Triangle Lectures in Combinatorics (TLC)

These are a series of combinatorial workshops, held once per semester, each on a Saturday. The series began with its first meeting in spring 2010. They rotate among the universities in and around the Research Triangle. Participants come from numerous colleges and universities within a few hours drive, and some from even farther away. These workshops are funded by the National Science Foundation and the National Security Agency, in particular enabling us to bring in four exciting speakers to give one hour talks each time as well as funding travel expenses for participants.

TLC steering committee: Patricia Hersh (NCSU), Ezra Miller (Duke), Gabor Pataki (UNC Chapel Hill), Scott Provan (UNC Chapel Hill), Nathan Reading (NCSU), and Seth Sullivant (NCSU).

Upcoming meeting: Saturday October 3, 2015, 9:15am -- 5pm.

Location: Duke University (Physics building, room 128).

Speakers: Yuliy Baryshnikov (UIUC), Susan Holmes (Stanford), Dana Randall (Georgia Tech), and John Shareshian (Washington University in St. Louis).

Organizing committee: Sayan Mukherjee, chair (Duke), John Harer (Duke), Patricia Hersh (NCSU), and Mauro Maggioni (Duke).

Recent 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).

Recent meeting: Saturday October 4, 2014, 9:15am -- 5pm.

Location: High Point University in High Point, North Carolina (one hour west of the Research Triangle by car).

Lecture Hall: 206 Cogdon Hall MAP (The map also contains parking information.)

Speakers: Richard Ehrenborg (Kentucky), Robert Ghrist (Penn), Luis Serrano (U Quebec a Montreal), and Rekha Thomas (U. Washington).

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 funding: We have funds available for travel and hotel for participants, especially for early career participants, thanks to grants from the NSF and the NSA. Much of this is restricted to U.S. 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 fill out the form here: Travel Funding Request Form If you have questions, please contact Nathan Reading (

Saturday Triangle Lectures in Combinatorics Schedule:

9:15-10am, coffee and bagels
10-11am, Rob Ghrist, Poincare Duality in Network Flow Optimization
11-11:30am, coffee break
11:30am-12:30pm, Luis Serrano, The immaculate basis of the non-commutative symmetric functions
12:30-2:30pm, lunch break
2:30-3:30pm, Rekha Thomas, The Euclidean distance degree of an algebraic variety
3:30-4pm, coffee break
4-5pm, Richard Ehrenborg, Euler enumeration
6pm, somewhat informal conference dinner at Sumela

Logistical information: The nearest airport is Greensboro/High Point (GSO), 20 minutes drive away. This is also known as the Piedmont Triad Airport. Within 90 minutes drive are both Raleigh-Durham (RDU) Airport and Charlotte (CLT) Airport. A cab from GSO costs about $30 each way. Hotels are about 1.5 to 3 miles from campus. Thus, a car rental may make sense for anyone flying to the meeting.

A block of hotel rooms has been reserved at the Courtyard Marriott High Point (336-882-3600), which is 1.7 miles from campus. The rooms are blocked off under "The TLC" until September 19th. However, if rooms are still available, they will honor our group rate of $119 per night (+tax) after this date. (The room includes two queen beds, wireless internet, and either a hot breakfast sandwich or two continental breakfast items, plus beverage, for 2 adults). There are many other local options including: -High Point Plaza , 1 mile from campus (more downtown). 336-889-8888 -Crestwood Suites, about 2.7 miles from campus (more economical). 336-886-5665.

Preregistered Participants (so far):

Geir Agnarsson, George Mason University
Ed Allen, Wake Forest
Taylor Allison, UNC Chapel Hill
Justin Allman, Virgina Tech
Jennifer Anderson, Marshall University
Emily Barnard, NCSU
Yue Cai, U Kentucky
Dustin Cartwright, U Tennessee
Shihwei Chao, Clemson University
Paul Cubre, Clemson University
Brian Davis, U Kentucky
Robert Davis, U Kentucky
Rafael S. Gonzalez D'Leon, U Kentucky
Anahita Davoudi, U Central Florida
Vivek Dhand, Charlottesville, VA
Richard Ehrenborg, U Kentucky
Norman Fox, U Kentucky
Amy Grady, Clemson University
Rob Ghrist, U Penn
Alex Happ, U Kentucky
Isaiah Harney, U Kentucky
Qijun He, Clemson
Patricia Hersh, NCSU
Qijun He, Clemson University
Cyrus Hettle, U Kentucky
Gabor Hetyei, UNC Charlotte
Elliot Hollifield, Wake Forest U
Chetak Hossain, NCSU
Wesley Hough, U Kentucky
Andy Jenkins, Clemson University
Yvonne Kemper, NIST
Florian Kohl, U Kentucky
Rebecca Kotsonism, Wake Forest U
David Lax, UNC Chapel Hill
David Mason, UNC Chapel Hill
Sarah Mason, Wake Forest
Marie Meyer, U Kentucky
Frank Moore, Wake Forest U
John Mosley, U Kentucky
Sayan Mukherjee, Duke
Vidit Nanda, U Penn
Yusra Naqvi, Rutgers
Sarah Nelson, U Kentucky
Elizabeth Niese, Marshall University
Jason Parsley, Wake Forest U
Gabor Pataki, UNC Chapel Hill
Svetlana Poznanovikj, Clemson U
Sarah Orchard, U Kentucky
Lindsay Piechnik, High Point
Richard Rimanyi, UNC Chapel Hill
Radmila Sazdanovic, NCSU
Luis Serrano, UQAM
Cliff Smyth, UNC Greensboro
Liam Solus, U Kentucky
Jeremiah Southwich, Wake Forest U
Kara Stasikelis, Clemson University
Seth Sullivant, NCSU
Breeanne Swart, The Citadel
Clifford Taylor, U Kentucky
Rekha Thomas, U Washington
Tzvetalin Vassilev, Nippising U.
Devin Wilmott, U Kentucky
Martha Yip, U Kentucky
Laurie Zack, High Point

Talk titles and abstracts:

Richard Ehrenborg (U Kentucky)

Title: Euler enumeration

Abstract: The flag vector contains all the face incidence data of a polytope, and in the poset setting, the chain enumerative data. It is a classical result due to Bayer and Klapper that for face lattices of polytopes, and more generally, Eulerian graded posets, the flag vector can be written as a cd-index, a non-commutative polynomial which removes all the linear redundancies among the flag vector entries. This result holds for regular CW complexes. We relax the regularity conditions to show the cd-index exists for manifolds whose boundary has a Whitney stratification. The setting of Whitney stratifications allows us to give shorter proofs of identities involving the cd-index and opens inequality questions for manifolds. This is joint work with Mark Goresky and Margaret Readdy.

Rob Ghrist (U Pennsylvania)

Title: Poincare Duality in Network Flow Optimization

Abstract: One of the classical cornerstones of optimization theory is LP (linear programming) duality, and one of its simplest applications is to the classical max-flow-min-cut theorem, which expresses a duality between optimal network flow values and optimal cut capacities. This talk argues that flow-cut duality is really topological in nature -- an expression of Poincare duality. A recent proof by S. Krishnan shows that Poincare duality on sheaves of semimodules implies a sheaf-theoretic extension of the classical max-flow-min-cut theorem. This greatly expands the types of problems to which flow-cut dualities can be applied, as well as explains the source of duality gaps in more advanced settings. This talk will give details.

Luis Serrano (U Quebec a Montreal)

Title: The immaculate basis of the non-commutative symmetric functions

Abstract: We introduce a new basis of the non-commutative symmetric functions whose elements have Schur functions as their commutative images. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions and decompose Schur functions according to a signed combinatorial formula. These bases have many interesting properties similar to those of the Schur basis, and we will outline a few of them.

Rekha Thomas (U Washington)

Title: The Euclidean distance degree of an algebraic variety

Abstract: It is a common problem in optimization to minimize the Euclidean distance from a given data point u to some set X. In this talk I will consider the situation in which X is an algebraic variety, a common situation in practice. The number of critical points of the objective function on X is called the Euclidean distance degree of X, and is an intrinsic measure of the complexity of this polynomial optimization problem. Using algebraic geometry we obtain formulas and methods to calculate this degree in many situations. I will explain the results and illustrate the formulas that can be obtained in several situations ranging from matrix analysis to control theory to computer vision. Joint work with Jan Draisma, Emil Horobet, Giorgio Ottaviani and Bernd Sturmfels.

Organizing Committee: Lindsay Piechnik (chair, High Point), Ed Allen (Wake Forest), Sarah Mason (Wake Forest), Seth Sullivant (NCSU), and Laurie Zack (High Point)

Past meetings:

Ninth meeting: February 22, 2014 at UNC-Chapel Hill.
Speakers: Shankar Bhamidi (UNC-Chapel Hill), Anders Buch (Rutgers), Pablo Parrilo (MIT), Eva Tardos (Cornell)
Organizing committee: Gabor Pataki (UNC-Chapel Hill), Lindsay Piechnik (High Point University), Scott Provan (UNC-Chapel Hill), Richard Rimanyi (UNC-Chapel Hill), Jack Snoeyink (UNC-Chapel Hill)
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Eighth meeting: September 21, 2013 at NCSU.
Speakers: George Andrews (Penn State), Matt Beck (San Francisco State University), Robin Pemantle (University of Pennsylvania), and Victoria Powers (Emory University)
Organizing committee: Michael Singer (NCSU), Carla Savage (NCSU), and Seth Sullivant (NCSU)
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Seventh meeting: February 9, 2013 at Wake Forest University.
Speakers: Louis Billera (Cornell), Rod Canfield (University of Georgia), Matthew Kahle (Ohio State University), Michelle Wachs (University of Miami)
Organizing committee: Sarah Mason (chair, Wake Forest University), Ed Allen (Wake Forest University), Alex Fink (NCSU), Patricia Hersh (NCSU)
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Sixth meeting: September 22, 2012 at NCSU.
Speakers: Allen Knutson (Cornell), Vin de Silva (Pomona College), Richard Stanley (MIT), Lauren Williams (UC Berkeley).
Organizing committee: Alex Fink (NCSU), Patricia Hersh (NCSU), Carla Savage (NCSU).
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Fifth meeting: February 11, 2012 at Duke University.
Speakers: Alex Fink (NCSU), Sergey Fomin (Michigan), Nets Katz (Indiana University), Isabella Novik (University of Washington).
Organizing Committee: Christine Berkesch (Duke), Sonja Mapes (Duke), Ezra Miller (Duke).
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Fourth meeting: November 5, 2011 at UNC Chapel Hill.
Speakers: Thomas Lam (Michigan), Jesus De Loera (UC Davis), Ezra Miller (Duke), Doron Zeilberger (Rutgers)
Organizing committee: Prakash Belkale (UNC Chapel Hill), Gabor Pataki (UNC Chapel Hill), Robert Proctor (UNC Chapel Hill), Scott Provan (UNC Chapel Hill), Richard Rimanyi (UNC Chapel Hill).
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Third meeting: April 9, 2011 at NCSU.
Speakers: Prakash Belkale (UNC Chapel Hill), Vic Reiner (University of Minnesota), John Stembridge (University of Michigan), Stephanie van Willigenburg (UBC).
Organizing committee: Hoda Bidkhori (NCSU), Alex Fink (NCSU), Patricia Hersh (NCSU), Carla Savage (NCSU).
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Second meeting: September 25, 2010 at Duke.
Speakers: Alexander Barvinok (University of Michigan), Anne Shiu (Duke), Sami Assaf (MIT), Persi Diaconis (Stanford).
Organizing committee: Patricia Hersh (NCSU), Sonja Mapes (Duke), Ezra Miller (Duke).
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First meeting: February 6, 2010 at NCSU.
Speakers: Carla Savage (NCSU), Bernd Sturmfels (UC Berkeley), Ed Swartz (Cornell), Laszlo Szekely (University of South Carolina).
Organizing committee: Patricia Hersh (NCSU), Ezra Miller (Duke), Scott Provan (UNC) Nathan Reading (NCSU).
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