TU Berlin

Fachgebiet Algorithmische AlgebraProf. Dr. Peter Bürgisser

Page Content

to Navigation


Prof. Dr. Peter Bürgisser


Technische Universität Berlin
Institut für Mathematik
Sekretariat MA 3-2
Straße des 17. Juni 136
10623 Berlin

Raum MA 317 (3. OG)
Institut für Mathematik


Beate Nießen
Raum MA 318
Tel.: +49 (0)30 314 - 25771


+49 (0)30 314 - 75902
+49 (0)30 314 - 25839

Während der Vorlesungszeit: Do, 15-16 Uhr.
Während der vorlesungsfreien Zeit: nach Vereinbarung.


Probabilistic Schubert Calculus
Citation key BL-Probabilistic-Schubert-Calculus
Author Peter Bürgisser and Antonio Lerario
Pages 1–58
Year 2020
DOI 10.1515/crelle-2018-0009
Journal Journal für die reine und angewandte Mathematik
Volume 760
Abstract Classical Schubert calculus deals with the intersection of Schubert varieties in general position. We present an attempt at developing such a theory over the reals. By the title we understand the investigation of the expected number of points of intersection of real Schubert varieties in random position. We define a notion of expected degree of real Grassmannians that turns out to be the key quantity governing questions of random incidence geometry. Using integral geometry, we prove a result that decouples a random incidence geometry problem into a volume computation in real projective space and the determination of the expected degree. Over the complex numbers, the same decoupling result is a consequence of the ring structure of the cohomology of the Grassmannian. We prove an asymptotically sharp upper bound on the expected degree of the real Grassmannians G(k,n). Moreover, if both k and n go to infinity, the expected degree turns out to have the same asymptotic growth (in the logarithmic scale) as the square root of the degree of the corresponding complex Grassmannian. This finding is in the spirit of the real average Bezout's theorem due to Shub and Smale. In the case of the Grassmannian of lines, we can provide a finer asymptotic.
Link to publication Link to original publication Download Bibtex entry


Quick Access

Schnellnavigation zur Seite über Nummerneingabe