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TU Berlin

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Leitung

Prof. Dr. Peter Bürgisser

Lupe

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

Büro
Raum MA 317 (3. OG)
Institut für Mathematik

Kontakt

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

eMail
peter.buergisser@offmath.tu-berlin.de

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

Sprechstunde
Während der Vorlesungszeit: Mittwochs 1200-1300.
Während der vorlesungsfreien Zeit: Nach Vereinbarung.

Publikationen

Probabilistic Schubert Calculus
Zitatschlüssel BL-Probabilistic-Schubert-Calculus
Autor Peter Bürgisser and Antonio Lerario
Jahr 2016
Monat 12
Zusammenfassung 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.
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