# Spring 2019

**Triangle Lectures in Combinatorics (TLC)**

**Seventeenth meeting:** Saturday, March 30, 2019

**Location:** Wake Forest University in Winston-Salem, NC

**Lecture Hall:** Manchester 16

**Speakers:** James Haglund (University of Pennsylvania), Adam Marcus (Princeton University), Jennifer Morse (University of Virginia), Nathan Reading (North Carolina State University)

**Registration and funding applications:**

To register and/or apply for funding, please fill out this **form**. If you have any questions, feel free to email Patricia Hersh at plhersh@ncsu.edu.

We are asking that participants pre-register if possible, as pre-registration is very helpful for planning our coffee breaks and obtaining funding to support these meetings. We have some funding available for some participants, especially for early-career participants. Most of this is restricted to US citizens and permanent residents, and what is available to others still requires that the participants be employed at a U.S. university. You will have the opportunity to apply for funding when registering for the conference.

**Conference schedule:**

9:15-10am, coffee and bagels

10-11am, Jennifer Morse, k-Schur functions in Catalania (slides)

11-11:30am, coffee break

11:30am-12:30pm, Nathan Reading, Lattice congruences of the weak order: Algebra, combinatorics, and geometry (slides)

12:30-2:30pm, lunch break (reservation at Zick’s)

2:30-3:30pm, Jim Haglund, Three faces of the Delta Conjecture (slides)

3:30-4pm, coffee break

4-5pm, Adam Marcus, Polynomial Techniques in Combinatorial Linear Algebra (slides)

5:30pm, informal conference dinner at Silo

**Parking:** Here is a marked campus map with parking suggestions and directions to the lecture hall.

**Hotel recommendations:** We have a block of rooms reserved at the Best Western Plus University Inn, which you can reserve through the link or by calling 1(800) 780-7234 and referring to the TLCC.

**Airport:** The nearest airport, for those flying, is the Triad Airport (serving Winston-Salem and Greensboro and High Point). The Raleigh-Durham airport is a 90+ minute drive away.

**Local organizing committee: **

Sarah Mason (Wake Forest), Ed Allen (Wake Forest), Patricia Hersh (NCSU), Cliff Smyth (UNC Greensboro)

**Participants:**

Edward Allen (Wake Forest University)

Jai Aslam (NCSU)

Cashous Bortner (NCSU)

Alex Chandler (NCSU)

Holly Paige Chaos (Wake Forest University)

Liza Escamilla (Lenoir-Rhyne University)

Timothy Goldberg (Lenoir-Rhyne University)

Ian Gossett (Wake Forest University)

Darij Grinberg (University of Minnesota)

Andres Guerrero-Guzman (Wake Forest University)

James Haglund (University of Pennsylvania)

Josh Hallam (Loyola Marymount University)

Sarah Helfert (Lenoir-Rhyne University)

Patricia Hersh (NCSU)

Gabor Hetyei (UNC Charlotte)

Ben Hollering (NCSU)

Daoji Huang (Cornell University)

Corrie Ingall (Wake Forest University)

Reed Jacobs

Joseph Johnson (NCSU)

Jongwon Kim (University of Pennsylvania)

Stephen Lacina (NCSU)

Ricky Liu (NCSU)

Xiaotian Liu (Wake Forest University)

Molly Lynch (NCSU)

Maria Macaulay (NCSU)

Adam Marcus (Princeton)

Colin Martin (Wake Forest University)

Sarah Mason (Wake Forest University)

Christopher McClain (WVU Tech)

Katherine Moore (Wake Forest University)

Jennifer Morse (University of Virginia)

Andrew Mosteller (Lenoir-Rhyne University)

Sarah Nelson (Lenoir-Rhyne University)

David Nichols (Wake Forest University)

Kathleen O’Hara

Lindsay Piechnik (High Point University)

Robert Proctor (UNC Chapel Hill)

Nathan Reading (NCSU)

Andrew Reeves (Lenoir-Rhyne University)

Hayley Russell (NCSU)

Dinesh Sarvate (College of Charleston)

Radmila Sazdanovic (NCSU)

Georgy Scholten (NCSU)

George Seelinger (University of Virginia)

Clifford Smyth (UNC Greensboro)

Carolyn Stephen (Wake Forest University)

Seth Sullivant (NCSU)

Cynthia Vinzant (NCSU)

Mirko Visontai (Google)

Michael Weselcouch (NCSU)

Seline Yang (Wake Forest University)

**Talk titles and abstracts:**

**Jennifer Morse ** (University of Virginia)

Title: k-Schur functions in Catalania

Abstract: We will discuss the inception, subsequent developments, and resolution of a symmetric function conjecture from the 1990’s. The k-Schur functions arose via computer experimentation with symmetric functions called Macdonald polynomials; they are symmetric functions with coefficients involving a t-parameter. Conjectures that k-Schur functions satisfy many strong and beautiful positivity properties compelled further study. In the special case when t=1, it was unexpectedly discovered that they are geometrically significant in an area called affine Schubert calculus and for computing Gromov-Witten invariants. However, the intricate combinatorics behind k-Schur functions involving the Bruhat order on the affine symmetric group made progress with generic t extremely hard to come by.

We recently discovered a new approach to the study of k-Schur functions; they are a subclass of Catalan functions, G-equivariant Euler characteristics of vector bundles on the flag variety defined by raising operators and indexed by Dyck paths. This perspective led us to settle decades old conjectures, providing tableaux enumeration formulas to do so.

Joint work with Blasiak, Pun, and Summers.

**Nathan Reading ** (North Carolina State University)

Title: Lattice congruences of the weak order: Algebra, combinatorics, and geometry

Abstract: The talk will begin with a crash course on congruences on a finite lattice, and the corresponding lattice quotients, and make the case that combinatorialists should care about them. To illustrate why I care, we’ll flip through some examples of lattice quotients of the weak order. I will sketch a complete combinatorial model for congruences on (and quotients of) the weak order on permutations and talk about how it generalizes to other Coxeter groups. The talk will conclude with a discussion of the connections to geometry, representation theory, and string theory and a mention of current and future work.

**Jim Haglund ** (University of Pennsylvania)

Title: Three faces of the Delta Conjecture

Abstract: The Delta Conjecture says that a certain symmetric function, expressed in terms of Macdonald polynomial operators, equals a weighted sum over Dyck lattice paths. It contains the well-known Shuffle Theorem of Carlsson and Mellit as a special case. There is also a third side to the problem that has emerged, centered around the goal of showing the symmetric function side also has a representation-theoretic interpretation. We will overview some of this work, including a recent conjecture of Mike Zabrocki which says the two sides of the Delta Conjecture equal the bigraded character of a generalization of the diagonal coinvariant ring. We will also discuss work of D’Adderio, Iraci, and Wyngaerd, who have used plethystic calculus to prove special cases of the Delta Conjecture.

**Adam Marcus ** (Princeton)

Title: Polynomial Techniques in Combinatorial Linear Algebra

Abstract: This talk will discuss recent innovations that use operations on polynomials to solve combinatorial problems in linear algebra. The most notable of these innovations is the “method of interlacing polynomials” which (as far as we know) is the only available technique for proving the existence of small probability eigenvalue events. I will then discuss recent applications of these techniques to problems in spectral graph theory, random matrix theory, and electrical engineering.