This working group will explore moduli spaces of stable maps from pointed rational curves to projective space. We will see examples which illustrate how these modern moduli spaces have had an incredible impact on algebraic geometry, in particular allowing one to obtain recursive solutions to classical enumerative problems. For example, we will see how to use Gromov-Witten theory to find the number of rational plane curves of degree d passing through 3d-1 general points.
For each of Monday through Thursday there
will be a short lecture followed by a group activity. For the activity, participants will be provided with a list
of problems, examples, and questions to work out and think about.
-lecture: Introduction to and motivation for
studying moduli problems and definition of the moduli space of stable n-pointed rational curves to
projective space;
-activity: the special case r=0 -- moduli of stable n-pointed rational
curves.
-lecture: More about stable maps including the boundary,
canonical morphisms and tautological classes;
-activity: examples.
-lecture:
Enumerative geometry using stable maps;
-activity:
Kontsevich's formula.
-lecture:
Gromov Witten invariants;
-activity:
enumerative calculations.
Friday
Participants will give short presentations with titles
along the lines of: What is this paper
about? or What is this
construction about? They
will explain a paper or a construction that they have learned about on their
own or in groups after hours during the week. Papers and ideas will be passed out on Monday so
participants will have lots of options.
Reading
List:
Joachim Kock and Israel Vainsencherck
Kontsevich's formula for
rational plane curves
Fulton and Pandharipande
Notes on stable
maps and quantum cohomology