KAIST Discrete Math Seminar

By modifying the definition of moments of ranks and cranks, we study the odd moments of ranks and cranks. In particular, we prove the inequality between the first crank moment M₁(n) and the first rank moment N₁(n):</p>

We also study new counting function ospt(n) which is equal to M₁(n) – N₁(n). We will also discuss higher order moments of ranks and cranks.
This is a joint work with G. E. Andrews and S. H. Chan.</div>

Weakly separated set families were first studied by Leclerc and Zelevinsky in the context of quantum flag variety. Two quantum Plücker coordinates quasi-commute whenever their indexing sets are weakly separated. It was conjectured that maximal such families always have the same size. Similar question was asked by Scott when she studied quantum Grassmannian. These conjectures were independently proved by Danilov-Karzanov-Koshevoy and Oh-Postnikov-Speyer using some planar graphs and by the author using truncation. In this talk, definitions and motivations for the weakly separated set families will be explained, including Oh-Postnikov-Speyer’s point of view on the subject. The proof of the purity conjecture using truncation will be provided, and related questions will be discussed.

An intertwine of two graphs G and H is a graph that has both G and H as a minor and is minor-minimal with this property. In 1979, Lovász and Unger conjectured that for any two graphs G and H, there are only a finite number of intertwines. This now follows from the graph minors project of Robertson and Seymour, although no ‘elementary’ proof is known.
In this talk, we consider intertwining problems for matroids. Bonin proved that there are matroids M and N that have infinitely many intertwines. However, it is conjectured that if M and N are both representable over a fixed finite field, then there are only finitely many intertwines. We prove a weak version of this conjecture where we intertwine ‘connectivities’ instead of minors. No knowledge of matroid theory will be assumed.
This is joint work with Bert Gerards (CWI, Amsterdam) and Stefan van Zwam (Princeton University).

List of speakers</p>

• Tuesday 10:30AM-12PM Seunghyun Seo (서승현), Kangwon National University, Refined enumeration of trees by the size of maximal decreasing trees
• Tuesday 1:30PM-3PM HwanChul Yoo (유환철), KIAS, Specht modules of general diagrams and their Hecke counterparts
• Tuesday 4PM-5:30PM Heesung Shin (신희성), Inha University, q-Hermite 다항식을 포함하는 두 항등식에 관하여
• Wednesday 10:30AM-12PM Soojin Cho (조수진), Ajou University, Skew Schur P-functions
• Wednesday 1:30PM-3PM Sangwook Kim (김상욱), Chonnam National University, Flag vectors of polytopes

Imagine that you are cooking chicken at a party. You will cut the raw chicken fillet with a sharp knife, marinate each of the pieces in a spicy sauce and then fry the pieces. The surface of each piece will be crispy and spicy. Can you cut the chicken so that all your guests get the same amount of crispy crust and the same amount of chicken?
We show that if the number of guests is a prime power, n=p^k. Then such partition is possible. We derive this from a more general statement about equipartitions of convex bodies with respect to a measure and d-1 continuous functionals on the space of convex bodies, where d is the dimension the convex body sits in.
Our proof uses optimal transport and equivariant topology.