David Eisenbud - US grants

The primary objective of this U.S.-Bulgaria cooperative mathematics project between Dr. David Eisenbud of Brandeis University and Dr. Luchezar Avramov of the University of Sofia is to jointly pursue several problems that lie at the heart of contemporary research in algebra and algebraic geometry. Efforts will address the theory of free resolutions as a means of deriving invariants of geometric structures algebraically. Homological methods will be used to extend contemporary experience in non-commutative algebra and general ring theory. Additionally, the researchers will examine the construction of cohomological invariants in relation to underlaying algebraic variety. Collectively, results of the new work should extend present knowledge in a number of related mathematical fields centering around algebraic geometry and commutative and non- commutative algebra, potentially yielding contributions to important topics like string theory and topology. This mathematics project fulfills the program objective of advancing scientific knowledge by enabling leading experts in the United States and Eastern Europe to combine complementary talents and pool resources in areas of strong mutual interest and competence.

This award supports cooperative research in algebra, geometry and computation between David Eisenbud of Brandeis University and Aron Simis of the Universidade Federal de Pernambuco in Brazil. The subjects of collaboration are: 1) Special Determinantal varieties, 2) Rees Algebras, 3) Computational Methods, and 4) Weierstrass Points. The first two subjects will make heavy use of computations made using a highly efficient computer algebra system oriented toward commutative algebra and algebraic geometry. In addition to the work on these two projects Eisenbud and Simis will do collaborative work on Macaulay development, using some of the techniques that Simis has developed for his projects. In addition, Rio is a center of work on Weierstrass points of curves. The principal investigators on this project will receive support for a graduate student as well as some equipment. These are both strong mathematicians who have each made substantial contributions to commutative algebra and algebraic geometry. The problems they pose to work on are central and important.

This research is concerned with problems in commutative algebra, algebraic geometry and computational issues in those fields. In commutative algebra, the principal investigator will study linear resolutions and Huneke's question on the structure of free resolutions. In algebraic geometry, he will consider a generalized Castelnuovo conjecture, the family of compactified Jacobians by projective stability, degeneracy of secant and tangent varieties by methods from differential geometry, as well as ribbons, K3 carpets and supermanifolds. The computational parts of this project are concerned with criteria for systems of parameters and development of the computer algebra system, Macaulay. This is research in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origins, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays, the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields, as well as, in theoretical computer science and robotics.

Eisenbud 9726460 This project is concerned with research in algebraic geometry, commutative algebra, symbolic computation, and computational statistics. In algebraic geometry, the main thrust is towards discovering connections between the geometry of sets defined by the vanishing of polynomials and the algebraic invariants derived by solving the linear equation over the polynomial ring whose coefficients are the defining equations. In commutative algebra, the principal investigator will work on the structure of syzygies in general, on the dimensions of the loci described by various sorts of equations, and on a problem coming from the part of number theory responsible for the recent proof of "Fermat's Last Theorem". In symbolic computation, he will work with the developers of Macaulay2 incorporating into scripts methods which will enable this software to address problems of importance to algebraic geometers. Finally, the principal investigator will study random walks on lattices, using techniques from algebraic geometry and commutative algebra to generate the walks and to study how fast they converge to their asymptotic distributions. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.

This project will establish a long-term collaboration between the Mathematics Department of San Francisco State University (SFSU), one of the country's largest centers of education for minority students, and the Mathematical Sciences Research Institute (MSRI), one of the world's leading mathematical research centers. Each semester this collaboration will revolve around an undergraduate topics course at SFSU based on one of the emphasis programs at MSRI that semester.

The SFSU-MSRI collaboration will be taught by a faculty member in the SFSU Mathematics Department (a different person each semester). This person will participate in the MSRI program associated with the chosen topic during the semester and will act as a bridge between SFSU students and the research mathematicians at MSRI. The SFSU-MSRI collaboration course will feature monthly talks by leading mathematicians who are visiting MSRI to participate in its program.

All lectures in the SFSU-MSRI collaboration course will be made available to the world through MSRI's streaming video technology. This technique, pioneered by MSRI for publishing mathematical lectures on the World Wide Web, uses one browser window to show the slides of the talk and another to show the speaker while the sound flows smoothly. Because research mathematicians rarely pitch work to undergraduates, as they will in the SFSU-MSRI collaboration course, the entire set of lectures will be made available on CD-ROM for dissemination to other universities.

So that SFSU students and faculty can have the library resources needed to prepare for MSRI programs, mathematics books will be purchased for the SFSU library. Each semester, SFSU will emphasize the purchase of books for the library related to the topic of that semester's SFSU-MSRI collaboration course.

This project also will support four SFSU graduate students each year. These students will participate in various activities at MSRI each semester.

SFSU will become an Academic Sponsor of MSRI. Membership benefits include sending two graduate students, expenses paid, to the MSRI Summer Program, inviting speakers from MSRI with expenses partially paid, receiving the preprint series, and financial support for conferences.

David Eisenbud will work on three groups of problems from commutative
algebra and algebraic geometry: The use of exterior algebra methods
in projective geometry and their extension to toric geometry; the study
of infinite free resolutions; and the study of codimensions of determinantal
(and related) ideals. This last includes the circle of problems around the
existence of vector bundles of low rank on projective spaces. His activity
will also include the development of fundamental algorithms in computational
algebraic geometry and their implementation in the Macaulay2 package
written by Grayson and Stilllman. By far the largest portion of the project
budget is devoted to the support of graduate students. This and the computational
tools produced by the project will broaden its impact.

Algebraic geometry deals with geometric forms defined by
simple equations (polynomials). The subject is central in mathematics
because these forms include the fundamental examples in most disciplines
of mathhematics; and it is important in the applications of mathematics
because these forms provide the models most often used in representing
nature in equations or in the computer. Over the last 150 years the group of
techniques called commutative algebra has been developed to study and unify this
and problems arising in number theory. Much more recently it was realized that
an extension of commutative algebra into the noncommutative domain of
exterior algebras gives a fruitful new approach to some old problems in this
field. Independently, computers and algorithms became powerful enough to
help research in algebraic geometry. Eisenbud's proposed work has to do
with some of the most classical problems in algebraic geometry as well
as with new uses of exterior algebra and computation in this domain, and
he will also continue his work developing computational tools for others
to use. The largest part of his grant will be devoted to the support
of graduate students, whom he will train in these techniques and methods.

Inaugurated in 2003, the Banff International Research Station (BIRS) has
developed, through a unique scientific partnership between the United
States and Canada, into a tremendous resource for the world's scientific
community. In 2006, the two countries will be joined by Mexico to make
BIRS the premiere research center for mathematical sciences in North
America.

Starting in 2006, BIRS will host an expanded schedule of weekly
workshops, pursuing an extremely broad program, embracing all aspects of
the mathematical sciences, from the most fundamental work on the great
problems of algebra, geometry and analysis to modern pure and applied
mathematics, theoretical and applied statistics, financial and industrial
mathematics, the mathematics of information technology and computer
science, bio-mathematics and math education. BIRS provides an environment
that optimizes creative interaction and the exchange of ideas, knowledge and
methods within the mathematical sciences and with related sciences and
industry. Recent PhD's and graduate students as well as members of
underrepresented groups are especially encouraged to organize and
participate in BIRS activities.

BIRS is administered by the Pacific Institute for the Mathematical
Sciences (PIMS) with the support of the Berkeley-based Mathematical
Sciences Research Institute (MSRI), the MITACS Network of Centres of
Excellence, as well as the Instituto de Matematicas at the National
University of Mexico (UNAM). The BIRS scientific program is developed by
an independent Scientific Advisory Board consisting of thirty of the most
eminent mathematical scientists in the world.

David Eisenbud will perform research on a number of
areas in which algebraic geometry mixes with homological commutative
algebra. One of these is the study of the Castelnuovo-Mumford regularity
of high powers of ideals. For an ideal in the homogeneous coordinate ring
of a projective variety that is of finite colength and is generated by forms
of a single degree, this "assymptotic regularity" is connected with the
properties of fibers of the corresponding map of varieties, and Eisenbud
will use this connection to investigate generic projections. A second
area is the study of schemes that are 2-regular. This work builds on the classification
for reduced schemes that was recently completed by Eisenbud, Green, Hulek and
Popescu. A third area has to do with the computation of higher direct images using
exterior algebra methods. Eisenbud will study certain varieties that appear naturally
in the deformation spaces of bundles in this way. All these areas are supported
by computations based on Groebner basis methods. Eisenbud will collaborate
with Stillman and Grayson in the development of the Macaulay2 program, which
is currently the best tool for computations in projective geometry. Finally, Eisenbud
will collaborate on a project with Diaconis, Holmes, and their students involving the
application of techniques in algebraic geometry to statistics.

An algebraic variety in complex n-space is a set defined by
the simultaneous vanishing of a collection of polynomial functions
of n variables. Many of the important geometric objects that appear
in mathematics can be defined this way. One way to study an algebraic
variety is to ask for the dimension of the vector space of polynomials
of degree at most d that vanish on the variety, as a function of d. This
is called the Hilbert function of the variety. David Hilbert showed how to
compute this function, and refine it, by computing something called a
free resolution of the variety. My work uses such free resolutions to study
varieties in many contexts. For example, if a variety of dimension n is embedded in
a vector space of dimension n+1,
then the variety is defined by just one equation, a relatively simple case.
Most varieties of dimension n can not be embedded in this way, but any
variety of dimension n can be mapped linearly onto a variety of the
same dimension in an n+1 dimensional space. One of the questions I will
study is the difference between the original variety and its image, for the
best possible embeddings.

This project studies two objects in the common ground between commutative algebra, combinatorics and algebraic geometry: the Cox rings of algebraic varieties and the Hilbert schemes of points. The Cox ring of an algebraic variety $X$ is a multigraded ring whose GIT quotients are the coordinate rings of images of $X$ via rational maps. For many interesting varieties, the so called Mori Dream spaces, the Cox ring is a finitely generated k-algebra. The project aims to develop tools to realize Cox rings of some Mori Dream spaces effectively, in terms of explicit coordinates and defining equations. Such descriptions are obtained via Koszul filtrations, via the combinatorics of Groebner bases or via new methods proposed.
Specific conjectures about the structure of the Cox rings of certain classes of varieties are proposed. The project also studies the Hilbert scheme of points in affine space, in particular the problem of describing the radical component (the closure of the set of radical ideals) and the singularities near monomial ideals.

An algebraic variety is a space defined by the simultaneous vanishing of a collection of polynomial functions in several variables. A very fertile classical technique for studying the geometry of an algebraic variety is to consider the various natural coordinate systems that this variety admits. The Cox ring of an algebraic variety is a way to study all these coordinate systems simultaneously. The first objective of this project is to provide descriptions of the Cox rings of some algebraic varieties. The second objective of this proposal is to study the Hilbert scheme of
points: the space of all possible configurations of n points in d-dimensional space. The geometry of this set is rather intricate and the aim of the project is to describe all possible ways in which the n-points can collide into one another and the geometry of the Hilbert scheme near these collisions.

The proposer will work on problems in commutative algebra, algebraic geometry, and symbolic computation. The common theme in these problems is that they are all related to free resolutions. First, the proposer will try to understand the cone of cohomology tables of vector bundles on varieties other than projective spaces. Even the case where the space is the product of two projective lines is challenging. In this case the proposer, with Frank Schreyer, has identified some extremal rays of the cone, and will try to show that they are all the extremal rays. The proposer will continue his collaboration with Persi Diaconis, trying to make useful probabilistic models for the 1-dependent processes corresponding to Koszul algebras that do not have Poincar'e-Birkhoff-Witt bases. Finally, the proposer will develop new algorithms for computation in commutative algebra, algebraic geometry, and the fields to which they can be applied.

A fundamental tool in many fields of mathematics and its applications involves finding solutions to systems of linear equations that vary with some parameters, solutions that themselves vary in a simple way--for example, as polynomial functions of the same parameters. The proposer's work extends the methods (both computational and theoretical) for using this tool, and the range of its applicability. One example of a relatively new kind of application of this technique is the study of so-called ``one-dependent'' statistical processes, which may be thought of as the results of successive experiments where the result of the next experiment depends on the result of the current one, but where knowing in addition the results of the past ones does not give any further information. (Imagine repeatedly tossing a coin that has a 2/3 chance of landing "heads" when tossed with heads up, and a 1/2 chance of landing "heads" when tossed with tails up. The sequence of heads and tails that results from tossing the coin many times, but always tossing it from the positions (heads or tails up) where it landed is a one-dependent process.) In collaboration with Persi Diaconis the proposer hopes to construct the probability laws of interesting new 1-dependent processes.

The Underrepresented Students in Topology and Algebra Research Symposium (USTARS) will take place at the University of California, Berkeley on April 11-13, 2014. USTARS is an annual symposium designed to bring together student researchers in algebra and topology, focusing on and showcasing research by underrepresented students and providing mentorship for the participants through critical educational and career transitions. In 2010, 29% of the mathematics doctorates were awarded to women, while less than 4% were awarded to minority students. USTARS promotes diversity in the mathematical sciences by encouraging women and minorities to attend and give talks. USTARS exposes all participants to the research and activities of underrepresented mathematicians, encouraging a more inclusive and collaborative mathematics community. Graduate students who attend USTARS are better equipped to seek academic positions and continue the cycle of mathematical research and collaboration. Undergraduate students are exposed to a wide variety of current research, ideas, and results, often more comprehensive than the topics available at their home institutions. A one-day workshop at the start of USTARS focuses on specific stages in professional careers and the critical transitions associated with each stage. Participants have the opportunity to network with underrepresented professors and students who may become future collaborators, colleagues, and/or mentors. USTARS participants continue to influence the next generation of students in positive ways by serving as much needed mentors and encouraging students in the mathematical sciences to advance themselves and participate in research and conference events. More information is available at: www.ustars.org

The Underrepresented Students in Topology and Algebra Research Symposium (USTARS) is a project proposed and guided by a national group of underrepresented mathematicians. At USTARS, student researchers convene in a structured meeting, along with professional mathematicians. Students deliver 30-minute research talks in structured parallel sessions. Graduate students give 75% or more of these presentations. Two distinguished graduate students and one invited faculty member are chosen to give 1-hour presentations. A poster session features invited undergraduate researchers. Topics of presentations given in prior years include: combinatorics, braid groups, graph theory, knot theory, representation theory number theory and algebraic geometry.

Algebraic Geometry is the study of systems of polynomial equations. As such, it has broad applications not only within mathematics but also in many fields of science from medicine to physics. The subject has developed for over 200 years, but there are many fresh problems and new directions. Commutative algebra is a subject that lies at the intersection of algebraic geometry and number theory. Since the advent of powerful and easily available computing resources, the possibilities for experimentation within commutative algebra, algebraic geometry, and their applications has multiplied. A byproduct of this project will be the further development of these computational tools. Cohen-Macaulay modules and sheaves play a role in commutative algebra and algebraic geometry that is a natural analogue of the role of finite dimensional representations in the case of finite dimensional algebras. The Principal Investigator will work on projects in commutative algebra, algebraic geometry, and computational methods that center around the theory of Cohen-Macaulay modules over particularly interesting classes of varieties: Toric varieties, complete intersections, and residual intersections. The PI will also continue to train graduate students in related research fields and actively be involved in several highly recognized outreach activities.

The Principal Investigator will work on Ulrich modules and Clifford Algebras. Cohen-Macaulay and Ulrich modules over quadratic hypersurfaces are well-understood from work of Knoerrer (over algebraically closed fields) and Buchweitz-Eisenbud-Schreyer over arbitrary fields. The PI will investigate deeper questions about Ulrich modules and other maximal Cohen-Macaulay modules on complete intersections of two quadrics using Clifford algebra techniques, extending Miles Reid's thesis, and making explicit work of Bondal-Orlov and Kapranov, as well as the theory of maximal Cohen-Macaulay modules over complete intersections developed by the proposer with Irena Peeva. Finally the PI will work on the cohomology of sheaves on toric varieties, extending techniques from exterior algebra algebras introduced in joint work with Daniel Erman and Frank-Olaf Schreyer for cohomology of sheaves on projective space and successfully extended to products of projective space to all toric varieties.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.