Computational Algebra and Convexity


Hal Schenck, Mike Stillman, Jan Verschelde
schenck@math.uiuc.edu
mike@math.cornell.edu.
jan@math.uic.edu.

This workshop will have two working groups; one which discusses recent work on graded betti numbers of free resolutions (henceforth the Boij-Soderberg working group), and one on primary decomposition (henceforth the Primary Decomposition working group). Details of the groups follow.

Boij-Soderberg working group

Objective. The main goal of the workshop is to understand the framework and methods of proof of the Boij-Soderberg conjecture, as proved in:

1. Boij-Soderberg: Betti numbers of graded modules and the Multiplicity Conjecture in the non-Cohen-Macaulay case.

2. Eisenbud-Floystad-Weyman: The Existence of Pure Free Resolutions.

3. Eisenbud-Schreyer: Betti Numbers of Graded Modules and Cohomology of Vector Bundles.

Participants will learn to apply tools and methods from homological algebra and representation theory to problems in commutative algebra. First, we'll spend a morning understanding the conjecture and setup of [1]. On the second day, we'll work thru [3], which proves the Boij-Soderberg conjecture; spectral sequences appear several times and there will be an overview talk on how to use and apply these gadgets. The first proof of the existence of pure free resolutions appeared in [2], using methods of Kempf-Kleiman-Lascoux-Weyman. While [3] gives a proof which avoids these methods, they are useful in their own right, and the second portion of the workshop will concentrate on this technique. This means we'll need to fill in many of the background results from representation theory, which are spread over references [4],[5],[6] below.

The majority of time will be spent in group work in understanding the topics of the day, with an emphasis on hands on computation and applying the tools and methods learned.

All participants will give a short talk (one overhead) on their own research on the first morning, to get people acquainted and familiar with everyone's interests. Each day, we'll all meet together in the late afternoon. The working group's will give a 25 minute talk, outlining what was studied and what was accomplished; this will be an overview allowing participants in the other working group have a picture of the problems, methods, and tools that are being studied by the other group. We will also have overview talks by David Eisenbud, Jan, Hal and Mike, mostly in the late afternoon. After dinner, there will be joint sessions, on Sage and technology in the classroom, on grant writing, and on communication (giving talks, writing papers, etc).

Additional references
4. Weyman: Cohomology of Vector Bundles and Syzygies
5. Fulton: Young Tableaux
6. Fulton-Harris:Representation Theory

Rough Outline
June 22
AM Common session with both groups-participant talks
PM Overview of the conjecture [1]
Talk: Open problems in commutative algebra (Schenck)

June 23
AM Spectral sequences: Theorem 3.1 of [3]
PM The existence of pure resolutions: Section 5 of [3]
Talk: Numerical Computations in Algebraic Geometry (Verschelde)

June 24
AM Representations of S_n, Schur functors
Talk 11-12: Using Macaulay2 to compute sheaves, cohomology (Stillman)
PM HIKE

June 25
AM Botts algorithm
PM Method of Kempf-Kleiman-Lascoux-Weyman, application in [2]

June 26
AM Computations using the method
PM Small group presentations "what we have learned"
Talk: TBA (Eisenbud)


Primary Decomposition working group

Computing a primary decomposition is a central problem in computational algebraic geometry. The objectives of this working group is to understand how this could be done numerically, with the aid of polyhedral methods in tropical algebraic geometry.

Rough Outline

  1. day 1: definition, examples, and problems

    objective: Every participant gives an example of a problem which requires the computation of a primary decomposition.
    The examples must be such that the embedded components are relevant to the problem. What if the input coefficients are approximate?

  2. day 2: numerical irreducible decomposition

    objective: Every participant tries the available methods and numerical software on the examples in the first day.
    Distrusting numerical software is part of the scientific method. How could we suspect from the output of the numerical software that there are embedded components?

  3. day 3: primary decomposition using Macaulay 2

    objective: Every participant tries to solve the example problems formulated on day 1 with Macaulay 2.
    Of course Macaulay 2 will solve all the examples we generated, but this is also the day of the hike, we can take it a bit easy.

  4. day 4: convexity and polyhedral methods

    objective: Understand how critical tropisms may lead to embedded components in a primary decomposition.
    Puiseux series are one of the roots of tropical geometry. The study of Puiseux series could occur earlier in the week.

  5. day 5: preparing for the AMS special session in January 2009

    objective: Every participant gives a short 10 to 15 minutes presentation about what is learned and accomplished.

References:

  1. W. Decker, G.-M. Greuel, G. Pfister: "Primary decomposition: algorithms and comparisons." In: G.-M. Greuel, B.H. Matzat, G. Hiss: Algorithmic Algebra and Number Theory. Springer Verlag, Heidelberg (1998), 187-220.
  2. Theo de Jong and Gerhard Pfister: "Local Analytic Geometry. Basic Theory and Applications." Vieweg, 2000.
  3. Anders N. Jensen, Hannah Markwig, and Thomas Markwig: "An algorithm for lifting points in a tropical variety."
  4. Anton Leykin: "Numerical Primary Decomposition." to appear in ISSAC 2008 proceedings.
  5. Andrew J. Sommese and Charles W. Wampler: "The numerical solution of systems of polynomials arising in engineering and science." World Scientific Press, 2005.
  6. Robert J. Walker: "Algebraic Curves." Princeton UP, 1950.

Basic daily schedule

7:30-9:00 breakfast
9:00-10:30 working groups
10:30 Coffee
10:30-12:00 working groups
12:00-1:30 lunch
1:30-3:30 working groups
3:30 Coffee
4:00-5:00 Day's progress talks (one per group: 25 min)
5:00-6:00 Common talk (S:Hal, M:Jan, R:David)
6:00-7:30 dinner
8:00-9:00 professional program (joint session: M,T,W)

Common joint sessions

6/22 AM: short participant presentations (one slide)
6/22 PM: Reception 6:30-7:30, followed by dinner
6/24 PM: Hike (no slides!)
6/26 PM: Small group presentations
6/26 PM: Banquet 6:30