Triangle Lectures in Combinatorics (TLC)

First meeting: February 6, 2010, SAS Hall 1102, NCSU

Slides from the talks (as pdf files):

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

Talk schedule:

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''

Dinner: there'll likely be some sort of informal dinner at a restaurant in the area, i.e. not a banquet, but something organized in a more ad hoc manner for those interested.

Friday Colloquium: there will also be a colloquium talk by Bernd Sturmfels entitled ``Spectrahedra'' on Friday, February 5th at 3pm in SAS Hall.

NEW: Information for the day of the conference:

Parking: You may park right outside SAS Hall for free. Here is a map of the campus with parking lots marked. We anticipate that you won't need any lot except the one by SAS Hall. SAS Hall is at the right of the map, just left of the compass, and its parking lot is a large one shaded red on the map.

Campus map: Here is a map that may be easier to navigate by.

Doors: The doors to SAS Hall will be unlocked, but only some of them. These are: the handicap access door at the main entrance from the parking lot and the handicap access door on the main entrance from the courtyard side.

The room: SAS Hall 1102 is the room immediately to your right when you enter from the parking lot. If you enter from the courtyard side, go down the long stairway or the elevators.

Hotel recommendations: within short walk of the math department are three hotels: Holiday Inn Brownstone (800-331-7919), Velvet Cloak Inn (919-828-0333) and Cameron Park Inn Bed and Breakfast (919-835-2171). Those with cars might also consider hotels farther away such as various choices on Wake Town Drive, which is near numerous good restaurants; some such hotels (all right next to each other) are Marriott Courtyard (919-821-3400), Hampton Inn (919-828-1813), or Extended Stay America (919-829-7271).

Abstracts for the talks:

Carla Savage, ``The geometry of lecture hall partitions and quadratic permutation statistics''

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.

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

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.

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

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.

Bernd Sturmfels, ``The convex hull of a space curve''

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).

Registered participants:

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