Rationally connected varieties and a theorem by Graber, Harris and Starr

Carolina Araujo

In this talk I shall discuss Rationally Connected Varieties.

In the first part of the talk I will define Rationally Connected Varieties and present some of their nice properties. I will also explain how and why these varieties should be viewed as higher dimensional analogs of rational curves and rational surfaces.

In the second part of the talk I will focus on a recent result by T. Graber, J. Harris and J. Starr. I will explain how their result fits in the picture described in the first part of the talk, and also discuss some ideas in the proof.

Why Hochschild? An algebraic geometer's point of view

Andrei Caldararu

In my talk I will survey some of the most important aspects of Hochschild homology and cohomology, from an algebraic-geometric point of view. The emphasis will be on how this is the correct (co)homology theory when one deals with derived categories, and in particular in the context of Kontsevich's homological mirror symmetry. I shall also try to discuss how Hochschild homology relates to Chen-Ruan orbifold cohomology, and connections with various Riemann-Roch-type formulas.
The arithmetic and the geometry of Kobayashi hyperbolicity

Izzet Coskun

The geometry and number theory of curves of genus greater than two differ fundamentally from those of genus zero and one. Hyperbolicity is one way of generalizing the complex analytic properties of curves of genus at least two to higher dimensional complex manifolds. Hyperbolic manifolds and curves of high genus are conjectured to exhibit similar number theoretic properties.

In this talk I will give various examples of hyperbolic manifolds due to Green, Siu and Yeung. I will discuss some geometric consequences of hyperbolicity that are conjectured to be equivalent to it. I will state the Lang conjectures about the number theoretic consequences of hyperbolicity and discuss the work of Caporaso, Harris and Mazur showing surprising implications of the conjectures for rational points on curves.

Multiplier ideals in algebraic geometry

Sam Grushevsky

In this talk we will define multiplier ideals both analytically and algebraically, state (and prove some of) their basic properties, following Demailly, Ein, Kohn, Lazarsfeld, Nadel, Siu, ....

As an example of application of multiplier ideal techniques, we will give a simple proof of Kollár's theorem that the theta divisor of a principally polarized abelian variety has no points of multiplicity higher than g. We will also state the theorem of deformational invariance of plurigenera and highlight some multiplier-ideal techniques used in Siu's proof.

Real algebraic curves and amoebas (following Mikhalkin)

David Lehavi

The main purpose of this talk is to define amoebas, and see some of their nice properties. The motivating problem is the classification of real plane algebraic curves that satisfy a special ``maximal'' intersection condition with 3 lines. If we fix 3 lines in the complex plane we (almost) set a torus action on the plane. The quotient of a 2-dimensional complex toric variety by the torus action is a 2-dimensional real polytope. An amoeba is the image of a curve under this torus action. We will analyze the relation between amoebas and the real structure, eventually proving the uniqueness of the topological type of ``maximal arrangements'' of real plane algebraic curves with respect to 3 lines. The lecture follows Mikhalkin AG/0010018.
Quotients by groupoids (following Keel-Mori)

Max Lieblich

A basic problem in algebraic geometry is the construction of moduli spaces -- spaces parametrizing a fixed type of variety (e.g., curves of genus g) or some type of object on a fixed variety (e.g., vector bundles on a fixed surface). It was realized early on that the construction of suitable parameter spaces is related to the construction of group quotients. This connection was not properly understood until the advent of stacks in groupoids (or ``groupoid actions''). We will describe a general result of Keel and Mori (proving a well-known ``folk theorem'') which yields the existence of quotients in many situations, ranging from quotients for proper actions by linear algebraic groups to coarse moduli spaces for separated Deligne-Mumford stacks. Even if one is interested only in group quotients, the method of Keel and Mori reveals that considering the more general notion of groupoid action significantly clarifies and simplifies the situation. (Those afraid of the word ``stack'' should rest assured that it will not appear during this lecture, in spite of its appearance in the abstract.)
Two Degeneration Techniques for Maps of Curves

Brian Osserman

In this talk, we discuss the theories of admissible covers and limit linear series, two techniques for using degenerations to study maps between curves. While the theories are superficially very similar, they have certain fundamental differences which we will explore.
Rigid analytic geometry and abelian varieties

Mihran Papikian

The purpose of the lecture will be the introduction of the basic notions of rigid analytic geometry, with the aim of discussing the non-archimedean uniformization of abelian varieties with purely toric reduction. The theory of rigid analytic spaces is a fundamental tool in modern number theory. For example, it plays an indispensable role in the theory of Drinfeld modular varieties, and provides the inspiration for the work of Mumford on degenerations.
Geometric invariant theory and projective toric varieties

Nick Proudfoot

Geometric invariant theory provides a method of defining quotients of algebraic varieties by group actions. When the variety in question is affine space, and the group is a torus, the GIT quotient is called a toric variety. I plan to use the GIT apporach to illustrate the relationship between toric varieties and polytopes, with many concrete examples.
Equivariant cohomology following Goresky, Kottwitz, and MacPherson

Julianna Tymoczko

Just as cohomology describes a variety based on the interplay of certain subvarieties, equivariant cohomology describes a variety carrying a group action based on the interplay of subvarieties preserved by the action. Goresky, Kottwitz, and MacPherson's paper ``Equivariant cohomology, Koszul duality, and the localization theorem'' presents, among many other things, a way to compute equivariant cohomology from certain combinatorial invariants. Starting with a brief discussion of equivariant cohomology, we will discuss these invariants and how to compute with them. This construction works only for certain varieties, called equivariantly formal and including many large families of familiar varieties. The talk will include many examples.
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