The Berkeley/Davis Combinatorics Gatherings are meetings held twice a year between the combinatorics communities at UC Berkeley and UC Davis. The purpose is to foster communication and collaboration between the departments. Thirty-minute talks are given by graduate students, postdocs, or faculty from each university, and plenty of time is allotted for sharing and working on open problems throughout the day.

The host institution alternates between Berkeley and Davis.


Location: 1015 Evans Hall, UC Berkeley
Time: Saturday, January 23rd, 2016. 10:00 AM -- 5:00 PM.
Organizers: Maria Monks Gillespie (monks {at} math.berkeley.edu) and Federico Castillo (fcastillo {at} math.ucdavis.edu)


9:00 -- 10:00 Welcome and Refreshments
10:00 -- 10:30 Dehn--Sommerville relations and the Catalan matroid Anastasia Chavez
10:30 -- 11:00 The influence of the Kirillov-Reshetikin crystal $B^{1,1}$ on the structure of simple cyclotomic KLR modules Henry Kvinge
11:00 -- 12:00 Open problems proposals Anyone with a problem!
12:00 -- 2:00 Problem solving session with lunch Small groups
2:00 -- 2:30 Elliptic Hall algebra and knots Eugene Gorsky
2:30 -- 3:00 Computing Linear Systems on Metric Graphs Bo Lin
3:00 -- 4:00 Tea
4:00 -- 5:00 Presentations on progress from afternoon

Confirmed UC Berkeley Participants

  • Madeline Brandt
  • Anastasia Chavez
  • Nic Ford
  • Bryan Gillespie
  • Maria Monks Gillespie
  • Steven Karp
  • Bo Lin
  • Olya Mandelshtam
  • Jeremy Meza
  • Christopher Miller
  • Khrystyna Serhiyenko
  • Emmanuel Tsukerman
  • Nicole Yamzon

Confirmed UC Davis Participants

  • Federico Castillo
  • Eugene Gorsky
  • Graham Hawkes
  • Oscar Kivinen
  • Henry Kvinge
  • Kirill Paramonov
  • Gicheol Shin
  • Lily Silverstein

Other Participants

  • John Guo (SF State)


Anastasia Chavez, Dehn--Sommerville relations and the Catalan matroid. The f-vector of a d-dimensional polytope P stores the number of faces of each dimension. When P is simplicial the Dehn--Sommerville relations condense the f-vector into the g-vector, which has length equal to the ceiling of (d+1)/2. Thus, to determine the f-vector of P, we only need to know approximately half of its entries. This raises the question: Which ceil((d+1)/2)-subsets of the f-vector of a general simplicial polytope are sufficient to determine the whole f-vector? We prove that the answer is given by the bases of the Catalan matroid.

Henry Kvinge, The influence of the Kirillov-Reshetikin crystal $B^{1,1}$ on the structure of simple cyclotomic KLR modules. Khovanov-Lauda-Rouquier (KLR) algebras were invented to categorify the negative half of the quantum Kac-Moody algebra associated to a symmetrizable Cartan data. It was later shown by Lauda-Vazirani that the simple modules of the cyclotomic KLR algebra, $R^{\Lambda}$, carry the structure of the highest weight crystal $B(\Lambda)$. It follows from this that any properties of $B(\Lambda)$ should be the shadow of some module-theoretic property of simple $R^{\Lambda}$-modules. In classical affine type, highest weight crystals (which are infinite) have the remarkable property that they can be constructed from the tensor product of the much more tractable perfect crystals (which are finite). In this talk I will describe the algebraic analogue of this phenomenon in terms of simple $R^{\Lambda}$-modules in the case where the perfect crystal is the Kirillov-Reshetkhin crystal $B^{1,1}$ and $\Lambda$ is the fundamental weight $\Lambda_i$. This is joint work with Monica Vazirani.

Eugene Gorsky, Elliptic Hall algebra and knots. Elliptic Hall algebra, introduced by Burban and Schiffmann, is an interesting algebra of operators on symmetric functions. I will describe this algebra and its connections to q,t-Catalan numbers and to polynomial invariants of torus knots. The talk is based on a joint work with Andrei Negut.

Bo Lin, Computing Linear Systems on Metric Graphs. A linear system on metric graphs is a set of effective divisors. It has the structure of a cell complex. We introduce the anchor divisors in it - they serve as the landmarks for us to compute the f-vector of the complex and find all cells in the complex. A linear system can also be identified as a tropical convex hull of rational functions. We compute the extremal generators of the tropical convex hull using the landmarks. We apply these methods to some examples - the canonical linear systems on $K_{4}$ and $K_{3,3}$.

Poset of partitions of 6 generated by Sage.