Twenty-sixth meeting: Saturday, April 19, 2025
Location: SAS 1102 at NC State
Parking: Free parking at the university during the weekend, starting on Friday at 5 pm. Google Maps directions for parking at SAS Hall.
Speakers:
Marcelo Aguiar (Cornell University)
Kyle Celano (Wake Forest University)
Ruth Luo (University of South Carolina)
Rosa Orellana (Dartmouth College)
Tentative Conference Schedule (all times in EST)
(Abstracts will be posted below)
09:00 - 10:00am, welcome
10:00 - 11:00am, Rosa Orellana
11:00 - 11:30am, coffee break
11:30 - 12:30pm, Ruth Luo
12:30 - 2:30pm, lunch break
02:30 - 3:30pm, Marcelo Aguiar
03:30 - 4:00pm, coffee break
04:00 - 5:00pm, Kyle Celano
Registration and Funding: Please, complete the following GoogleForm, which contains questions regarding funding too
Lunch break: There are many restaurants and take-out places around campus. In particular, in Hillsborough street you can find Bul-box, Lemon & lime, Guasaca, Chipotle, Jasmin & Olive, Player’s retreat, David’s dumplings, among others. Also in the Cameron Village, which is ~15 minutes walking-distance, you can find other places like CAVA, Cantina 18, Kale me crazy, and many more.
Organizing Committee: Laura Colmenarejo (NCSU) and Clifford Smyth (UNC Greensboro)
Funding provided by NSF
Titles and Abstracts
Marcelo Aguiar: Descents for cones of a real hyperplane arrangement
Reading a permutation from left to right and recording the positions where the values decrease yields the descent set of the permutation. More generally, descents may be attached to elements of a finite Coxeter group. In this talk we will discuss a different extension of the notion of descents, to the setting of real hyperplane arrangements. The notion on which this is based is the Tits product, an operation that exists on the set of faces of such an arrangement. The talk will center around this fundamental notion and include a discussion of faces, cones, and descents, together with lots of illustrations. No previous familiarity with hyperplane arrangements will be assumed.
Kyle Celano: A new proof that K^{-1}K=I
Of course, a matrix times its inverse is the identity. However, when the matrix and its inverse have combinatorial interpretations, one can ask for a combinatorial proof. The Kostka matrix K is the transition matrix between two symmetric function bases, and has a combinatorial interpretation in terms of semistandard Young tableaux. Eğecioğlu and Remmel produced a combinatorial proof of KK^{-1} = I, but were unable to find a proof for K^{-1}K=I. Partial progress on this was made by Sagan and Lee and the full proof by Loehr and Mendes of the identity requires the use of the bijective matrix algebra and the Eğecioğlu and Remmel map. In this talk, we present a full combinatorial proof of K^{-1}K=I independent of the Eğecioğlu and Remmel map utilizing a new combinatorial interpretation of K^{-1} found by Allen and Mason. This requires us to leave the land of symmetric functions and journey into noncommutative symmetric function territory before eventually returning home. Joint work with E. E. Allen and S. K. Mason.
Ruth Luo: Extremal problems for cycles in graphs and hypergraphs
Extremal combinatorics studies the maximum size a combinatorial object can have while forbidding a given substructure. Equivalently, it studies the minimum size of an object that forces the existence of the substructure. One of the most well studied areas in this field is that of Hamiltonian cycles in graphs. In the first half of this talk, we give a brief historical overview of extremal problems for cycles in graph theory. In the second half, we introduce analogues of cycles in hypergraphs called Berge cycles. We then discuss some recent developments in extremal hypergraph theory and how these relate to or differ from classical results for graphs.
Rosa Orellana: Reconstructing trees from their chromatic symmetric function
In 1995, Stanley introduced the chromatic symmetric function (CSF) as a generalization of the chromatic polynomial, which counts the number of proper colorings of a graph. A central question is determining what structural properties of a graph can be recovered from its CSF. Properties such as connectivity, the number of connected components, the presence of cycles, and the number of triangles are easily identifiable. However, while the CSF encodes many structural properties, it does not fully determine a graph, as non-isomorphic graphs with the same CSF exist—though all known examples contain cycles. The tree isomorphism conjecture posits that no two non-isomorphic trees share the same CSF. In this talk, I will discuss progress toward resolving this conjecture using the star basis of symmetric functions.