Combinatorics: Advances, Trends & Speculations (CATS 2010)

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Combinatorics 2010: Advances, Trends and Speculations (CATS 2010 Workshop)

University of Kentucky, Lexington Kentucky March 26, 2010


This one day workshop showcases current research advances in four of the main areas of combinatorics: Coxeter groups, topological combinatorics, polytopes and manifolds, and quasisymmetric functions. There is much cross-pollination between these areas, with techniques from Hopf algebras, commutative algebra and algebraic geometry being used to solve problems in discrete geometry and enumerative combinatorics.

The cutting-edge nature of the four keynote talks will set the tone for the workshop. Between the talks the participants will have extended opportunities to interact and exchange new ideas in these key areas.

This workshop is being held in conjunction with the AMS Special Session, "Advances in Algebraic & Geometric Combinatorics".

Invited Speakers


Richard Ehrenborg University of Kentucky
Margaret Readdy University of Kentucky


Marion Anton, Centre College
Marisa Belk, Cornell University
Saul Blanco, Cornell University
Ben Braun, University of Kentucky
Jonathan Browder, University of Washington
Shihwei Chao, Clemson University
Eric Clark, University of Kentucky
Scott Cook, Washington University in St. Louis
Elena Dimitrova, Clemson University
Anton Dochtermann, Dartmouth College
Felix Effenberger, Universität Stuttgart
Alexander Engström, Berkeley
Stefan Forcey, Tennessee State University
Andy Frohmader, Cornell University
David Hawes, University of Kentucky
Gábor Hetyei, University of North Carolina at Charlotte
Chris Hillar, MSRI
JiYoon Jung, University of Kentucky
Eric Kahn, Bloomsburg University
Theodoros Kyriopoulos, University of Kentucky
Beth Kelly, University of Kentucky
Steve Klee, University of Washington
Carly Klivans, University of Chicago
Sam Kolins, Cornell University
Matjaž Konvalinka, Vanderbilt University
Manoj Kummini, Purdue University
Lori Layne, Clemson University
Aaron Lauve, Texas A&M University
Carl Lee, University of Kentucky
Kurt Luoto, University of British Columbia
Matthew Macauley, Clemson University
Sonja Mapes, Duke University
Uwe Nagel, University of Kentucky
Eran Nevo, Cornell University
Megan Owen, North Carolina State University
Pasha Pylyavskyy, University of Michigan
Nathan Reading, North Carolina State University
Bryce Richards, Clemson University
Michael Slone, Preservation & Digital Programs, UK Libraries
Bernd Sturmfels, UC Berkeley
Rebecca Swanson, Indiana University
Ed Swartz, Cornell University
Daniel Wells, University of Kentucky
Walter Whiteley, York University
Russ Woodroofe, Washington University in St. Louis
Catherine Yan, Texas A&M University
Alex Yong, University of Illinois at Urbana Champaign
Ruriko Yoshida, University of Kentucky
Thomas Zaslavsky, Binghamton University
Matt Zeckner, University of Kentucky


Registration is free. However, we ask participants to register by e-mailing one of the organizers.
Please indicate whether or not you will attend the workshop dinner.

Workshop Schedule (pdf)

Friday, March 26th
Classroom Building 106

9:00 am Invited address: Sara Billey, "A stratification of branched polymer space" slides (pdf)

A mathematical model of polymers (from chemistry) has lead to some very interesting questions in probability and algebraic geometry. We will introduce the basic concepts, describe some new results and give several open problems. This is joint work with Dave Anderson.

10:00 am Coffee and discussion

11:00 am Invited address: Isabella Novik, "Lower bound theorems for spheres, manifolds, and pseudomanifolds" arXiv:0711.1348v4 [math.CO] & arXiv:1004.5100v1 [math.CO]

The celebrated lower bound theorem (due to Barnette, 1973 and Kalai, 1987) asserts that among all simplicial manifolds of a fixed dimension and with a fixed number of vertices, (the boundary complex) of a certain polytope, called a stacked polytope, simultaneously minimizes all the face numbers. In this talk we will discuss various strengthenings and generalizations of this theorem for the classes of (i) balanced simplicial spheres, (ii) general simplicial manifolds with and without boundary, and (iii) certain pseudomanifolds with isolated singularities. We will also briefly outline several commutative algebra techniques and new results that are needed for the proofs. Part (i) is joint with Michael Goff and Steve Klee; parts (ii) and (iii) are joint with Ed Swartz.

12:00 pm Lunch

2:00 pm Invited address: Patricia Hersh, "A combinatorial topological toolkit for stratified spaces" arXiv:0711.1348v4 [math.CO]

There are quite a few stratified spaces, i.e., spaces made of pieces each homeomorphic to $R^n$ for some n, of current interest in such areas as combinatorial representation theory, (real) Schubert calculus, algebraic statistics, and total positivity theory, the last of which are closely related to the theory of canonical bases and the theory of cluster algebras. A common feature of these spaces is that they may often be regarded as the image of a map from a much simpler space of parameters. I will discuss some recently developed tools for determining topological structure of such spaces in a rather strong sense, namely homeomorphism type. This will include topological collapsing lemmas as well as a new criterion for determining whether a finite CW complex is regular with respect to a choice of characteristic maps. While the proofs are topological, using these tools can be essentially combinatorial, as I will illustrate on the main application to date.

3:00 pm Coffee and discussion

4:00 pm Invited address: Stephanie van Willigenburg, "Quasisymmetric refinements of Schur functions" slides (pdf)

In this talk we introduce a basis for quasisymmetric functions, called quasisymmetric Schur (QS) functions, which partition Schur functions in a natural way. Furthermore, we show how these QS functions refine many Schur function properties including

* Kostka numbers
* Pieri rules
* The Littlewood-Richardson rule.

Extending the definition of QS functions, we define skew QS functions, which likewise partition skew Schur functions. We observe how these skew functions arise in the study of both the noncommutative Schur functions of Rosas-Sagan, and the unrelated ones of Fomin-Greene.

This is joint work with Christine Bessenrodt, Jim Haglund, Sarah Mason, and especially Kurt Luoto, who will speak in the AMS Special Session on a noncommutative Littlewood-Richardson rule arising from skew QS functions and their relation to free Schur functions.

6:00 pm Workshop dinner: Rincon Mexicano, 818 Euclid Avenue

Saturday and Sunday, March 27th -- 28th

See: Advances in Algebraic & Geometric Combinatorics

This workshop is supported by the National Science Foundation DMS-1024407.

We would also like to thank the University of Kentucky Department of Mathematics for partial support of this workshop.

Last updated May 10, 2010