The Graduate Summer School bridges the gap between a general graduate education in mathematics and the specific preparation necessary to do research on problems of current interest. In general, these students will have completed their first year, and in some cases, may already be working on a thesis. While a majority of the participants will be graduate students, some postdoctoral scholars and researchers may also be interested in attending.

The main activity of the Graduate Summer School will be a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures will not duplicate standard courses available elsewhere. Each course will consist of lectures with problem sessions. Course assistants will be available for each lecture series. The participants of the Graduate Summer School meet three times each day for lectures, with one or two problem sessions scheduled each day as well.

Course Titles and Descriptions:

The 2008 Summer Session in Analytic and Algebraic Geometry will consist of eight graduate level lecture series.  On any day during the summer session, three lectures will be offered.  Graduate students are asked to attend the lectures as well as two daily problem sessions associated with the lecture and led by a graduate TA.

Bo Berndtsson, Chalmers University of Technology

L2 Methods for the ∂ Equation

The course will give the basic existence theorems and related L2 Methods for the ∂ Equation.

1. The case of one complex variable. Beginning theory in several variables. Discussion of plurisubharmonic functions and pseudoconvex domains. The Hormander estimates in the case of $(0,1)$-forms in domains in $C^n$ and estimates of Donnelly-Fefferman type.
2. Holomorphic hermitian line bundles on complex manifolds, and $L^2$-theory for $(n,q)$-forms with values in a line bundle, for compact or complete Kahler manifolds.
3. The Hodge theorems. Singular metrics on line bundles and the Kawamata-Viehweg theorem.

John D'Angelo, University of Illinois at Urbana-Champaign

Topics

We will warm up by discussing some issues arising when studying degenerate critical points. We will then discuss real hypersurfaces in complex Euclidean spaces, with particular emphasis on those with degenerate Levi form. We will spend some time on developing properties of the Levi form. We use elementary commutative algebra to describe a notion of point of finite type; in the pseudoconvex case, points of finite type are analogous to the non-degenerate case of strongly pseudoconvex points. In some sense the strongly pseudoconvex points behave like regular points of a holomorphic mapping, and points of finite type behave like branch points. We then explain how points of finite type arise in the study of regularity for the Cauchy-Riemann equations. The material will be almost completely self-contained.

Jean-Pierre Demailly, Université de Grenoble

Analytic Approach of the Minimal Model Program and of the Abundance Conjectures

1. Applications of the Ohsawa-Takegoshi theorem in algebraic geometry, e.g.
   - analytic Zariski decompostion
   - equisingular approximation and generalized Hard Lefschetz theorem
   - proof of the conjecture on the invariance of plurigenera
2. Analytic approach of the minimal model program
3. Canonical metrics and abundance conjectures

Christopher Hacon, University of Utah

Higher Dimensional Minimal Model Program

In this series of talks we will discuss recent progress in the Minimal Model Program which in particular implies the existence of minimal models for varieties of general type and arbitrary dimension.

János Kollár, Princeton University

Introduction to Minimal Models and Flips

The series of talks will give an introduction to the ideas and techniques of the Minimal Model Program, also called Mori’s program. The main text we use is "Birational geometry of algebraic varieties" by J. Kollár and S. Mori.

Robert Lazarsfeld, University of Michigan

An Introduction to Multiplier Ideals

I will present the basic theory of multiplier ideals and their applications. My hope is to get as far as Siu’s theorem on the deformation-invariance of plurigenera for varieties of general type. Most of the material will come from Chapters 9 and 11 of my book on Positivity in Algebraic Geometry.

Mircea Mustaţă, University of Michigan

Resolution of Singularitie

The plan is to explain Hironaka’s resolution of singularities. Recent simplifications of the proof, due to Włodarczyk, Kollár, and several other people, have now made the resolution algorithm accessible to the working algebraic geometer. Moreover, several ideas and constructions in this algorithm present an independent interest, and will hopefully find other applications as well.

The only prerequisite for these lectures is a basic knowledge of algebraic geometry, at the level of Hartshorne's "Algebraic geometry".

Dror Varolin, SUNY at Stony Brook

L2 Methods in Complex Geometry

The goal of this course is to equip the student with tools, adapted from several complex variables, to handle problems in complex geometry,especially problems related to recent developments in algebraic geometry. We will begin with the adaptation of Hormander's Theorem to the setting of holomorphic line bundles on Stein and projective manifolds. We will then discuss the Ohsawa-Takegoshi Extension Theorem and Skoda's Theorem, which are two powerful results one obtains from the L^2 technique. We will conclude with a recent semi-positivity theorem of Berndtsson and Paun, and if time permits we will discuss applications of the results proved in the course.

Participants in the Graduate Summer School also may wish to become involved in the Undergraduate Summer School, attend parts of the Research Program, or participate in the programs of the Education component. Graduate students are expected to participate in Institute-wide activities such as the "Cross Program Activities" and may be asked to contribute some time to volunteer projects related to running the Summer Session.

A limited number of graduate students who have not completed the basic courses may attend. These students will attend some graduate level courses and may be involved as teaching assistants in other programs or work as audio-visual assistants.

The Graduate Summer School is supported by National Science Foundation grant no. 0437137.