Jonathan Cox (UIUC), Stacks of Stable Maps and Gromov-Witten Invariants
Abstract: This fundamental paper of K. Behrend and Yu. Manin will be discussed. The paper proves important basic facts about moduli stacks of stable maps and, in special cases, the "virtual fundamental class" and the Gromov-Witten Invariants.
Xinyun Zhu (UIUC), Gorensten threefold singularities with small resolutions via invariant theory for weyl groups
Abstract: This paper by Katz and Morrison will be discussed.
Josh Mullet (UIUC), More on K3 - fibered Calabi-Yau Threefolds
Jonathan Cox (UIUC), Intersections of Q-Divisors on Kontsevich's Moduli Space \bar{M}_0,n(P^r,d) and Enumerative Geometry
Abstract:We will discuss this paper of Rahul Pandharipande. In a previous talk we proved a few facts about these Q-divisors. The definitions and facts will be restated. This time we will actually compute intersections of Q-divisors and demonstrate applications to enumerative geometry. This is an important beginning step in understanding the Chow ring of the moduli space of genus 0 stable maps.
David Murphy (UIUC), Elementary Embeddings of Groups
Abstract: We will discuss elementary G-embeddings (normal G-varieties having exactly two G-orbits, one open and isomorphic to G and the other closed of codimension one). These may be related to one-parameter subgroups of G, and hence to its Tits building. If time permits, I will mention how an arbitrary G-embedding can be covered by elementary ones, which may be the key to classifying G-compactifications.
David Gepner (UIUC), An Outline of A^1-Homotopy Theory
Josh Mullet (UIUC), K3-Fibered Calabi-Yau Threefolds Over the Projective Line
Abstract: We consider Calabi-Yau threefolds which have a fibration over P^1 with general fiber a K3 surface. We will prove some generalities and then give some examples.
Jonathan Cox (UIUC), Q-divisors on Kontsevich's Moduli Space \bar{M}_0,n(P^r,d)
Abstract: We will explore results from Pandharipande's paper "Intersections of Q-divisors on Kontsevich's Moduli Space \bar{M}_0,n(P^r,d) and Enumerative Geometry." Basically, he tells us how to do intersection theory of divisors on the moduli space of stable maps, assuming we restrict to genus zero and take rational coefficients. The results allow many enumerative computations. However, in this talk we will cover only Section 1, which focuses on the vector space structure of the Picard group. Intersecting divisors with curves in the moduli space is a key technique.
David Murphy (UIUC), Buildings and group compactifications
Abstract:After briefly introducing the basic notions of Tits' buildings, we present Mumford's construction of an embedding of a semi-simple group found in his book Toroidal Embeddings I. Some familiarity with toric geometry and algebraic groups will be helpful, but the essential ideas will be defined. If time permits, I will indicate how this construction might be generalized to arbitrary group compactifications.
Xinyun Zhu (UIUC), Versal deformation and superpotentials for rational curves in smooth threefolds
Jonathan Cox (UIUC), Riemann-Roch Theorems and Enumerative Geometry
Abstract: What happens to a bundle when you push it forward? Riemann-Roch theorems answer this question. This information, together with appropriate exact sequences of bundles, gives relations in Chow ring of a space which then allow computation of certain integrals. The integrals give enumerative information about the space. I will give examples for Grassmannians and, at least conjecturally, for moduli spaces of stable curves.
Xinyun Zhu (UIUC), Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds
Josh Mullet (UIUC), Introduction to K3 Surfaces
Abstract: In this leisurely talk we will consider some definitions, examples and basic properties of K3 surfaces.
David Murphy (UIUC), Stability of principal bundles
Abstract: We compare various notions of stability for principal bundles and show that over a compact Riemann surface of genus greater than 2, there exist principal SL(2)-bundles that are Ad-stable.
Jonathan Cox (UIUC), Fulton-Pandharipande's construction of the coarse moduli space of stable maps
Abstract: We will begin with a brief seminar organizational meeting for about five minutes. We then review the construction from the paper "Notes on Stable Maps and Quantum Cohomology," focusing on sections 3 and 4. (Relevant definitions will be recalled.) These spaces are important in mirror symmetry (subject of a course this semester) and string theory (subject of a RAP this semester).