Triangle Lectures in Combinatorics (TLC)
Eighth meeting: September 21, 2013
Location: North Carolina State University
Lecture Hall: SAS Room 1102

Slides from the talks:

Recent meeting: September 21, 2013

Location: North Carolina State University

Lecture Hall: SAS Room 1102

Speakers: George Andrews (Penn State), Matt Beck (San Francisco State University), Robin Pemantle (University of Pennsylvania), and Victoria Powers (Emory University)

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 apply for funding, send email to (Patricia Hersh) including: (1) name, (2) affiliation, (3) career stage (e.g. grad student, assistant professor, etc.), (4) mileage (if driving) or expected airfare, (5) expected hotel cost (and whether this is for 1 or 2 nights), other expenses, (6) other expenses (e.g. taxi to/from airport), (7) citizenship/residency status, (8) other sources of funding you have available if we cannot provide full expense reimbursement, and (9) a sentence or two of justification for why it will be valuable for you to attend. We will make funding decisions about once per month while funds still remain.

Friday Schedule:

Friday Afternoon NCSU Math Colloquium: George Andrews, The enigmatic lost notebook of Ramanajan, followed by Colloquium Dinner.

Saturday Triangle Lectures in Combinatorics Schedule:

9:15-10am, coffee and bagels
10-11am, Matthias Beck
11-11:30am, coffee break
11:30am-12:30pm, Robin Pemantle
12:30-2:30pm, lunch break
2:30-3:30pm, Victoria Powers
3:30-4pm, coffee break
4-5pm, George Andrews
Saturday evening, somewhat informal conference dinner

Suggested hotels and parking information: some recommended hotels are Doubletree by Hilton, Brownstone (easy walk to conference at 1707 Hillsborough, 919-828-0811), Cameron Park Inn B and B (also easy walk but more expensive at 211 Groveland Ave, 919-835-2171), Clarion Hotel (downtown hotel at 320 Hillsborough, 919-832-0501). Parking is available for free in the lot adjacent to SAS Hall just to the east of SAS Hall.

Preregistered participants (so far):

Geir Agnarsson, George Mason University
George Andrews, Penn State
Melissa Bechard, U South Carolina
Matthias Beck, San Francisco State University
Daniel Bernstein, NCSU
Sarah Birdsong, UNC Charlotte
Brandon Bock, NCSU
Ed Boehnlein, U South Carolina
Timothee Bryan, NCSU
Yue Cai, U Kentucky
Shihwei Chao, Clemson
Ann Clifton, U South Carolina
Garner Cochran, U South Carolina
Noah Daleo, NCSU
Lola Davidson, U. Kentucky
Ruth Davidson, NCSU
Aaron Dutle, U South Carolina
Hamza Ghadyali, Duke
Nicole Gin, NCSU
Anant Godbole, East Tennessee State University
Brent Gorbutt, George Mason University
Amy Grady, Clemson
Alexis Harper, NCSU
Jonathan Hauenstein, NCSU
Qijun He, Clemson
Dustin Hedmark, U Kentucky
Patricia Hersh, NCSU
Gabor Hetyei, UNC Charlotte
Chetak Hossain, NCSU
Wesley Hough, U Kentucky
Travis Johnston, U South Carolina
Austin Jones, NCSU
JiYoon Jung, Marshall University
Florian Kohl, U Kentucky
Darla Kremer
David Lax, UNC Chapel Hill
Shirley Law, Washington College (MD)
Alan Liddell, NCSU
Jack Love, George Mason University
Matthew Macaulay, Clemson
Emily Meehan, NCSU
Sam Mendelson, George Mason University
Ezra Miller, Duke
Kailash Misra, NCSU
Michael Mossinghoff, Davidson College
Sarah Nelson, U Kentucky
Asamoah Nkwanta, Morgan State University
Kathy O'Hara
Chris O'Neill, Duke
Sarah Nelson, U Kentucky
Robin Pemantle, U Penn
Lindsay Piechnik, High Point University
Shira Polster, NCSU
Victoria Powers, Emory
Svetlana Poznanovikj, Clemson
Nathan Reading, NCSU
Richard Rimanyi, UNC Chapel Hill
Danny Rorabaugh, U South Carolina
Joseph Rusinko, Winthrop University
Carla Savage, NCSU
Radmila Sazdanovik, NCSU
Taylor Short, U South Carolina
Michael Singer, NCSU
Heather Smith, U South Carolina
Liam Solus, U Kentucky
Salvatore Stella, NCSU
Blair Sullivan, NCSU
Seth Sullivant, NCSU
Hayato Ushijima-Mwesigwa, Clemson
Ryan Vinroot, College of William and Mary
Mirko Visontai, KTH SWEDEN
Josephine Yu, Georgia Tech

Talk titles and abstracts (received so far):

Speaker: George Andrews, Penn State University

Title for TLC talk: Seventh order mock theta functions and beyond

Abstract: In 1986, Hecke type expansions (involving indefinite quadratic forms) were found for the fifth and seventh order mock theta functions of Ramanujan. The treatment of the fifth order functions relied on the classical q-hypergeometric series hierarchy. The treatment of the seventh order functions relied on ad hoc recurrence arguments. In this talk we discuss a recent, unified approach to these problems which has the advantage of introducing an infinite family of results whose first two instances are the fifth and the seventh order mock theta functions.

Title for George Andrews' Colloquium: The enigmatic lost notebook of Ramanajan

Abstract: In 1976, I visited the Trinity College Library in Cambridge to examine papers from the estate of G.N. Watson. In one box, there was a collection of more than 100 pages written by Ramanujan. Examination of the pages revealed that this work must have been written during the last year of his life in 1919-1920. I dubbed this manuscript __ Ramanujan's Lost Notebook. Up until that moment, the only information available about this time in Ramanujan's life came from a single letter that Ramanujan wrote to Hardy in early 1920. This incredible document and its quite amazing mathematics has formed a major theme in my life. In this talk, I will discuss some of the results in the Lost Notebook and their subsequent implications. If time allows, I will describe a few of the related non-mathematical incidents that have taken place.

Speaker: Matthias Beck, San Francisco State University

Title: Graph polynomials and their geometry

Abstract: Enumerative graph theory is abundant with polynomials. For example, one proves in any introductory graph theory course that the number of proper k-colorings of a given graph G is a polynomial in k, the chromatic polynomial of G. Many graph polynomials yield surprising connections; in the case of the chromatic polynomial, these include evaluations that give the number of acyclic orientations of a graph and a decomposition formula in terms of order polynomials of certain partially ordered sets. We will survey old and new theorems on graph polynomials. Our goal is to weave a unifying thread through various results on graph polynomials by looking at them through the lens of discrete geometry.

Speaker: Robin Pemantle, U. Penn (joint work with Rick Kenyon)

Title: Double-dimer configurations and the hexahedron recurrence

Abstract: This talk concerns the hexahedron recurrence, a new recurrence relation in Z^3 which is similar to the octahedron and cube recurrences. Solutions to the hexahedron recurrence are partition functions for configurations on a certain graph, and have a natural interpretation in terms of cluster algebras. There is an explicit correspondence between monomials in the Laurent expansions arising in the recurrence with double-dimer configurations on a certain graph. We compute limit shapes for these double-dimer configurations. As a special case, we recover the Kashaev difference equation arising in the Ising model star-triangle relation. This reveals the cluster nature underlying the Ising model and implies a Laurent phenomenon for the Kashaev difference equation.

Speaker: Victoria Powers, Emory

Title: Certificates of Positivity: Theory, Practice, and Applications

Abstract: Suppose S is a subset of R^n and a polynomial f in R[x] = R[x_1,...,x_n] is positive, or nonnegative, on S. By a certificate of positivity for f on S we mean an algebraic expression for f, usually involving sums of squares of real polynomials, from which one can deduce the positivity condition immediately. For example, if f is a sum of squares in R[X], then f is nonnegative on all of R^n and a representation f = g_1^2 + ... + g_k^2 is a certificate of positivity for f on R^n.

This talk concerns the theory and practice of certificates of positivity, i.e., results about the existence of certificates and results about finding and counting certificates. We will give the history of the subject, starting with Hilbert's seminal work in the late 19th century, and describe some of the many applications.

Conference organizing committee: Michael Singer (NCSU), Carla Savage (NCSU), and Seth Sullivant (NCSU)