The Triangle Lectures in Combinatorics is a series of combinatorial workshops held each semester on a Saturday in the Research Triangle region of North Carolina, funded by the National Science Foundation.  The workshop this spring, the 15th installment of the Triangle Lectures in Combinatorics (TLC), will be hosted by North Carolina State University in Raleigh, North Carolina on Saturday, March 24, 2018.  It will include four one hour invited talks as well as coffee breaks and ample time for discussions throughout the day.

 

To apply for participant travel funding: See the conference web site,  http://www.math.ncsu.edu/TLC/   where there is a link to the application form.  The second round of funding awards will be made soon after February 25th.  It is also possible some additional applications for funding received somewhat beyond that date could also be funded, depending upon remaining availability of funds at that point.  

Practical information:  See the conference web site,  http://www.math.ncsu.edu/TLC/   where practical information (hotels, parking, campus map, etc.) may be found, and where more such information may be posted as the meeting gets closer.

Talk titles and abstracts:Richard Kenyon (Brown University)

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 University)

Title: Flow polytopes in combinatorics and algebra

Abstract: The flow polytope F_G(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,…, v_n) in Zⁿ. The points of F_G(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.

Ileana Streinu (Smith College)

Title: Characterizing 3D rigidity: combinatorial and geometric obstructions

Abstract: In spite of many conjectures and approaches spanning the last 30 years, a combinatorial characterization of 3D rigid graphs (generic bar-and-joint frameworks) remains elusive. I will discuss a few approaches, all of which fall short of solving the problem due to various combinatorial or geometric obstructions. They include “classical” results (the non-matroidal nature of the underlying “3n-6” sparsity counts and the failure of natural inductive constructions), as well as more recent ones: (1) the existence of arbitrarily large nucleation-free graphs (joint work with M. Sitharam and J. Cheng) and (2) a generalization from the finite to the periodic setting, in which a (rather unusual) definition of genericity has led to a combinatorial characterization via quotient graphs (joint work with C. Borcea).

Seth Sullivant (North Carolina State University)

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.

We would appreciate if you could forward this announcement to others you know who may be interested in participating.

We hope you will be able to attend!