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