Algecom7


Supported by NSF grant DMS-1146096



 

 



Algebra, 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.

In Fall 2012 we held the second NSF funded ALGECOM at Purdue. We
invited Jerzy Weyman (F, Northeastern), Ragnar Olaf Buchweitz (F,
Toronto), Ben Wyser (P, UIUC). We also invited and had scheduled
Christine Berkesch (P&U, Duke). However, Prof. Berkesch needed to
withdraw a few days before hand. We replaced her with Ralph Kaufmann
(F, Purdue).

In addition we held a graduate student poster session featuring 4
posters Oliver Pechenik (G, UIUC), Matheus Brito (G&U, IUPUI), Michael
Dipasquale (G, UIUC) and Dominic Searles (G, UIUC). We plan to
continue this poster session idea.




Date:   Oct 20, 2012

Location: Department of Mathematics, room 175, at Purdue University in West Lafayette

For a map, click here. Coffee-breaks  will be held in the math library on the third floor.


Speakers  and schedule:

Coffee and pastries          9h00

 

Jerzy Weyman (Northeastern) 10:00h-11:00h

Title: Finite free resolutions and Kac-Moody Lie algebras

Abstract. Let us recall that  a  format (r_n,\ldots ,r_1) of the free complex
0-->F_n-->F_{n-1}-->\ldots F_0
over a commutative Noetherian ring is the sequence of ranks r_i  of the i-th differential d_i.
 We will assume that  rank F_i =r_i+r_{i+1}. We say that an acyclic  complex
F_{gen} of a given format over a given ring R_{gen} is generic if for every
complex G of this format over a Noetherian ring S there exists a homomorphism
f:R_{gen}--> S such that G=F_{gen}\otimes_{R_{gen}} S.

For complexes of length 2 the existence of the generic acyclic complex was
established by Hochster and Huneke in the 1980's. It is a normalization of the
ring giving a generic complex (two matrices with composition zero and rank
conditions).

I will discuss the ideas going into the proof of the following result:

Associate to a triple of ranks (r_3, r_2, r_1) a triple (p,q,r)=(r_3+1,
r_2-1, r_1+1). Associate to (p,q,r) the graph T_{p,q,r} (three arms of lenghts
p-1, q-1, r-1 attached to the central vertex). Then there exists a Noetherian
generic ring for this format if and only if T_{p,q,r} is a Dynkin graph. In other
cases one can construct in a uniform way a non-Noetherian generic ring, which
carries an action of the Kac-Moody Lie algebra corresponding to the graph
T_{p,q,r}.




Ragnar-Olaf Buchweitz (Toronto) 11:30h-12:30h

Title: Maximal Cohen--Macaulay modules and noncommutative geometry

Abstract: The classical McKay correspondence relates maximal Cohen-Macaulay modules on
Kleinian singularities to representations of the finite subgroups of SL(2,C)
and to the exceptional divisors in the desingularization.
It can also be interpreted (Kapranov-Vasserot 1999) as an algebraic
description of the derived category of coherent sheaves on the desingularization.

This approach has been vastly generalized, first by Bridgeland-King-Reid
(2001), then by D.Orlov (2009) and most recently by Amiot-Iyama-Reiten (2012).

The potential of these developments in Commutative Algebra has not yet been
explored,
and we hope this talk will entice some to take a closer look.




Coffee and snacks          14h00


Ralph Kaufmann (Purdue) 14:30h-15:30h

Title: Graphs, algebras and cohomology

Abstract:  We discuss how graphs, especially trees appear in the description
of certain algebras.
This observation has three levels. First, there are specific
cohomology groups and algebras basically given by trees.
The reason is often that they naturally index cells or divisors for certain
spaces, especially moduli spaces.
Secondly algebras themselves, like Lie, pre-Lie, associative etc algebras can
also be described in terms of certain trees and graphs.
This is related to the first occurrence, as we explain. Finally, The graphs
describing these classes of algebras themselves again
form algebras, like Lie or BV algebras. We will progress through these levels
giving the representative examples.




Ben Wyser (UIUC) 16:00h-17:00h

Title: Geometry and combinatorics of K-Orbits on the flag manifold.

Abstract: The orbits of a symmetric subgroup on a flag manifold ("K-orbits") are
of importance in the representation theory of real Lie groups, and have been
studied extensively from this perspective. The closures of such orbits are
generalizations of Schubert varieties, and any geometric and/or combinatorial
question one has about Schubert varieties can equally well be posed about these
more general orbit closures. However, while the geometry and combinatorics of
Schubert varieties have been studied exhaustively, even apart from their role in
representation theory, K-orbits and their closures have received far less
attention from these perspectives. I will discuss the K-orbit analogue of a story
which is well understood in the case of Schubert varieties. Namely, I will
describe how one can compute representatives for the torus-equivariant cohomology
classes of K-orbit closures, and how these formulas can be interpreted as Chern
class formulas for certain types of degeneracy loci. This is in parallel with the
well-known story, due to Lascoux-Schutzenberger, Fulton, Pragacz, Graham, et.
al., of double Schubert polynomials as representatives for the torus-equivariant
classes of Schubert varieties, and their interpretation as Chern class formulas
for the classes of degeneracy loci associated to flagged vector bundles.


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List of participants:

Arnold Yim (G, Purdue)
Michael Dipasquale (G, UIUC)
Jimmy Shan (G, UIUC)
Botong Wang (P, Notre Dame)
Youngho Yoon (G, Notre Dame)
Wenbo Niu (P, Purdue)
Yi Zhang (P, Purdue)
Ragnar O. Buchweitz (F, Toronto)
Saugata Basu (F, Purdue)
Alexander Yong (F, UIUC)
Ben Wyser (P, UIUC)
Dominic Searles (G, UIUC)
Oliver Pechenik (G, UIUC)
Amita Malik (G&U, UIUC)
Chayapa Darayon (G&U, UIUC)
Matheus Brito (G&U, IUPUI and UNICAMP, Brazil)
Evgeny Mukhin (F, IUPUI)
Vitaly Tarasov (F, IUPUI)
Andrei Gabrielov (F, Purdue)
Bill Butske (F, Rose-Hulman)
Matt Toeniskoetter (G, Purdue)
Abhinishek Parab (G, Purdue)
Christopher Drupieski (F, DePaul University)
Peter Tingley (F, Loyola Chicago)
Uli Walther (F, Purdue)
Alexandre Eremenko (F, Purdue)
Jerzy Weyman (F, Northeastern)
Ralph Kaufmann (F, Purdue)



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Parking: Please park in the garage next to the math building on University street. The easiest access is coming from the south via State Street. Weekend parking is free.

Lodging: We put on hold a block of 10 guestroom at Union Club Hotel, see
http://www.union.purdue.edu/HTML/UnionClubHotel/
at the rate of $99(standard queen) per night plus tax.
You can get rooms for up to 4 people with a double deluxe room ($144).
Make reservations by calling (800) 320-6291 
or (765) 494-8913. Please, identify your group affiliation: Algecom day
to get the discounted rates.
The reservations have to be made by 5:00pm October 5, 2012.

We have another block of 5 rooms in the Hilton, see
http://www.hiltongardeninn.com/en/gi/hotels/index.jhtml?ctyhocn=LAFWLGI
at the rate of $98 per night plus tax. Make reservations by calling
1-765-743-2100. Please, identify your group affiliation: ALGE
to get the discounted rates.
The reservations have to be made by 5:00pm September 28, 2012.


Banquet:
6pm--?? at Nine Irish Brothers of West Lafayette, http://www.nineirishbrothers.com