Triangle Lectures in Combinatorics (TLC)

Fifteenth meeting: Saturday, March 24, 2018, 9am-5pm
Location: North Carolina State University, Raleigh
Lecture Hall: SAS Room 1102

Speakers: Richard Kenyon (Brown), Karola Mészáros (Cornell), Gabor Hetyei (UNC-Charlotte), and Seth Sullivant (NCSU)

Preregistration: please send email to (Patricia Hersh) to preregister. This is very helpful in our planning how much coffee, etc. to have at coffee breaks and for our obtaining funding to support these meetings.

Participant Travel Expense Reimbursement: we have some funding available for some participants, especially for early-career participants. Most of this is restricted to US citizens and permanent residents, and what is available to others still requires that the participants be employed at a U.S. university. To apply for funding please fill out the form here.

Conference schedule:
9:15-10am, coffee and small breakfast
10-11am, Richard Kenyon (Brown),  Harmonic functions and the chromatic polynomial (slides)
11-11:30am, coffee break
11:30am-12:30pm, Karola Mészáros (Cornell),  Flow polytopes in combinatorics and algebra (slides)
12:30-2:30pm, lunch break
2:30-3:30pm, Gabor Hetyei (UNC-Charlotte),  Counting partitions of a fixed genus (slides)
3:30-4pm, coffee break
4-5pm, Seth Sullivant (NCSU),  Maximum agreement subtrees (slides)
6pm, informal conference dinner

Practical details:

Parking: You may park right outside SAS Hall for free. Here is a map of the campus. On Saturdays, you can park anywhere on campus that is not specifically marked as being restricted (e.g. handicap spots are still off limits). We are hopeful that you won't need any lot except the one by SAS Hall. SAS Hall is at the upper right of the map, and the parking lot is near the intersection of Stinson Drive and Boney Dr. A good back-up option for parking is the Coliseum Parking Deck.

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: We have a block of rooms reserved at the Doubletree by Hilton Raleigh - Brownstone (919-828-0811) accessed by this link. Also within a short walk of the math department is Aloft Raleigh (919-828-9900). About 1.5 miles away in downtown Raleigh (also walkable, but somewhat long walk) is the Holiday Inn Raleigh Downtown (919-832-0501). Those with cars might also consider hotels farther away such as Holiday Inn Express (919-854-0001) 3741 Thistledown Drive (near Centennial Campus) as well as various hotel choices on Wake Town Drive, which is near numerous good restaurants; some such hotels (all right next to each other on Wake Towne Drive) are Marriott Courtyard (919-821-3400), Hampton Inn (919-828-1813), or Extended Stay America (919-829-7271).

Airport: Raleigh-Durham International Airport is 20-30 minutes drive from NCSU. Taxi fare is about $30.

Registered Participants (so far):

Aaron Abrams (Washington and Lee U.)
Phillip Andreae (Meredith College)
Ed Allen (Wake Forest U.)
Cashous Bortner (NCSU)
Michael Bremen-McKay (NCSU)
Jane Coons (NCSU)
Benjamin Cooper (Shaw U.)
Jessie Copher (NCSU)
Darij Grinberg (U. Minnesota)
Josh Hallam (Wake Forest U.)
Patricia Hersh (NCSU)
Gabor Hetyei (UNC Charlotte)
Ben Hollering (NCSU)
Chetak Hossain (NCSU)
Rick Kenyon (Brown)
Stephen Lacina (NCSU)
Susanna Lange (U. Kentucky)
Lionel Levine (Cornell)
Ricky Liu (NCSU)
Molly Lynch (NCSU)
Aida Maraj (U. Kentucky)
Sarah Mason (Wake Forest)
Emily Meehan (Gallaudet U.)
Karola Meszaros (Cornell)
Wesley Nelson (NCSU)
Gabor Pataki (UNC Chapel Hill)
Lindsay Piechnik (High Point U.)
Rodney Reid
Tim Reid (NCSU)
Radmila Sazdanovic (NCSU)
Dan Scofield (NCSU)
Avery St. Dizier (Cornell)
Grace Stadnyk (NCSU)
Michael Strayer (UNC Chapel Hill)
Seth Sullivant (NCSU)
Bill Trok (U. Kentucky)
Julianne Vega (U. Kentucky)
Shira Viel (NCSU)
Cynthia Vinzant (NCSU)
Michael Weselcouch (NCSU)

Local organizing committee: Patricia Hersh (NCSU), Ricky Liu (NCSU), and Cynthia Vinzant (NCSU)

Talk titles and abstracts:

Richard Kenyon (Brown)

Title: Harmonic functions and the chromatic polynomial

Abstract: The chromatic polynomial X(n) of a graph counts the number of proper colorings with n colors. For each negative integer n we show how to compute |X(n)| as the degree of a certain rational mapping. This mapping arises from the "Dirichlet problem" of finding a harmonic function with fixed boundary values. Our techniques also allows us to equate |X(n)| with a certain set of acyclic orientations of a related graph. This is joint work with Wayne Lam.

Karola Mészáros (Cornell)

Title: Flow polytopes in combinatorics and algebra

Abstract: The flow polytope FG(v) is associated to a graph G on the vertex set {1,..., n} with edges directed from smaller to larger vertices and a netflow vector v=(v1,..., vn) in Zn. The points of FG(v) are nonnegative flows on the edges of G so that flow is conserved at each vertex. Postnikov and Stanley established a remarkable connection of flow polytopes and Kostant partition functions two decades ago, developed further by Baldoni and Vergne. Since then, flow polytopes have been discovered in the context of Schubert and Grothendieck polynomials and the space of diagonal harmonics, among others. This talk will survey a selection of results about the ubiquitous flow polytopes.

Gabor Hetyei (UNC-Charlotte)

Title: Counting partitions of a fixed genus

Abstract: We show that, for any fixed genus g, the ordinary generating function for the genus g partitions of an n-element set into k blocks is algebraic. The proof involves showing that each such partition may be reduced in a unique way to a primitive partition and that the number of primitive partitions of a given genus is finite. We illustrate our method by finding the generating function for genus 2 partitions, after identifying all genus 2 primitive partitions, using a computer-assisted search. This is joint work with Robert Cori.

Seth Sullivant (NCSU)

Title: Maximum agreement subtrees

Abstract: Probability distributions on the set of trees are fundamental in evolutionary biology, as models for speciation processes. These probability models for random trees have interesting mathematical features and lead to difficult questions at the boundary of combinatorics and probability. This talk will be concerned with the question of how much two random trees have in common, where the measure of commonality is the size of the largest agreement subtree. The case of maximum agreement subtrees of pairs of random comb trees is equivalent to studying longest increasing subsequences of random permutations, and has connections to random matrices. This elementary talk will try to give a sense of what is known (not very much) and what is unknown (lots!) about this problem.