This is a student-run seminar on quantum cohomology. The ultimate goal is to understand this exciting area of mathematics better. We'll begin by constructing the Kontsevich moduli space of stable maps and by proving Kontsevich's formula for rational plane curves. We hope to devote the rest of the semester to applications and relations to enumerative geometry, symplectic geometry, mirror symmetry and the like.
Upcoming talks:
Speaker: Howard Nuer
Date: Tuesday, April 26
Title: Tying it all together with quantum cohomology
Abstract: Quantum cohomology! It miraculously encodes the data of Gromov-Witten invariants into the structure of a ring. We'll close out the semester by seeing how this works and more.
Previous talks:
Speaker: Howard Nuer
Date: Tuesday, April 19
Title: Reconstruction theorem for projective space
Abstract: We'll take a look at how genus-0 Gromov-Witten invariants can be computed recursively by GW-invariants either of lower degree or with fewer marked points.
Speaker: Howard Nuer
Date: Tuesday, April 12
Title: Gromov-Witten theory and Quantum cohomology
Abstract: With Kontsevich's formula and space of stable maps at our disposal, we begin our study of Gromov-Witten theory and quantum cohomology.
Speaker: Howard Nuer
Date: Tuesday, April 5
Title: Enumerative geometry and Kontsevich's formula
Abstract: How many degree-d rational plane curves pass through 3d-1 given points in general position? Kontsevich's formula answers this longstanding question using stable maps and quantum cohomology. With Kontsevich's space of stable maps at our disposal, we'll begin to investigate Gromov-Witten invariants and their relations. Along the way we'll get to see many geometric results afforded to us by this machinery, like the result on plane curves above.
Speaker: Sjuvon Chung
Date: Tuesday, March 29
Title: Getting the Gist of things
Abstract: Today we explore Kontsevich's space of stable maps: its boundary, which is similar to the case of stable curves, and collections of natural morphisms with which the space comes equipped. We'll tie everything together with as many examples as possible. The goal will be to set the stage for tackling problems in enumerative geometry via stable maps.
Speaker: Sjuvon Chung
Date: Tuesday, March 22
Title: Beginning the construction of Kontsevich's space
Abstract: At last we arrive at Kontsevich's space of stable maps. In Howie's last talk we began to see how all our work culminates to the construction of this moduli space. We'll continue this, specifically focusing on the case of genus-zero maps into projective space.
Speaker: Howard Nuer
Date: Tuesday, March 8
Title: Continuing towards Kontsevich's space
Abstract: More on stable maps and Kontsevich stability before we tackle Kontsevich's moduli space.
Speaker: Howard Nuer
Date: Tuesday, March 1
Title: First steps towards Kontsevich's moduli space
Abstract: Today we'll take our first steps towards constructing Kontsevich's moduli space. We will start with discussions on stable maps.
Speaker: Sjuvon Chung
Date: Tuesday, February 23
Title: Stable maps
Abstract: We'll wrap up our discussion of the boundary of the moduli of stable curves before we move on to the next topic: stable maps.
Speaker: Howard Nuer
Date: Tuesday, February 16
Title: More on moduli of stable curves
Abstract: We'll pick up where we left off on stable curves and their moduli.
Speaker: Sjuvon Chung
Date: Tuesday, February 9
Title: Moduli of stable curves and their boundaries
Abstract: We continue our discussion of stable curves with the aim of describing their moduli space boundaries.
Speaker: Sjuvon Chung
Date: Tuesday, February 2
Title: Stable pointed curves
Abstract: We begin our discussion of stable pointed curves and their moduli spaces.
Speaker: Howard Nuer
Date: Tuesday, January 26
Title: Introduction to moduli and stable curves
Abstract: We'll begin with establishing a precise notion of what we want and need in a moduli space for a given moduli functor, and then we'll go on to discuss stable pointed curves. We hope to cover the construction of a (fine) moduli space of genus-0 stable pointed curves, and prove a relation that is essential in the proof of the WDVV equation of Gromov-Witten invariants.