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

August 30th | Hunter Spink Stanford |
## Log-concavity of Matroid h-vectors(Joint with Andrew Berget and Dennis Tseng) For any matroid M, we compute the Tutte polynomial using mixed intersection numbers of classes in the combinatorial Chow ring of M arising from Grassmannians. Using this, we resolve and strengthen an old conjecture of Dawson (independently proved by Ardila-Denham-Huh) on the log-concavity of the h-vector of the independence complex of M. |

September 6th | Holiday - no talk | |

September 13th | Zhongyang Li University of Connecticut |
## Asymptotics of random perfect matchingsI will discuss random perfect matchings on a large class of graphs, which form Schur processes. I will then talk about the limit shape (Law of Large Numbers) and convergence of height fluctuations to Gaussian Free Field (Central Limit Theorem) for random perfect matchings on these graphs in the scaling limit. |

September 20th | Martha Precup Washington University in St. Louis |
## Hessenberg varieties and the Stanley--Stembridge conjectureHessenberg varieties are subvarieties of the flag variety with important connections to representation theory, algebraic geometry, and combinatorics. In 2015, Brosnan and Chow, and independently Guay-Paquet, proved the Shareshian--Wachs conjecture, which links the combinatorics of chromatic symmetric functions to the geometry of Hessenberg varieties via a symmetric group action on the cohomology ring of regular semisimple Hessenberg varieties. This talk will give an overview of that story and present a set of linear relations satisfied by the multiplicities of certain permutation representations in that representation. As an application of these results, we prove an inductive formula for the multiplicity coefficients corresponding to partitions with a maximal number of parts. This talk is based on joint work with M. Harada. |

September 27th | Mei Yin University of Denver |
## Interval Parking Functions, Edge-Labeled Spanning Trees, and the Bruhat vs. Pseudoreachability OrdersThe topic of parking functions has wide applications in probability, combinatorics, group theory, and computer science. One generalization of the classical parking function is the interval parking function, where each car has an interval rather than a single spot of preference. We classify features of interval parking functions, build a bijection between interval parking functions and edge-labeled spanning trees of the complete graph, and discuss the Bruhat vs. pseudoreachability orders on interval parking functions. Partially based on joint work with Emma Colaric, Ryan DeMuse, and Jeremy Martin. |

October 4th | Greta Panova University of Southern California |
## Linear extensions of posets and lattice paths: inequalities and log-concavityLinear extensions (total orders) of partially ordered sets generalize classical combinatorial notions like permutations and Standard Young Tableaux, but remain largely mysterious in full generality. The 1/3-2/3 conjecture and Kahn-Saks conjectures on sorting probabilities, and the cross-product inequality conjecture are still widely open. Stanley's and Kahn-Saks log-concavity results have been proven only via the Aleksandrov-Fenchel inequalities. Linear extensions can be interpreted as lattice paths in certain monotone regions in Z^d, giving them yet another geometric flavor. I will describe how this interpretation lead to the proof of the Kahn-Saks conjecture for Young diagrams, the cross-product conjecture for width two posets, and also purely combinatorial proofs of Stanley's and Kahn-Saks log-concavity theorems for width two posets. As an application of the method we obtain log-concavity of exit probabilities of random walks in a large class of domains. Based on a series of papers with Swee Hong Chan and Igor Pak. |

October 11th | Swee Hong Chan University of Califonia, Los Angeles |
## Combinatorial Atlas for log-concavityThe study of log-concave inequalities for combinatorial objects has seen much progress in recent years. One such progress is the solution to the strongest form of Mason's conjecture (independently by Anari et. al. and Brándën-Huh) that the f-vectors of matroid independence complex is ultra-log-concave. In this talk, we discuss a new proof of this result through linear algebra, and discuss generalizations to greedoids and posets. |

October 18th | Daoji Huang University of Minnesota |
## Bijective combinatorics in Schubert calculus with pipe dreams and bumpless pipe dreamsPipe dreams and bumpless pipe dreams both give combinatorial formulas for Schubert polynomials. As a result, many important identities in Schubert calculus can be understood through pipe dreams and/or bumpless pipe dreams. Since the discovery of bumpless pipe dreams, a direct weight-preserving bijection between pipe dreams and bumpless pipe dreams has been of great interest. In this talk, we present such a bijection, and establish its canonical nature by showing that it preserves Monk's rule. We also remark that the technical recipe used in this bijection has been useful in a few other contexts, including a combinatorial rule for Schubert structure constants in the separated descent Schubert problem. |

October 25th | Sean Griffin University of California, Davis |
## Springer fibers and the Delta ConjectureSpringer fibers are a family of varieties that have remarkable connections to representation theory and combinatorics. In particular, Springer constructed an action of the symmetric group on the cohomology ring of a Springer fiber, and used it to geometrically construct the Specht modules (in type A), which are the irreducible representations of the symmetric group. In this talk, I will introduce a new family of varieties generalizing the Springer fibers, and show how they are connected to the Delta Conjecture from algebraic combinatorics (which was recently partially proven). We’ll then use these varieties to geometrically construct the induced Specht modules. This is joint work with Jake Levinson and Alexander Woo. |

November 1st | GaYee Park University of Massachusetts, Amherst |
## Minimal skew semistandard Young tableaux and the Hillman-Grassl CorrespondenceStandard tableaux of skew shape are fundamental objects in enumerative and algebraic combinatorics and no product formula for the number is known. In 2014, Naruse gave a formula as a positive sum over excited diagrams of products of hook-lengths. In 2018, Morales, Pak, and Panova gave a q-analogue of Naruse's formula for semi-standard tableaux of skew shapes. They also showed, partly algebraically, that the Hillman-Grassl map restricted to skew shapes gave their q-analogue. We study the problem of making this argument completely bijective. For a skew shape, we define a new set of semi-standard Young tableaux, called the minimal SSYT, that are equinumerous with excited diagrams via a new description of the Hillan-Grassl bijection and have a version of excited moves. Lastly, we relate the minimal skew SSYT with the terms of the Okounkov-Olshanski formula for counting SYT of skew shape. This is joint work with Alejandro Morales and Greta Panova. |

November 8th | Stoyan Dimitrov University of Illinois, Chicago |
## Moments of permutation statistics and central limit theorems (joint work with Niraj Khare)We show that if a permutation statistic can be written as a linear combination of bivincular patterns, then its moments can be expressed as a linear combination of factorials with constant coefficients. This generalizes a result of Zeilberger. We use an approach of Chern, Diaconis, Kane and Rhoades, previously applied on set partitions and matchings. In addition, we give a new proof of the central limit theorem (CLT) for the number of occurrences of classical patterns, which uses a lemma of Burstein and H\"{a}st\"{o}. We give a simple interpretation of this lemma and an analogous lemma that would imply the CLT for the number of occurrences of any vincular pattern. Furthermore, we obtain explicit formulas for the moments of the descents and the minimal descents statistics. The latter is used to give a new direct proof of the fact that we do not necessarily have asymptotic normality of the number of pattern occurrences in the case of bivincular patterns. |

November 15th | Luis Ferroni University of Bologna |
## Valuative invariants for large classes of matroidsIn this talk we will exhibit a systematic way of computing valuative invariants for the class of sparse paving and paving matroids. Such families are conjecturally predominant among all matroids. In particular, this permits to obtain nice expressions for the volume, the Tutte polynomial, the Kazhdan-Lusztig polynomial, the Ehrhart polynomial, the gamma polynomial, the Derksen-Fink invariant and many other valuative invariants for all such matroids. As two remarkable applications, we obtain examples of matroid polytopes that are not Ehrhart positive and we show that some large classes of matroids are Kazhdan-Lusztig non-degenerate and Kazhdan-Lusztig gamma-positive. |

November 22nd | Jessica De Silva California State University, Stanislaus |
## Erdös-Ko-Rado GraphsThe Erdös-Ko-Rado (EKR) theorem states that the maximum size of an intersecting family of r-element subsets of [n] can be attained by taking all subsets containing some fixed element. This classical result in extremal combinatorics has been rephrased in the language of graph theory as an extremal property regarding independent sets of a graph. A family of independent r-sets of a graph G is an r-star if every set in the family contains some fixed vertex v. A graph is r-EKR if the maximum size of an intersecting family of independent r-sets can be attained by an r-star. In this context, the classical EKR theorem determines the values of r for which the empty graph satisfies the r-EKR property. This talk will highlight that there is much to be discovered in the search for EKR graphs, along with motivation and early findings for classes of very well-covered graphs. |

November 29th | Nate Harman, University of Michigan |
## (Some) Algebra and Combinatorics of the Restriction ProblemThe restriction problem in combinatorial representation theory asks for a combinatorial interpretation of the branching coefficients for restricting from GL_n to S_n. First, I will remove the representation theory and formulate a purely combinatorial problem which would answer this question. Then I will add in extra representation theory, and discuss a hidden action of a certain infinite dimensional lie algebra on a space closely related to this problem. |