Supported by NSF grant DMS-1146096Algebra, Geometry and Combinatorics Day (AlGeCom) is a one day,
informal meeting of mathematicians from the
University of Illinois, Purdue University, IUPUI, and nearby universities, with interests in algebra,
geometry and combinatorics (widely interpreted).
Further details will be posted here as they become available. Or you may
contact the University of Illinois organizers
Hal Schenck and Alexander Yong, or the Purdue organizers
Uli Walther
and Saugata Basu or IUPUI organizer Evgeny Mukhin.
The ALGECOM conference has had two events since the grant was awarded.
In Spring 2012 we invited Allen Knutson (Cornell), Ravi Vakil
(Stanford), Daniel Erman(postdoc, Michigan) and Alexandra Seceleanu
(postdoc, Nebraska) to speak.
Date: Apr 28, 2012
Location: Altgeld Hall, at UIUC
For a map, click here. Coffee-breaks will be held in the hallway on the first floor. Parking: There will be a marathon in town on algecom day. The marathon passes by the conference site (Altgeld Hall) but the last wave is at 7:25AM. By 8:30-8:45 AM the area should be mostly clear. Coming to the conference from the NORTH of campus should be mainly trouble free. Specifically, if arriving from the highway (e.g., coming from Purdue), get off at the Lincoln ave exit and drive south to University Ave. Then driving west on University until Wright street. Then you can just go south on Wright to Green Street (where Altgeld Hall is). There is a parking structure at 6th street/John and 6th street/Daniel that is free on weekends. http://www.parking.illinois.edu/campus_map/parkingmap.pdf Street parking is available as well; the marathon should not affect legality of any usual parking spots near campus. HOWEVER, it will be trickier to come to the campus by car from the south of campus. See the maps here http://illinoismarathon.com/course.php
Speakers and schedule: Coffee and pastries 8h30
Alexandra Seceleanu (Nebraska) 10:00h-11:00h Title: Bounding Projective Dimension
Abstract: Given a homogeneous ideal in a polynomial ring one can
measure its computational complexity in several ways. One of these is
the projective dimension, i.e. the minimal length of a graded free
resolution. There is great interest, originally prompted by a question
of Stillman, in finding bounds on the projective dimension in terms of
the degrees and number of generators. This is a problem with
connections to bounding regularity as well.
While Stillman's question remains open in full generality, I will
show that no polynomial function can be used to bound projective
dimension. I will also talk about how one can proceed in finding
bounds if the ideal satisfies special conditions. Throughout the talk
I will highlight a number of contributions of former or current UIUC
and Purdue mathematicians to this circle of ideas.
Dan Erman (Michigan)
11:15h-12:15h
Title: Duality in Boij-Soederberg Theory
Abstract:
Boij-Soederberg Theory is the study of two types of invariants: those coming
from free resolutions on a polynomial ring and those coming from sheaf
cohomology on projective space. Eisenbud and Schreyer first observed that
these invariants are related, and I'll motivate this relationship by looking
closely at a simple example. Then I'll describe the construction of a
duality pairing that leads to precise duality results relating free
resolutions and sheaf cohomology. This is joint work with David Eisenbud.
Allen Knutson(Cornell) 2:00h-3:00h
Title: Three geometric reasons to associate a juggling pattern to a matrix
Abstract: Given a matrix of rank k with n columns, I'll explain how to
associate a juggling pattern with k balls and periodicity n.
This leads to a stratification of the Grassmannian of k-planes in n-space.
This stratification arises naturally if one considers
(1) the Frobenius splitting on the Grassmannian over a field of characteristic p,
(2) the deformation of the Grassmannian to a noncommutative space, or
(3) the totally nonnegative part of the real Grassmannian.
Moreover, many interesting subvarieties of the Grassmannian
(in particular Schubert, Richardson, and those arising in Vakil's
"geometric Littlewood-Richardson rule") are closures of these strata,
which are irreducible, normal, Cohen-Macaulay, and have rational
singularities.
This work is joint with Thomas Lam and David Speyer.
Ravi Vakil (Stanford) 3:30h-4:30h
Title: Stabilization of discriminants in the Grothendieck ring
Abstract: We consider the ``limiting behavior'' of {\em discriminants}, by which we
mean informally the closure of the locus in some parameter space of some
type of object where the objects have certain singularities. We focus on
the space of partially labeled points on a variety $X$, and linear systems
on $X$. These are connected --- we use the first to understand the second.
We describe their classes in the "ring of motives", as the number of points
gets large, or as the line bundle gets very positive. They stabilize in an
appropriate sense, and their stabilization can be described in terms of the
motivic zeta values. The results extend parallel results in both arithmetic
and topology. I will also present a conjecture (on ``motivic stabilization
of symmetric powers'') suggested by our work. Although it is true in
important cases, Daniel Litt has shown that it contradicts other hoped-for
statements. This is joint work with Melanie Wood. (This is less technical
than it sounds,!
and I will define everything from scratch.)
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List of participants:
Alexander Yong (F, UIUC)
Uli Walther (F, Purdue)
Ravi Vakil (F, Stanford)
Daniel Erman (P, Michigan)
Jimmy Shan (G, UIUC)
Michael DiPasquale (G, UIUC)
Paolo Mantero (G, Purdue)
Andrei Gabrielov (F, Purdue)
Jianrong Li (G, Purdue)
Vitaly Tarasov (F, IUPUI)
Evgeny Mukhin (F, IUPUI)
Ser-Wei Fu (G, UIUC)
Dominic Searles (G, UIUC)
Arnold Yim (G, Purdue)
Sal Barone (G, Purdue)
Javid Validashti (P, UIUC)
Saugata Basu (F, Purdue)
Botong Wang (G, Purdue)
Wenbo Niu (P, Purdue)
Aisha Arroyo (G&U, UIUC)
Justin Chen (G, Purdue)
Chayapa Darayon (G&U, UIUC)
Oliver Pechenik (G, UIUC)
Youngsu Kim (G, UIUC)
Alexandra Seceleanu (G&U, Nebraska)
Allen Knutson (F, Cornell)
Bruce Reznick (F, UIUC)
Abhishek Parab (G, Purdue)
Tom Nevins (F, UIUC)
Gabriele La Nave (F, UIUC)
Bill Haboush (F, UIUC)
Jinwon Choi (G, UIUC)
Jinhyung To (G, UIUC)
Hal Schenck (F, UIUC)
Parking:
Lodging: Banquet: April 28 is the day of the
Illinois marathon, see here for info on parking and driving in town.
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