ICTP Mathematics Seminar on Friday 7 June - L. Goettsche

[math]

M A T H E M A T I C S   S E M I N A R S   2013


Friday, 7 June, at 14:00

Lothar Goettsche (ICTP)

Refined curve counting (V)


Abstract:
The subject of these lectures is enumerative geometry of curves on 
algebraic surfaces and its relation to Hilbert schemes of points, real 
algebraic geometry and tropical geometry.
The generating function of the Euler numbers of Hilbert schemes of 
points on curves has been used by Pandharipande and Thomas to give a 
mathematical definition of Gopakumar-Vafa invariants, and by 
Kool-Shende-Thomas to give a proof of a conjecture of mine which 
describes the generating function of the numbers of δ-nodal singular 
curves in a linear system of dimension δ on any surface. Shende and me 
gave a conjectural refinement of this conjecture. Here the number of 
curves is replaced by a polynomial, whose meaning is still mysterious, 
but it can be seen to be related to tropical geometry and real algebraic 
geometry. In this series of lectures we want to explain this circle of 
ideas, and also use the opportunity to introduce a number of important 
concepts and techniques, generating functions, cobordism ring, 
localization, Severi degrees, Welschinger invariants, tropical geometry, 
Heisenberg algebra.


VENUE: Luigi Stasi Seminar Room, first floor , ICTP Leonardo Building