Dan Abramovich - US grants

This award supports the research of Professor D. Abramovich in the field of algebraic geometry. Abramovich will continue his study of the Gieseker-Morrison compactification of the moduli space of vector bundles of rank 2 on semistable curves, with the goals of identifying the vector bundles corresponding to points on the boundary, extending the construction to higher ranks and comparing with other compactifications. He will also continue his work on Lang's conjecture in characteristic p, and will study integral and torsion points on elliptic curves and integral points on algebraic stacks. Algebraic geometry is one of the oldest parts of modern mathematics. In the past ten years, it has blossomed to the point where it has solved problems that have stood for centuries. Originally, it treated figures in the plane defined by the simplest of equations, namely polynomials. Today, the field utilizes methods not only from algebra, but also from analysis and topology; conversely, it is extensively used in those fields. Moreover, it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics.

Abramovich 9700520 In this project, the principal investigator will study semistable reduction problems, making use of the Alteration method of de Jong, the Torifying Blowup of Abramovich and de Jong, and Toroidal Geometry of Kempf, Knudsen, Mumford and Saint-Donat. He will work on generalizing his results on fibered powers to the logarithmic case, and continue the study of stably integral points on abelian varieties, using the compactified moduli spaces of Alexeev and Nakamura. This project falls into the general area of arithmetic geometry, a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful having recently solved problems that withstood generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers and compact disks.

The area of study of this project lies within algebraic geometry, the branch of mathematics devoted to geometric shapes called algebraic varieties, defined by polynomial equations. Algebraic geometry has significant applications in coding, industrial control, and computation. But the topics of this project are more closely related to applications in theoretical physics, where physicists consider algebraic varieties as a piece of the fine structure of our universe. This is especially true with the first topic, moduli theory. This theory studies a remarkable phenomenon in which the collection of all algebraic varieties of the same type is manifested as an algebraic variety, called a moduli space, in its own right. Thus in algebraic geometry, the metaphor of thinking about a community of "organisms" as itself being an "organism" is not just a metaphor but a rigorous and quite useful fact. The other topic studied in this project is birational geometry, which is devoted to a certain abstract relationship, called birational equivalence, among algebraic varieties, which lies at the foundation of algebraic geometry.

The investigator will continue studying problems in moduli theory; the main foci of the project are Moduli spaces of stable logarithmic maps and Artin fans. Additional topics include the degeneration formula for KKO invariants, the birational geometry of torus quotients, and logarithmic Kodaira dimensions of fibered powers in relation to uniformity of stably integral points.

Algebraic Geometry Summer Institute (Proposal DMS-0456683.)

A large scale summer institute in algebraic geometry, three weeks in
duration, will be held during July-August 2005 at the University of
Washington, Seattle. Organizational support will be provided by the
American Mathematical Society and the Clay Mathematics Institute. Each
week will have a general topic as a focus and at the same time will
include a component of core "classical" algebraic geometry. The foci
will be: Mathematical Physics in Algebraic Geometry; Combinatorics,
Commutative Algebra and Classical Algebraic Geometry; and Arithmetic Algebraic
Geometry. Three plenary morning lecture series will be held in each
week, lecturers having been selected as leading researchers who are at the
same time exemplary expositors. In addition, leading specialists will be
invited to serve as seminar leaders, and will select high impact
lectures given by the best lecturers for afternoon seminars. Additional
activities will include the CMI distinguished lectures and an evening graduate
program.

Algebraic geometry, the study of geometric shapes defined by polynomial
equations, is a vast generalization of the high-school topic "analytic
geometry". It has enjoyed much cross-fertilization in past decades with
areas in pure mathematics, applied mathematics, computing and
theoretical physics. One way American algebraic geometers have maintained the
vibrancy of the field is by holding a large scale, NSF-funded summer institute
about every decade, an institute which aims to bring participants up to
the forefront of research in the salient direction the field has taken
in the previous years, and to project directions which the field is
expected to take in the following years. These institutes have had a crucial
formative role on the subject, with enormous impact especially on the
emerging careers of young researchers. This summer institute project
aims to follow in their footsteps.

Moduli theory is one of the central parts of algebraic geometry, cantered around the phenomenon that in many cases objects of algebraic geometry (for instance algebraic curves) are parametrized by an object of algebraic geometry (such a sthe moduli space of curves). The past 15 years have witnessed an explosion of activity in this area. It interacts extensively with other area: besides the rest of algebraic geometry, it has very important intersections with complex analysis, symplectic geometry, string theory, homotopy theory, geometric topology.

A workshop and conference titled ``Aspects of Moduli Theory'' is organized, to be held June 15-28 at the de Giorgi Center in Pisa, Italy. This activity is meant to offer a wide view of many aspects of the field. It will be split into two parts. The first week will consist of a summer school, in which high level experts will teach five courses on recent and very important advances in the subject. During the first part of each course he teacher will present the necessary background material, making it accessible to graduate students and beginning researchers. During the second week a regular conference will take place.

The program aims to impact a considerable number of advanced graduate students and post-doctoral fellows who will attend this program.

Algebraic geometry has a strong and broad representation at the research institutions of the Northeastern states. AGNES is a series of biannual workshops that intends to further the interaction and collaborations between the algebraic geometers in the area. Each workshop is held over a weekend at one of the participating institutions. The workshops include research talks by renowned experts and junior researchers, both from outside the area and within. Professional development sessions and introductory pre-talks are aimed particularly at graduate students. Every workshop culminates with an open problem session. This gives an opportunity to disseminate recent results and developments, and exchange ideas and views about future directions of algebraic geometry.

Algebraic geometry is the study of spaces defined by polynomial equations. Many of the spaces occurring in nature are of this type, and for this reason algebraic geometry has found diverse applications in the sciences. In particular, there are strong connections with recent work in theoretical physics (string theory). This grant will support a series of algebraic geometry conferences in the Northeastern states. The key aims of the series are to expose graduate students to a broad spectrum of research in the field and to improve communication between the many algebraic geometers in the northeast.

This project will support six semiannual AGNES weekend workshops in Algebraic Geometry, the first of which is to be held at Stony Brook University, April 25 - 27, 2014. Algebraic geometry is a dynamic subject with vital connections to physics, applied mathematics, and fields such as number theory, differential geometry and representation theory. The Northeastern United States are a particular nexus, with a density of researchers and an output rivaling any region in the world. AGNES weekend workshops bring together a spectrum of algebraic geometers from the region and beyond, as well as representatives from allied fields invigorating algebraic geometry. There is diversity both in speakers and participants, from senior experts to burgeoning young researchers. There is a special emphasis on students: AGNES hosts "students-only" introductory pre-talks, as well as professional development sessions. Through lectures, panels, problem sessions, poster sessions, and informal discussion periods, AGNES informs the community of developments, AGNES fosters new research collaborations, and AGNES trains junior algebraic geometers.

Algebraic geometers study the geometry of shapes defined by the polynomial equations of Algebra. In fact, the geometric shapes occurring in nature -- in science, in engineering, etc. -- are frequently "algebraic", explaining the vital importance of algebraic geometry in physics and applied mathematics. This grant promotes research, dissemination and training in algebraic geometry through a series of conferences held throughout the Northeastern United States. There is an emphasis on supporting graduate students, highlighting the broad array of work in the area and its many connections to allied areas, and fostering communication in our broad research community.

AGNES website is available at http://www.agneshome.org/