Triangle Lectures in Combinatorics (TLC)
Seventh meeting: February 9, 2013
Location: Wake Forest University (Winston-Salem, North Carolina)
Lecture Hall: Manchester Hall, Room 016 (within the Calloway Center -- not the same as Manchester Athletic Center)

Practical information about hotels, airport, taxis and about parking and information about dining options.

Slides from the talks:
Louis Billera's talk as pdf file
Rod Canfield's talk as pdf file
Matthew Kahle's talk as pdf file
Michelle Wachs' talk as pdf file

9:15-10am, coffee and bagels
10-11am, Louis Billera
11-11:30am, coffee break
11:30am-12:30pm, Rod Canfield
12:30-2:30pm, lunch break
2:30-3:30pm, Matthew Kahle
3:30-4pm, coffee break
4-5pm, Michelle Wachs

Louis Billera (Cornell)
Rod Canfield (University of Georgia)
Matthew Kahle (Ohio State University)
Michelle Wachs (University of Miami)

Please pre-register.
This should take you 30 seconds or less using the online preregistration form , or a few minutes longer if you apply for funding.

Participant funding available.
Graduate students and early career mathematicians are particularly encouraged to apply for funding. To apply, use the online preregistration form . We will need the following information: (1) estimated roundtrip mileage and/or airfare and local travel expenses, (2) estimated hotel cost, (3) a sentence or two of justification for why it would be useful for you to participate, and (4) other possible funding sources. Funding decisions will be made shortly after the 15th of each month up until the conference or until all funds have been committed.

Titles and abstracts:

Louis Billera, Balanced and unbalanced collections: from mathematical economics through quantum physics

Abstract: A collection of nonempty subsets of the set {1,...,n} is said to be balanced if the convex hull of the indicator functions of these sets in the n-cube meets the diagonal. The collection is unbalanced otherwise.

Balanced collections were defined 50 years ago by Lloyd Shapley (who was awarded a Nobel Prize in Economics last December) in his study of cores of cooperative games. Minimal balanced collections played an important role in determining when such games arise from economic trading models. One can view minimal balanced collections as generalized partitions.

Maximal unbalanced collections arose recently in physics in the study of thermal field theory, a combination of quantum field theory and statistical mechanics. They are also closely related to the study of threshold Boolean functions, threshold collections and voting games. We consider a hyperplane arrangement whose regions correspond to unbalanced collections.

I will say a bit about the applications of balanced and unbalanced collections and give relations between the various questions they ask. The talk will be more a survey than a recitation of new results, although some new approaches will be described. In particular, there are many questions here that modern algebraic combinatorics ought to be able to answer.

Rod Canfield, Compositions and infinite matrices

Abstract: Suppose $a_n$, $n \ge 0$, is a sequence of positive integers of combinatorial interest; typically the $a_n$ count something. The existence of the limit $\lim_{n\rightarrow\infty} (a_n)^{1/n}$ can sometimes be difficult to prove. When the limit is known to exist, there arises the challenge to pinpoint it in some way, for example as the root of a polynomial or transcendental equation. In the study of locally restricted compositions we find many such limits which can be linked to the eigenvalues of infinite matrices. (A composition of $n$ is a $k$-tuple $(c_1,\dots,c_k)$ of positive integers whose sum is $n$; local restrictions are conditions such as ``no two adjacent parts are equal,'' or ``the parts alternate in magnitude.'') Directions for further study will be indicated. This is collaborative research with Ed Bender and Jason Gao.

Matthew Kahle, Configuration spaces: combinatorics, topology, and physics

Abstract: Configuration spaces of points are well-studied spaces in algebraic topology, algebraic geometry, geometric group theory, and combinatorics. Give the particles thickness, and you have what physicists might describe as phase space for a hard spheres gas. When the points are points, the topology of the configuration space is well understood but hardly anything is known when points have thickness. Then changes in the topology as the thickness varies could be thought of as topological phase transitions.

I will report on recent work understanding these kinds of changes in topology, including work in progress with Bob MacPherson.

Michelle Wachs, Eulerian numbers, chromatic quasisymmetric functions and Hessenberg varieties

Abstract: We consider three distinct topics of independent interest; one in enumerative combinatorics, one in symmetric function theory, and one in algebraic geometry. The topic in enumerative combinatorics concerns a q-analog of a generatlization of the Eulerian numbers, the one in symmetric function theory deals with a refinement of Stanley's chromatic symmetric functions, and the one in algebraic geometry deals with a representation of the symmetric group on the cohomology of the regular semisimple Hessenberg variety of type A. Our purpose is to explore some connections between these topics and consequences of these connections. This talk is based on joint work with John Shareshian.

List of preregistered participants (so far):

Ed Allen, Wake Forest University
Taylor Allison, UNC Chapel Hill
Fawwaz Batayneh, Clemson
Jonathan Beagley, George Mason University
Kenneth Berenhaut, Wake Forest University
Christine Berkesch, Duke
Louis Billera, Cornell University
Sarah Birdsong, UNC Charlotte
Kayla Blyman, University of Kentucky
Yue Cai, U. Kentucky
Rod Canfield, University of Georgia
Shihwei Chao, Clemson
Shaoshi Chen, NCSU
Ruth Davidson, NCSU
Robert Davis, University of Kentucky
Alex Fink, NCSU
Norman Fox, University of Kentucky
Jennifer Galovich, Virginia Tech
Jennifer Gamble, NCSU
Nicole Gin, NCSU
Rafael S. Gonzelez D'Leon, University of Miami
Brent Gorbutt, George Mason University
Darij Grinberg, MIT
Qijun He, Clemson
Patricia Hersh, NCSU
Gabor Hetyei, UNC Charlotte
John Hutchens, NCSU
Austin Jones, NCSU
Matthew Kahle, Ohio State University
Chris Kirkland, NCSU
Andrey Kuney, NCSU
Shirley Law, NCSU
David Lax, UNC Chapel Hill
Matthew Macauley, Clemson
Sarah Mason, Wake Forest University
Emily Meehan, NCSU
Sam Mendelson, George Mason University
Ezra Miller, Duke
Walter Morris, George Mason University
John Mosley, University of Kentucky
Michael Mossinghoff, Davidson College
Sayan Mukherjee, Duke
Sarah Nelson, University of Kentucky
Asamoah Nkwanta, Morgan State University
Christopher O'Neill, Duke
Elliot Paquette, U. Washington
Lindsay Piechnik, High Point University
Shira Polster, NCSU
Svetlana Poznanovik, Clemson
Scott Provan, UNC Chapel Hill
Nathan Reading, NCSU
Joe Rusinko, Winthrop University
Carla Savage, NCSU
Michael Singer, NCSU
John Steenberger, Duke
Seth Sullivant, NCSU
Clifford Taylor, U Kentucky
Nate Tryon, NCSU
Bethany Turner, NCSU
Hayato Ushijima-Mwesigwa, Clemson
Mirko Visontal, KTH
Michelle Wachs, University of Miami
Rika Yatchak, NCSU

Organizing committee: Sarah Mason (chair, Wake Forest University), Ed Allen (Wake Forest University), Alex Fink (NCSU), Patricia Hersh (NCSU)