# Spring 2016

**Triangle Lectures in Combinatorics (TLC)**

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

**Location:** Sullivan Science Center 101, UNC Greensboro.

**Speakers:**

Penny Haxell (U. Waterloo)

Jeff Kahn (Rutgers)

Greta Panova (U. Penn)

Peter Winkler (Dartmouth)

**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:** 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:**

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.

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.

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.

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.