# TLC Fall 2023

**Twenty-third meeting: Saturday, November 18, 2023**

** Location: **SAS 1102, North Carolina State University, Raleigh

**Parking:**Local parking on SAS is free during the event

**Speakers: **

David Galvin (University of Notre Dame), Pamela E. Harris (University of Wisconsin at Milwaukee), Jianping Pan (NCSU), and Fan Wei (Duke University)

** Tentative Conference Schedule** (all times in EST)

(Abstracts can be found below)

09:00 – 10:00am, welcome

10:00 – 11:00am,

**David Galvin**

11:00 – 11:30am, coffee break

11:30 – 12:30pm,

**Fan Wei**

12:30 – 2:30pm, lunch break

02:30 – 3:30pm,

**Jianping Pan**

03:30 – 4:00pm, coffee break

04:00 – 5:00pm,

**Pamela E. Harris**

** Registration:**To register, please fill out this form. This form includes a section with questions regarding funding.

** Lunch break: **There are many restaurants and take-out places around campus. In particular, in

**you can find Bul-box, Lemon & lime, Guasaca, Chipotle, Jasmin & Olive, Player’s retreat, David’s dumplings, among others. Also in the**

*Hillsborough street***, which is ~15 minutes walking-distance, you can find other places like CAVA, Cantina 18, Kale me crazy, and many more.**

*Cameron Village*** Organizing Committee:** Laura Colmenarejo (NCSU) and Clifford Smyth (UNC Greensboro)

### Titles and Abstracts

**David Galvin:** Counting independent sets and colorings

In the late 1980’s, motivated by a problem from combinatorial group theory, Andrew Granville asked “at most how many independent sets can a regular graph admit?” This has proven to be a remarkably fruitful question, and one amenable to a remarkable variety of techniques, including graph containers, entropy and linear programming.

Although Granville’s initial question has by now been completely resolved (first by Kahn, for regular bipartite graphs, then by Zhao for all regular graphs), many related open questions remain.

I’ll discuss this problem, and some of its relatives, including maximizing the count of H-colorings of a regular graph, minimizing the count of H-colorings of a tree, and maximizing the count of independent sets in a regular graph in the presence of a bound on the largest independent set in the graph.

**Pamela E. Harris: **Finding needles in haystacks: Boolean intervals in the weak order of $\mathfrak{S}_n$

Finding and enumerating Boolean intervals in $W(\mathfrak{S}_n)$, the weak order of symmetric group $\mathfrak{S}_n$, can feel like trying to find needles in a haystack. However, through surprising connection to the outcome map of parking functions we provide a complete characterization and enumeration for Boolean intervals in $W(\mathfrak{S}_n)$. We show that for any $\pi\in\mathfrak{S}_n$, the number of Boolean intervals in $W(\mathfrak{S}_n)$ with minimal element $\pi$, is a product of Fibonacci numbers. This is joint work with Jennifer Elder, Jan Kretschmann, and J. Carlos MartĂnez Mori.

**Jianping Pan**: The Many Facets of Permutations

Permutations are fundamental objects in mathematics, biology and many other fields. Different presentations reveal different aspects of permutations, for instance, the one-line notation describes how a permutationâ€™s pattern goes up and down while the reduced words highlight their construction through adjacent transpositions. In this talk, I will talk about certain tableaux, posets, and polytopes from permutations and the interplay between them. This talk is based on joint works with Banaian, Chepuri, Gunawan$^3$, Russel$^2$ and Tenner$^2$.

**Fan Wei: **Graph homomorphism density inequalities

Graph limits is a recently developed powerful theory in studying graphs from a continuous perspective. In this talk, we will show it is undecidable to prove general inequalities of homomorphism densities into (edge weighted) graphs. Then I will show how the perspective of graph limits helps with graph homomorphism inequalities and how to make advances in a common theme in extremal combinatorics: when does randomness give nearly optimal bounds?