The UC Berkeley combinatorics seminar |
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DATE |
SPEAKER |
TITLE (click to show abstract) |

January 30th | Max Wimberley, UC Berkeley |
## Introduction to the Schiffmann Algebra and its ApplicationsDiscovered by Schiffmann and Vasserot in 2008 as the Hall Algebra of an elliptic curve, the Schiffmann algebra has proven to be a remarkably useful tool in unifying and generalizing many topics in symmetric function theory, including the diagonal harmonics and the shuffle conjecture. My talk will focus on providing an accessible exposition of the algebra, its presentations and its symmetries, grounded in computation. As time allows, I will discuss my own investigation into a particular Poisson degeneration of the algebra and its relationship to a conjectured formula for the "triagonal harmonics," the version of the diagonal harmonics with three alphabets of variables instead of two. |

February 6th | Tamsen McGinley, Santa Clara University |
## Crystals Graphs, Perforated Tableaux, and Combinatorial ObjectsCrystal graphs of biwords are directed graphs whose nodes are biwords and whose edges are determined by "crystal operators". These are used to model irreducible representations of GL_n . We present a new model for the nodes of a crystal graph, called a "perforated tableaux", on which the crystal operators are easily defined. We show that several well-known combinatorial objects, including semi-standard Young tableaux, Littlewood-Richardson fillings, and the Robinson-Schensted-Knuth correspondence, naturally appear in the setting of perforated tableaux. |

Friday, February 10th, Evans 740, 4 PM |
Sarah Brauner, University of Minnesota |
## Configuration spaces and combinatorial algebrasIn this talk, I will discuss connections between configuration spaces, an important class of topological space, and combinatorial algebras arising from the theory of reflection groups. In particular, I will present work relating the cohomology rings of some classical configuration spaces—such as the space of n ordered points in Euclidean space—with Solomon’s descent algebra and the peak algebra. The talk will be centered around two questions. First, how are these objects related? Second, how can studying one inform the other? This is joint, on-going work with Marcelo Aguiar and Vic Reiner. |

February 13th | No Seminar | |

February 20th | Academic Holiday - No Seminar | |

February 27th | Bernd Sturmfels, UC Berkeley |
## Algebraic Statistics, Scattering, and Geometric CombinatoricsWe discuss themes at the interface of algebraic statistics and particle physics that rely heavily on methods from geometric combinatorics. Our journey starts with maximum likelihood estimation for discrete models, it visits scattering amplitudes and moduli spaces, and it concludes with a recent article on likelihood degenerations. Our actors include matroids, hyperplane arrangements, very affine varieties, and tropical numerics. |

March 6th | Gene B. Kim, Stanford |
## Golf and gamblingDaily fantasy sports (DFS) is a multi-billion dollar industry. Despite the resemblance to sports betting, it has been declared as a game of skill in most US states; at its essence, DFS is a game of probabilistic optimization. We will discuss how to use data and Markov chains to model golf and use them to potentially multiply our investments by 10,000+ (and maybe even win a million dollars!) |

Tuesday March 7th, Evans 748, 4 PM |
Bruce Sagan, Michigan State University |
## Stirling numbers for complex reflection groupsThe ordinary Stirling numbers count set partitions and permutations of {1,2,...,n} by number of subsets and number of cycles, respectively. We show how to generalize these concepts to a complex reflection group. The ordinary Stirling numbers are recovered in type A. It turns out that often these Stirling numbers can be expressed in terms of elementary and homogeneous symmetric functions. We also make a connection with super coinvariant algebras. All terminology concerning Stirling numbers, symmetric functions, and complex reflection groups will be defined. This is joint work with Joshua Swanson. |

March 13th | Ray Li, UC Berkeley |
## Binary Longest Common SubsequencesThe longest common subsequence (LCS) is a fundamental measure of the similarity of two strings. I will discuss two basic questions on the LCS of binary strings, one combinatorial and one algorithmic. The combinatorial question is motivated by coding theory. A binary code of (asymptotically) positive rate is a subset of {0,1}^n with size exponentially large in n. In any positive rate binary code, one can find two strings (codewords) sharing the same majority bit and thus a common subsequence of length at least n/2. Surprisingly, whether this trivial constant 1/2 could be improved was a longstanding open problem in coding theory. I will talk about a recent result that solves this question: for an absolute constant c>0, any positive rate binary code has two strings with a common subsequence of length (1/2+c)n. In coding theory language, there are no positive rate binary codes correcting a (1/2-c) fraction of worst-case deletions. I will talk about our tools, which include regularity arguments for strings and a structural lemma, very roughly analogous to a Fourier transform, that classifies binary strings by their oscillation patterns. I will then discuss how these combinatorial tools help give better approximation algorithms for LCS. |

March 20th | Lucy Martinez, Rutgers University |
## The G-Shi Arrangement: Pak-Stanley Labels and Games on PathsGiven a simple graph G, one can define a hyperplane arrangement called the G-Shi arrangement. The Pak-Stanley algorithm labels the regions of this arrangement with (G•)-parking functions in a way that some (G•)-parking functions may appear more than once. These repetitions of Pak-Stanley labels are a topic of interest in the study of (G•)-Shi arrangements and (G•)-parking functions, as well as the many combinatorial objects they are connected to. In this talk, we will learn about the combinatorial model called the "Three Rows Game" which is used to completely characterize the repetitions of the Pak-Stanley labels for path graphs. |

March 27th | Spring Break - No Seminar | |

April 3rd | Joseph Pappe, UC Davis |
## Promotion and Cyclic Sieving Phenomenon for Fans of Dyck Paths and Vacillating TableauxUsing chord diagrams, we construct a diagrammatic basis for the space of invariant tensors of certain Type B representations. This basis carries the property that rotation of the chord diagrams intertwines with the natural action of the longest cycle in the symmetric group on the tensor factors. Our approach involves constructing an injection from the set of r-fans of Dyck paths (resp. vacillating tableaux) of length n into the set of chord diagrams on [n] that intertwines promotion and rotation. From this, we give a cyclic sieving phenomenon on r-fans of Dyck paths (resp. vacillating tableaux). This is joint work with Stephan Pfannerer, Anne Schilling, and Mary Claire Simone. |

April 10th | Yan Zhuang, Davidson College |
## Shuffle-Compatibility: From Linear to CyclicA permutation statistic st is said to be shuffle-compatible if the distribution of st over the set of shuffles of two disjoint permutations π and σ depends only on st(π), st(σ), and the lengths of π and σ. This notion is implicit in Stanley’s work on P-partitions, and was first explicitly studied by Gessel and Zhuang in 2018, who developed a unifying framework for shuffle-compatibility in which quasisymmetric functions play an important role. Since then, shuffle-compatibility has become an active topic of research. The first half of my talk will give an overview of the theory of shuffle-compatibility from my joint work with Ira Gessel; the second half will focus on more recent work—joint with Jinting Liang and Bruce Sagan—on shuffle-compatibility of cyclic permutation statistics, in which the role played by quasisymmetric functions is replaced by the cyclic quasisymmetric functions introduced by Adin, Gessel, Reiner, and Roichman. |

April 17th | Dustin Ross, SFSU |
## Putting the “volume” back in “volume polynomials”Recent developments in tropical geometry and matroid theory have led to the study of “volume polynomials” associated to tropical varieties, the coefficients of which record all possible degrees of top powers of tropical divisors. In this talk, I’ll discuss a volume-theoretic interpretation of volume polynomials of tropical fans; namely, they measure volumes of polyhedral complexes obtained by truncating the tropical fan with normal hyperplanes. I’ll also discuss how this volume-theoretic interpretation inspires a general framework for studying an analogue of the Alexandrov-Fenchel inequalities for degrees of divisors on tropical fans, providing a new method for understanding and generalizing the log-concavity of characteristic polynomials of matroids. Much of the work I’ll report on is in collaboration with Anastasia Nathanson, Lauren Nowak, and Patrick O’Melveny. |

April 24th | Mark Haiman, UC Berkeley |
## Loehr-Warrington and moreIn 2007, Nick Loehr and Greg Warrington conjectured a combinatorial formula, as a sum of terms indexed by systems of nested Dyck paths, for the image of a Schur function under the action of the operator \nabla from the theory of Macdonald polynomials. This generalizes the earlier Shuffle Conjecture, which is the case when the Schur function is an elementary symmetric function. After the proof of the Shuffle Conjecture by Carlsson and Mellit, the Loehr-Warrington conjecture was the longest standing (and to me the most intriguing) open problem in 'q,t -Catalan combinatorics.' I'll explain the context and statement of the Loehr-Warrington conjecture, then give an overview of how it follows from recent collaborative work by Blasiak, Morse, Pun, Seelinger and me. Along the way, we'll learn a little about the q-symmetric functions known as LLT (Lascoux - Leclerc - Thibon) polynomials, the elliptic Hall algebra of Burban and Schiffmann, and the beautifully combinatorial way that one can construct elements which represent LLT polynomials in any of the many copies of the algebra of symmetric functions that live inside the Schiffmann algebra. |

May 1st | Anastasia Chavez, St. Mary's College of CA |
## The valuation polytope of the zig-zag posetThe summer 2021 Latinx Mathematical Research Community (LMRC) served as a catalyst for several research projects in various areas of mathematics. This talk will introduce the research of one such project. Geissinger defined the valuation polytope as the set of all [0,1]-valuations on a finite distributive lattice. We study the valuation polytope, V(Z_n), arising from the height 2 up-down poset on n elements, referred to as the zig-zag poset Z_n. Dobbertin showed that the valuation polytope of any poset can be described as the convex hull of vertices characterized by all the chains of that poset. It follows that the dimension of the valuation polytope is the number of elements of the corresponding poset. We discuss combinatorial results of V(Z_n) such as its normalized volume, the existence of a unimodular triangulation, and facet enumeration. This is joint work with Federico Ardila, Anastasia Chavez, Jessica De Silva, Pamela E. Harris, Jose Luis Herrera Bravo, and Andrés R. Vindas-Meléndez. |

Wednesday May 3rd, 1-2 PM, Evans 939 |
Matt Hogancamp, Northeastern University |
## How to compute some superpolynomialsKhovanov-Rozansky homology is an invariant of knots and links, taking values in triply graded vector spaces. Taking Poincare series results in a 3-variable series, sometimes called the superpolynomial. In this talk I will give an updated perspective on a combinatorial technique for computing these invariants, developed in joint work with Ben Elias. Along the way we will see the appearance of (q,t)-Catalan and some of its cousins. |

Thursday May 4rd, 4-5 PM, Evans 939 |
Vaidy Sivaraman, Mississippi State University |
## Forbidden induced subgraph characterizations of hereditary graph classesA graph class that is closed under induced subgraphs is called hereditary. How can we determine the set of forbidden induced subgraphs for a given hereditary graph class? Can we at least tell if it is finite? Some recent work on certain graph classes in the vicinity of perfect graphs will be discussed. Several open problems concerning the relationship between the chromatic number and the clique number (so-called chi-boundedness) of graphs in hereditary graph classes will be mentioned. Joint work with Thomas Zaslavsky, Daniel Slilaty, Rebecca Whitman, and Deven Gill. |

May 8th | Susanna Fishel, Arizona State University |
## Maximal length chains in certain tubing posetsTubing posets were defined by Forcey and also by Ronco. They are orientations of the 1-skeleton of graph associahedra. For the complete graph on n vertices, the poset is the weak order on the symmetric group. When the graph is the path on n vertices, the poset is the Tamari lattice. We discuss the maximal length chains of certain tubing posets and their image under the Coxeter-Knuth bijection. We assume no prior knowledge of any of the topics in the talk. This is joint work with Samantha Dahlberg. |