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.
Topic:
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.
Literatur:
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.
Vortragsprogamm:
Das Vortragsprogramm finden Sie
hier.
Die Vorträge können auf Deutsch oder auf Englisch präsentiert werden.
Tutorium: Dr. Mara Ungureanu
