# Abstracts

**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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