Bernd Sturmfels (colloquium), Spectrahedra

Carla Savage, The geometry of lecture hall partitions and quadratic permutation enumeration

Ed Swartz, f-vectors, descent sets and the weak order

Laszlo Szekely, A new tool for asymptotic enumeration: the Lovasz Local Lemma?

Bernd Sturmfels, The convex hull of a space curve

9-10am, coffee and bagels

10-11am, Carla Savage (North Carolina State University),``The geometry of lecture hall partitions and quadratic permutation statistics''

11-11:30am, coffee break

11:30am-12:30pm, Ed Swartz (Cornell University), ``f-vectors, descent sets and the weak order''

12:30-2:30pm, lunch break

2:30-3:30pm, Laszlo Szekely (University of South Carolina), ``A new tool for asymptotic enumeration: the Lovasz Local Lemma?''

3:30-4pm, coffee break

4-5pm, Bernd Sturmfels (UC Berkeley), ``The convex hull of a space curve''

NEW:

Abstract: Lecture hall partitions were introduced in 1997 by Bousquet-M{\'e}lou and Eriksson. Since then, these intriguing combinatorial objects and their various generalizations have been shown to be related to Bott's formula in the theory of affine Coxeter groups, Euler's partition theorem, the q-Chu-Vandermonde identities, the q-Gauss summation, and the little Goellnitz partition theorems.

In this talk, we take a geometric view of lecture hall partitions and a related family, the anti-lecture hall compositions, to settle some open questions about their enumeration. The method uncovers an intrinsic connection between these families of partitions/compositions and distributions of certain quadratic permutation statistics.

This is joint work with Katie Bright.

Abstract: In the last few years Chari's convex ear decomposition has been applied to several apparently unrelated order complexes of posets to derive new insight into their f-vectors. An unexpected consequence of these investigations is that their flag f-vectors are closely related to a largely unexplored interaction between descent sets and the weak order of finite Coxeter groups. The goal of the lecture is to explain how face enumeration on these posets naturally leads to a zoo of open problems involving descent sets and the weak order.

Abstract: Since 1974, the Lovasz Local Lemma has been the tool to find the proverbial needle in the haystack. It turns out that it can be used for asymptotic enumeration as well, as a substitute for the Chen-Stein method.

The Lovasz Local Lemma is known to have an extension for cases where the dependency graph requirement is relaxed to negative dependency graph (Erdos-Spencer 1991). The difficulty is to find relevant negative dependency graphs that are not dependency graphs. We provide two generic constructions for negative dependency graphs, in the space of random matchings of complete and complete bipartite graphs.

As applications, we easily derive asymptotic enumeration results for permutations, Latin rectangles, and regular graphs. These results in enumeration are not as strong as the best current results, but are much better than what was known in 1974.

This is joint work with Lincoln Lu.

Abstract: The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and a scroll of stationary bisecants. We express the degree of this surface in terms of the degree, genus and singularities of the curve. We present methods for computing their defining polynomials, and we exhibit a wide range of examples. Most of these are innocent-looking trigonometic curves such as (cos(t), sin(2t), cos(3t)). This is joint work with Kristian Ranestad (arXiv:0912.2986).

2-4 people from George Mason University

5-7 people from East Tennessee State University

Ed Allen, Wake Forest University

Moa Apagodu, Virginia Commonwealth University

Alyssa Armstrong, NCSU

Erin Bancroft, NCSU

Katie Bright, NCSU

Rod Canfield, University of Georgia

Manoj Chari, SAS

Josh Cooper, University of South Carolina

Eva Czabarka, University of South Carolina

Ruth Davidson, NCSU

Anant Godbole, East Tennessee State University

Nathan Gray, NCSU

Ruth Haas, Smith College

Patricia Hersh, NCSU

Gabor Hetyei, UNC Charlotte

J.T. Hird, NCSU

John Hutchins, NCSU

Min Kang, NCSU

Chirag Lakhani, NCSU

Craig Larson, Virginia Commonwealth University

James Lawrence, George Mason University

Yue Li, NCSU Operations Research

Matthew Macaulay, Clemson

Sonja Mapes, Duke

Sarah Mason, Wake Forest University

Jed Mihalisin, Meredith College

Ezra Miller, Duke

Kailash Misra, NCSU

Carlos Nicolas, UNC-Greensboro

Megan Owen, NCSU

Gabor Pataki, UNC Chapel Hill

Bob Proctor, UNC Chapel Hill

Scott Provan, UNC Chapel Hill

Nathan Reading, NCSU

Margaret Readdy, University of Kentucky

Carla Savage, NCSU

Mike Schuster, NCSU

Michael Singer, NCSU

Camilla Smith Barnes, Sweet Briar College

Cliff Smyth, UNC-Greensboro

John Steenbergen, Duke

Ernie Stitzinger, NCSU

Bernd Sturmfels, Berkeley

Seth Sullivant, NCSU

Ed Swartz, Cornell

Laszlo Szekely, U. South Carolina

Gopal Viswanathon, NCSU Computer Science