Triangle Lectures in Combinatorics (TLC)

Thirteenth meeting: Saturday February 27, 2016, 9:15am -- 6pm.

Location: Sullivan Science Center 101, UNC Greensboro.

Penny Haxell (U. Waterloo)
Jeff Kahn (Rutgers)
Greta Panova (U. Penn)
Peter Winkler (Dartmouth)

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: All of the participant funding awards for this meeting have already been made at this time.

Conference schedule:
9:15-10am, coffee and small breakfast
10-11am, Jeff Kahn,  Thresholds and "thresholds"
11-11:30am, coffee break
11:30am-12:30pm, Greta Panova,  Lattice models and symmetric functions
12:30-2:30pm, lunch break
2:30-3:30pm, Peter Winkler,  Permutons
3:30-4pm, coffee break
4-5pm, Penny Haxell,  Matchings in hypergraphs

Logistical information: Here is an information packet with suggested restaurants, a campus map, etc.

The closest airport is Piedmont Triad, which is 10 miles from the university. Other possible airports are Charlotte Douglas (100 miles) or Raleigh-Durham (66 miles). We recommend the Biltmore hotel in downtown Greensboro (1 mile from campus) or the Courtyard Marriott on Wendover Avenue (7 miles from campus). There are many other decent hotels along I-40 a few miles south of campus. We would appreciate if you could go ahead and make your own reservations.

Parking: Parking will be in the McIver Deck. Attendees will need to pick up an arrival parking card from the gate upon entering, but the exit gate will automatically rise up when they go to leave. (No parking pass or payment will be needed for them to exit.)

Preregistered Participants (so far):
Alie Alhajjar, George Mason U.
Ed Allen, Wake Forest
Khalid Almnesi, Morgan State U.
Abdullah Alqarni, Morgan State U.
Zeinab Bandpey, Morgan State U.
Emily Barnard, NCSU
Fidel Barrera-Cruz, Georgia Tech
Karthik Chandrasekhar, U Kentucky
Jue Cheng, Wake Forest
Greg Clark, USC
Joshua Cooper, USC
Ruth Davidson, UIUC
Brian Davis, U Kentucky
Chris Edgar, USC Columbia
David Galvin, Notre Dame
Mark Ginn, Appalachian State U.
Anant Godbole, ETSU
Jerry Griggs, USC
He Guo, Georgia Tech
Joshua Hallam, Wake Forest U.
Arran Hamm, Winthrop U.
Penny Haxell, U. Waterloo
Patricia Hersh, NCSU
Gabor Hetyei, UNC Charlotte
Shilpa Jayarajan, Wake Forest
Ashley Jones, UNC Greensboro
Jeff Kahn, Rutgers
Lauren Keough, Davidson College
Julie Lang, USC
Ricky Liu, NCSU
Sarah Mason, Wake Forest
Olsen McCabe, U Kentucky
Thomas McConville, MIT
Marie Mickley, Wake Forest
Walter Morris, George Mason U.
Michael Mossinghoff, Davidson
Kathy O'Hara
Greta Panova, U Penn
Gabor Pataki, UNC Chapel Hill
Josiah Reiswig, USC
James Rudzinski, UNC Greensboro
Sandi Rudzinski, UNC Greensboro
Aaron Shank, UNC Greensboro
Alexander Sharif UNC Greensboro
Michael Singer, NCSU
Caprice Stanley, NCSU
Michael Strayer, UNC Chapel Hill
Spencer Saunders, USC
Heather Smith, Georgia Tech
Jeremiah Southwick, Wake Forest
Anton Strizhov, USC
Seth Sullivant, NCSU
Cliff Smyth, UNC Greensboro
Rupei Xu, UT Dallas
Zhiyu Wang, USC
Hays Whitlatch, USC
Peter Winkler, Dartmouth
Xiaowei Wu, UNC Greensboro

Organizing committee: Clifford Smyth, chair (UNC Greensboro), David Galvin (Notre Dame), and Patricia Hersh (NCSU)

Talk titles and abstracts:

Penny Haxell (U. Waterloo)

Title: Matchings in hypergraphs

Abstract: A matching in a hypergraph is a set of disjoint edges. It is a well-known difficult problem to give good lower bounds on the maximum size of a matching in a hypergraph in terms of other natural parameters. Here we discuss tools for this, with a focus on the special case of tripartite hypergraphs: those for which the vertex set can be partitioned into three parts, such that each edge contains exactly one vertex from each part. For example, if a tripartite hypergraph is r-regular (meaning that each vertex is in exactly r edges) with n vertices in each class then it has a matching of size at least n/2, and this is tight for certain special hypergraphs. We investigate how this bound can be improved for all other hypergraphs.

Jeff Kahn (Rutgers)

Title: Thresholds and "thresholds"

Abstract: For a family F of subsets of a finite set X, we're interested in understanding when a "p-random" subset Y of X (gotten by retaining elements independently with probability p) is likely to lie in F. Traditionally one takes F to be increasing (closed under taking supersets), in which case the probability that Y belongs to F increases with p, but there are also many natural families that don't quite fit this framework. We'll try to give some feel for this subject and mention some of the recent progress, or lack thereof.

Greta Panova (U. Penn)

Title: Lattice models and symmetric functions

Abstract: We will discuss lattice models arising from statistical mechanics: lozenge tilings (dimer covers of a hexagonal lattice/plane partitions in special cases), 6-vertex model, dense loop model. They are rich in combinatorial and probabilistic properties. We will describe their limiting behavior as the lattice size goes to 0 and see the arising phenomena -- limit shapes (surfaces), arctic circles, distributions from Random Matrices near the boundary. Proving such probabilistic properties brings in yet another field in play. It can be done with some asymptotic analysis of symmetric functions -- objects from representation theory and algebraic combinatorics, and related combinatorial models of Young tableaux, plane partitions and Gelfand-Tsetlin schemes. Thanks to these methods, we can also consider lozenge tilings with various global symmetries and recover the same probabilistic behavior.

Peter Winkler (Dartmouth)

Title: Permutons

Abstract: Permutons are limit objects for permutations---technically, a permuton is a probability measure on the unit square with uniform marginals. By finding a permuton that maximizes a certain integral, we (with Rick Kenyon, Dan Kral and Charles Radin) can (sometimes) count and describe large permutations that satisfy certain properties---for example, permutations that have specified pattern densities.