Sommersemester 2019: Introduction to Quantum Cohomology

Prof. Dr. Katrin Wendland
Dr. Mara Ungureanu

Wann und wo: Di 14 - 16, SR 125, Ernst-Zermelo-Str. 1

Vorbesprechung: Montag, 04.02.2019, 12:15, SR318, Ernst-Zermelo-Str. 1
Um teilzunehmen, kommen Sie bitte in die Vorbesprechung des Seminares; eine Teilnehmerliste wird nicht vorab ausliegen.

One of the oldest avenues of research in algebraic geometry is enumerative geometry, whose aim is to compute the number of objects satisfying certain geometric conditions. One beautiful example of an enumerative problem is that of determining the number Nd of rational curves of degree d passing through 3d-1 points in the projective plane P2. The numbers N1=N2=1 were known already from antiquity, while N3=12 was computed in 1848 by Steiner, albeit with methods that lacked a rigorous foundation. Despite the advances in intersection and deformation theory in the 20th century, which resulted in many classical enumerative problems being solved, the determination of the numbers Nd proved to be more difficult than expected. The turning point came in the 90s, when a connection between enumerative geometry and string theory was discovered. The breakthrough was the realisation that the counts of various enumerative problems can be organised in terms of certain physical quantities (correlation functions of some topological quantum field theories) and computed using the product rules of a deformation of the de Rham cohomology, namely quantum cohomology.
The purpose of the seminar is to understand the derivation via quantum cohomology of the Kontsevich formula that yields the numbers Nd for an arbitrary d. In doing so, we shall introduce the concept of moduli spaces in algebraic geometry and discuss some basics of deformation theory (which will explain why 3d-1 is the appropriate number of points to consider). We shall then reformulate the problem in terms of moduli spaces of stable maps to P2, define Gromov-Witten invariants, and set up the necessary axiomatics of topological quantum field theories and quantum cohomology.

Die Links führen auf Webseiten, von denen aus dem Universitätsnetz die jeweiligen Referenzen zugänglich sind. Falls kein Link gesetzt ist, finden Sie die Referenz in der Bibliothek des Mathematischen Institutes Freiburg.

Das Vortragsprogramm finden Sie hier.
Die Vorträge können auf Deutsch oder auf Englisch präsentiert werden.

Tutorium: Dr. Mara Ungureanu