Chiu-Chu Melissa Liu
Gromov-Witten invariants, Fan-Jarvis-Ruan-Witten invariants, and Mixed-Spin-P fields

Lecture 1: Moduli of stable maps and Gromov-Witten invariants
Abstract: We will introduce stable maps and their moduli, and define Gromov-Witten (GW) invariants
of non-singular complex projective varieties
in terms of the fundamental class (in nice cases)
or the Behrend-Fantechi virtual fundamental class (in the general case).
We will give explicit examples when the target is a projective space or a hypersurface in a projective space.

Lecture 2: Stable maps with fields
Abstract: We will introduce moduli of stable maps to a projective  space with fields,
and describe an alternative definition of GW invariants of Calabi-Yau hypersurfaces
in projective  spaces in terms of the Kiem-Li cosection localized virtual cycle.
This  is an exposition of "Gromov-Witten invariants of stable maps with  fields"
by Huai-Liang Chang and Jun Li.

Lecture 3: Witten's top Chern class and Fan-Jarvis-Ruan-Witten invariants
Abstract: The Fan-Jarvis-Ruan-Witten (FJRW) invariants of a quasi-homogeneous polynomial
are virtual counts of solutions to the Witten's equation associated to the quasi-homogeneous polynomial.
We will describe an algebraic definition of FJRW invariants (in the "narrow" sector) of a Fermat polynomial
x_1^n+ ... + x_n^n in terms of the Kiem-Li cosection localized virtual cycle.
This is a special case of the general construction in "Witten's top Chern class via cosection localization"
by Huai-Liang Chang, Jun Li, and Wei-Ping Li.

Lecture 4: Landau-Ginzburg/Calabi-Yau correspondence
Abstract: We will describe the Landau-Ginzburg/Calabi-Yau correspondence relating the FJRW invariants
the Fermat polynomial W_n = x_1^n+... + x_n^n and GW invariants of the Calabi-Yau hypersurface
{W_n=0} in P^n.

Lecture 5: Mixed-Spin-P Fields
Abstract: We will describe the theory of Mixed-Spin-P (MSP) fields which interpolates
the FJRW theory of the Fermat polynomial W_n and GW theory of the Calabi-Yau hypersurface {W_n=0} in P^n.
We will present some computations of GW invariants and FJRW invariants using localization
on MSP moduli spaces.
This is based on joint work with H.-L. Chang, J. Li, and W.-P. Li.