Floer theory.
Numbers on barcodes and torsion theory
Montag, 25.11.24, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
Morse function on a manifold M is called strong if all its critical points have different critical values. Given a strong Morse function f and a field F we construct a bunch of elements of F, which we call Bruhat numbers (they're defined up to sign). More concretely, Bruhat number is written on each bar in the barcode of f. It turns out that if homology of M over F is that of a sphere, then the product of all the numbers is independent of f. We then construct the barcode and Bruhat numbers with twisted (a.k.a. local) coefficients and prove that mentioned product equals the Reidemeister torsion of M. In particular, it's again independent of f. This way we link Morse theory to the Reidemeister torsion via barcodes. Based on a joint work with Petya Pushkar. \n