**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 plhersh@ncsu.edu (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 plhersh@ncsu.edu (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)