Guest lectures and seminars - Page 26
A tropical curve is a graph embedded in R^2 satisfying a number of conditions. Mikhalkin's celebrated correspondence theorem establishes a correspondence between algebraic curves on a toric surface and tropical curves. This translates the difficult question of counting the number of algebraic curves through a given number of points to the question of counting tropical curves, i.e. certain graphs, with a given notion of multiplicity through a given number of points which can be solved combinatorially. To get an invariant count, real rational algebraic curves are counted with a sign, the Welschinger sign and there is a real version of the correspondence theorem. Furthermore, Marc Levine defined a generalization of the Welschinger sign that allows to get an invariant count of algebraic curves defined over an arbitrary base field. For this one counts algebraic curves with a certain quadratic form.
In the talk I am presenting work in progress joint with Andrés Jaramillo Puentes in which we provide a version Mikhalkin's correspondence theorem for an arbitrary base field, that is a correspondence between algebraic curves counted with the above mentioned quadratic form and tropical curves counted with a quadratic enrichment of the multiplicity. Then I will explain how to use this quadratic correspondence theorem to do the count of algebraic curves over an arbitrary base field.
C*-algebra seminar talk by Lucas Hataishi (University of Oslo)
Following Givental, enumerative mirror symmetry can be stated as a relation between genus zero Gromov-Witten invariants and period integrals. I will talk about a relative version of mirror symmetry that relates genus zero relative Gromov-Witten invariants of smooth pairs and relative periods. Then I will talk about how to use it to compute the mirror proper Landau-Ginzburg potentials of smooth log Calabi-Yau pairs.