Supported by NSF grant DMS1146096
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. Coffeebreaks will be held in the math library on the third floor.
Speakers and schedule: Coffee and pastries 9h00
Jerzy Weyman (Northeastern) 10:00h11:00h Title: Finite free resolutions and KacMoody Lie algebras
Abstract. Let us recall that a format (r_n,\ldots ,r_1) of the free complex 0>F_n>F_{n1}>\ldots F_0 over a commutative Noetherian ring is the sequence of ranks r_i of the ith 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_21, r_1+1). Associate to (p,q,r) the graph T_{p,q,r} (three arms of lenghts p1, q1, r1 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 nonNoetherian generic ring, which carries an action of the KacMoody Lie algebra corresponding to the graph T_{p,q,r}.
RagnarOlaf Buchweitz (Toronto) 11:30h12:30h
Title: Maximal CohenMacaulay modules and noncommutative geometryAbstract: The classical McKay correspondence relates maximal CohenMacaulay 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 (KapranovVasserot 1999) as an algebraic description of the derived category of coherent sheaves on the desingularization.
This approach has been vastly generalized, first by BridgelandKingReid (2001), then by D.Orlov (2009) and most recently by AmiotIyamaReiten (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.
Ralph Kaufmann (Purdue) 14:30h15:30h
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, preLie, 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) 16:00h17:00h
Title: Geometry and combinatorics of KOrbits on the flag manifold.Abstract: The orbits of a symmetric subgroup on a flag manifold ("Korbits") 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, Korbits and their closures have received far less attention from these perspectives. I will discuss the Korbit 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 torusequivariant cohomology classes of Korbit closures, and how these formulas can be interpreted as Chern class formulas for certain types of degeneracy loci. This is in parallel with the wellknown story, due to LascouxSchutzenberger, Fulton, Pragacz, Graham, et. al., of double Schubert polynomials as representatives for the torusequivariant classes of Schubert varieties, and their interpretation as Chern class formulas for the classes of degeneracy loci associated to flagged vector bundles.
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, RoseHulman) 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) 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) 3206291 or (765) 4948913. 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 17657432100. 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

