Triangle Lectures in Combinatorics

Eleventh meeting: Saturday February 21, 2015, 9:15am -- 5pm.

Location: North Carolina State University

Lecture Hall: SAS Room 1102

Speakers: Matthew Baker (Georgia Tech), Henry Cohn (Microsoft New England), Lionel Levine (Cornell), and Anne Schilling (UC Davis).

Local organizing committee: Ricky Liu (NCSU), Seth Sullivant (NCSU), and Cynthia Vinzant (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 U.S. citizens, and what is available to others still requires that the participants be employed at a U.S. university. To request funding please fill out the form here.

Saturday Triangle Lectures in Combinatorics Schedule:

9:15-10am, coffee and bagels
10-11am, Matthew Baker, Tropical Geometry and Torsor Structures on Spanning Trees
11-11:30am, coffee break
11:30am-12:30pm, Anne Schilling, Crystal approach to affine Schubert calculus
12:30-2:30pm, lunch break
2:30-3:30pm, Lionel Levine, Circles in the Sand
3:30-4pm, coffee break
4-5pm, Henry Cohn, The physics of error-correcting codes
5:30pm, somewhat informal conference dinner at David's Dumpling and Noodle Bar, a short walk from SAS Hall.

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: within short walk of the math department are several hotels, including the Doubletree by Hilton Raleigh - Brownstone (919-828-0811) and the Cameron Park Inn Bed and Breakfast (919-835-2171). About 1.5 miles away in downtown Raleigh (also walkable, but somewhat long walk) is the Clarion Raleigh Hotel (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.

Preregistered Participants (so far):

Henry Adams (Duke)
Justin Allman (Wake Forest)
Spencer Backman (Georgia Tech)
Matt Baker (Georgia Tech)
Daniel Bernstein (NCSU)
Brandon Bock (NCSU)
Timothee Bryan (NCSU)
Henry Cohn (Microsoft New England)
Gabor Hetyei (UNC Charlotte)
Swee Hong Chan (Cornell)
Suzanne Crifo (NCSU)
Vivek Dhand (Charlottesville, VA)
Amy Grady (Clemson)
Qijun He (Clemson)
Patricia Hersh (NCSU)
Gabor Hetyei (UNC Charlotte)
Andy Jenkins (Clemson)
Garrett Johnson (Wake Technical Community College)
David Lax (UNC Chapel Hill)
Lionel Levine (Cornell)
Ricky Liu (NCSU)
Colby Long (NCSU)
Olsen McCabe (U Kentucky)
Emily Meehan (NCSU)
Jed Mihalisin (Raleigh)
Ezra Miller (Duke)
Kailash Misra (NCSU)
Sayan Mukherjee (Duke)
Gabor Pataki (UNC Chapel Hill)
Lindsay Piechnik (High Point University)
Shira Polster (NCSU)
Robert Proctor (UNC Chapel Hill)
Nathan Reading (NCSU)
Carla Savage (NCSU)
Radmila Sazdanovic (NCSU)
Anne Schilling (UC Davis)
Daniel Scofield (NCSU)
Ryan Shifler (Virginia Tech)
Farbod Shokrieh (Cornell)
Michael Singer (NCSU)
Grace Stadnyk (NCSU)
Kara Stasikelis (Clemson)
Caprice Stanley (NCSU)
Ernie Stitzinger (NCSU)
Seth Sullivant (NCSU)
Ashleigh Thomas (Duke)
Cynthia Vinzant (NCSU)
Camron Withrow (Virginia Tech)
Chi Ho Yuen (Georgia Tech)
Anila Yadavalli (NCSU)

Talk Titles and Abstracts:

Matthew Baker (Georgia Tech)

Title: Tropical Geometry and Torsor Structures on Spanning Trees

Abstract: A connected finite graph G has an associated Jacobian group, which is a finite group Jac(G) whose cardinality is the number of spanning trees in G. It also has a tropical Jacobian, which is a real torus of dimension equal to the genus (or first Betti number) of G. The group Jac(G) is naturally a subgroup of the tropical Jacobian. There is a natural torsor for the tropical Jacobian which has a canonical decomposition into polyhedral cells, one for each spanning tree of G. Comparing the volume of the torus to the volumes of the individual cells, one obtains a new proof of the Matrix-Tree Theorem. The resulting picture contains a lot of other information as well. For example, the vertices of the cells are naturally a torsor for Jac(G), and translating these vertices by a generic vector makes the set of spanning trees into a torsor. For planar graphs, both the rotor-routing torsor of Chan-Church-Grochow and the Bernardi torsor of Baker-Yao admit this kind of geometric description. This picture also gives rise to some new families of combinatorial bijections between spanning trees and elements of the Jacobian group. This is joint work with Yang An, Greg Kuperberg, Farbod Shokrieh, and Chi Ho Yuen.

Henry Cohn (Microsoft Research)

Title: The physics of error-correcting codes

Abstract: Why are classical error-correcting codes such as Hamming, Golay, and Reed-Solomon codes so important and widely used? In this talk, based on joint work with Yufei Zhao, we'll explore how discrete models of physics shed light on this question. No special background in physics or coding theory will be assumed.

Lionel Levine (Cornell)

Title: Circles in the Sand

Abstract: I will describe the role played by an Apollonian circle packing in the scaling limit of the abelian sandpile (a.k.a. chip-firing) on the square grid Z^2. The sandpile solves a certain integer optimization problem. Associated to each circle in the packing is a locally optimal solution to that problem. Each locally optimal solution can be described by an infinite periodic pattern of sand, and the patterns associated to any four mutually tangent circles obey an analogue of the Descartes Circle Theorem. Joint work with Wesley Pegden and Charles Smart.

Anne Schilling (UC Davis)

Title: Crystal approach to affine Schubert calculus

Abstract: We apply ideas from crystal theory to affine Schubert calculus, flag Gromov—Witten invariants, positroid varieties, and Hall—Littlewood polynomials. By defining operators on certain decompositions of elements in the type-$A$ affine Weyl group, we produce a crystal reflecting the internal structure of Specht modules associated to permutation diagrams (for which the representatives are stable Schubert polynomials, or Stanley symmetric functions). We show how this crystal framework can be applied to study the product of a Schur function with a $k$-Schur function. Consequently, we prove that a subclass of 3-point Gromov—Witten invariants of complete flag varieties for $\mathbb{C}^n$ enumerate the highest weight elements under these operators. Included in this class are the Schubert structure constants in the (quantum) product of a Schubert polynomial with a Schur function $s_\lambda$ for all $|\lambda^c| <n$. Another by-product gives a highest weight formulation for fusion coefficients of the Verlinde algebra and our results apply to the Schubert decomposition of positroid varieties. This is joint work with Jennifer Morse (arXiv:1408.0320).