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

January 25th | Eugene Gorsky UC Davis |
## Algebraic weaves and braid varietiesIn the talk I will define braid varieties, a class of affine algebraic varieties associated to positive braids, and develop a Soergel-like diagrammatic calculus for correspondences between them. The braid varieties are closely related to augmentation varieties for Legendrian links, and the correspondences provide an algebraic machinery to study exact Lagrangian fillings of such links. This is a joint work with Roger Casals, Mikhail Gorsky and Jose Simental Rodriguez. |

February 1st | Christopher Ryba UC Berkeley |
## Stable characters from permutation patternsFor a fixed permutation σ∈Sk, let Nσ denote the function which counts occurrences of σ as a pattern in permutations from Sn. We study the expected value (and d-th moments) of Nσ on conjugacy classes of Sn and prove that the irreducible character support of these class functions stabilizes as n grows. This says that there is a single polynomial in the variables n,m1,…,mdk which computes these moments on any conjugacy class (of cycle type 1^m1,2^m2,⋯) of any symmetric group. Our proof is, to our knowledge, the first application of partition algebras to the study of permutation patterns. This is joint work with Christian Gaetz. |

February 8th | Camille Combe Universite de Paris |
## A hierarchy of operads on wordsThe associative symmetric operad is an operad on permutations. It is an algebraic structure endowed with a composition operation allowing us to insert a permutation into another one. This structure is rich both under a combinatorial and an algebraic point of view. In this context, Aguiar and Livernet have constructed alternative bases of this operad relating it with the combinatorics of the weak order on permutations. In this talk, I will present a family of analogous operads, defined on some families of words of integers. These sets of words, called cliffs, can be put in correspondence with some usual combinatorial sets (permutations, increasing trees, Fuss-Catalan objects, etc.). The construction of this family of operads is detailed and some properties are presented. One of the peculiarities of some operads of this hierarchy is that, despite to their relative simplicity, some are infinitely generated and have nonquadratic and nonhomogeneous nontrivial relations. Joint work with Samuele Giraudo. |

February 15th | Academic Holiday - no talk | |

February 22nd | Nikhil Srivastava UC Berkeley |
## Support of Closed Walks and Second Eigenvalue Multiplicity of GraphsWe show that the multiplicity of the second adjacency matrix eigenvalue of any connected bounded degree graph is O(n/polylog(n)), improving the previously known O(n/polyloglog(n)) bound of Jiang et al. The key ingredient in the proof is a new probabilistic result: in any bounded degree graph, the number of distinct vertices traversed by a simple random walk of length 2k conditioned to return to its starting vertex is typically at least poly(k). Joint work with T. McKenzie and P. Rasmussen. |

March 1st | Jonathan Novak UCSD |
## Schur polynomials via matrix integralsOne could make a strong argument that Schur polynomials are the most fundamental objects in algebraic combinatorics. Consequently, everyone who works in the area (or wants to) has been (or will be) very familiar with them. However, there is an aspect of Schur polynomials which many combinatorialists are unaware of, namely that they can be represented as certain unitary matrix integrals called Harish-Chandra/Itzykson-Zuber integrals. This fact, which is essentially a manifestation of the Kirilov orbit philosophy in representation theory, gives a whole new set of tools for working with Schur polynomials which is particular useful for doing asymptotics. In my talk I will explain the relationship between Schur polynomials and HCIZ integrals, and tell you about some of its ramifications. |

March 8th | Alejandro Morales UMass Amherst |
## Refinements and symmetries for volumes of flow polytopesFlow polytopes are an important class of polytopes in combinatorics whose lattice points and volumes have interesting properties and relations. The Chan-Robbins-Yuen (CRY) polytope is a flow polytope with normalized volume equal to the product of consecutive Catalan numbers. Zeilberger proved this by evaluating the Morris constant term identity, but no combinatorial proof is known. There is a refinement of this formula that splits the largest Catalan number into Narayana numbers, which Mészáros gave an interpretation as the volume of a collection of flow polytopes. We introduce a new refinement of the Morris identity with combinatorial interpretations both in terms of lattice points and volumes of flow polytopes. Our results generalize Mészáros's construction and a recent flow polytope interpretation of the Morris identity by Corteel-Kim-Mészáros. We prove the product formula of our refinement following the strategy of the Baldoni-Vergne proof of the Morris identity. This is joint work with William Shi. |

March 15th | Raul Penaguiao SFSU |
## Feasible regions meets pattern avoidance - The long awaited part three of feasible regionsGlebbov, Hoppen and others introduced the notion of feasible regions for permutation patterns. Given a fixed integer $k$, the feasible region is a set in $\mathbb{R}^{S_k}$ defined as follows: for a sequence of permutations $\sigma_k$ with growing size, compute the limit of the proportion of occurrences of each pattern of size $k$ in $\sigma_n$, obtaining a vector indeed by $S_n$. The feasible region arises as the set of such limits. Many interesting problems were studied in this context, like computing the dimension of the feasible region and its extreme points. Sometimes full descriptions can be given, but an overarching result is missing. If we consider consecutive patterns instead of classical patterns, we get simpler results, and we can characterize the feasible region: it is a polytope, and the vertices are given by cycles of a particular graph called overlap graph. Finally, we will talk about the feasible region for consecutive patterns resulting from considering permutations avoiding certain patterns. These new feasible regions, called the pattern-avoiding feasible regions, are now governed by different versions of the overlap graph, and we propose a unified characterization for all pattern-avoiding feasible regions avoiding one pattern. Along the way, we discuss connections of this work with the problem of packing patterns in pattern-avoiding permutations. |

March 22nd | Spring Recess - no talk | |

March 29th | Foster Tom UC Berkeley |
## A combinatorial Schur expansion of triangle-free horizontal-strip LLT polynomialsAbstract: In recent years, Alexandersson and others proved combinatorial formulas for the Schur function expansion of the horizontal-strip LLT polynomial $G_\lambda(x;q)$ in some special cases. We associate a weighted graph $\Pi$ to $\lambda$ and we use it to express a linear relation among LLT polynomials. We apply this relation to prove an explicit combinatorial Schur-positive expansion of $G_\lambda(x;q)$ whenever $\Pi$ is triangle-free. We also prove that the largest power of $q$ in the LLT polynomial is the total edge weight of our graph. |

April 5th | Jeremy Meza UC Berkeley |
## An exploration of LLT polynomials in classical Lie typeLLT polynomials were first introduced by Lascoux, Leclerc, and Thibon in their study of plethystic substitutions of Hall-Littlewood polynomials, and have since then enjoyed many applications. Grojnowski and Haiman later extended LLT polynomials to arbitrary Lie type, albeit with an algebraic description, and showed that for GL_n, their definition coincides with the combinatorial formula in terms of k-cores and k-quotients. I will give details on work in progress toward extending these tableau combinatorics of LLT polynomials to other classical Lie types, in particular to Sp_{2n}. Along the way we will encounter a medley of objects, including non-symmetric Hall-Littlewood polynomials, oscillating tableaux, symplectic tableaux and Demazure characters. |

April 12th | Zajj Daugherty CUNY |
## Degenerate two boundary BMW algebrasThe Birman-Murakami-Wenzl (BMW) algebras and their generalizations arise both as diagram algebras and as algebras of operators that preserve symmetry in tensor products of simple modules for symplectic and orthogonal Lie algebras and quantum groups. Their diagrammatic presentations are that of tangles, potentially in spaces with one or more puncture. They are fundamental to realizing certain knot and link invariants, but also have ties into lattice models in Statistical Mechanics. In this talk, we will introduce specifically the degenerate two-boundary BMW algebras and explore some of their algebraic and combinatorial structure. |

April 19th | Kelly Isham UC Irvine |
## On n-arcs in projective spaceAn n-arc in (k-1)-dimensional projective space is a set of n points so that no k lie on a hyperplane. Understanding the number of n-arcs has been an important area of study in several fields including finite geometry and coding theory. In 1988, Glynn gave an algorithm to count n-arcs in the projective plane of order q in terms of simpler combinatorial objects called superfigurations. Formulas for the number of n-arcs are known for $n \le 9$. In this talk, I will discuss new results about the number of 10-arcs in the projective plane of order q. I will then introduce a generalization of Glynn's framework and use this to produce an algorithm to count n-arcs in 3-dimensional projective space. This talk includes joint work with Nathan Kaplan and Max Weinreich. |

April 26th | Yulia Alexandr UC Berkeley |
## Logarithmic Voronoi cellsA statistical model is a subset of a probability simplex. The maximum likelihood estimation (MLE) is a map that takes an empirical data point and assigns it a point in the model that maximizes the log-likelihood function defined by the data point. Finding the maximum likelihood estimate for a given data point is a problem that can be solved using optimization tools. On the other hand, given a point in the model, we may study the fiber of the MLE map corresponding to that point, known as its logarithmic Voronoi cell. Each logarithmic Voronoi cell lives inside its log-normal polytope, and these log-normal polytopes corresponding to the points of our model fill the probability simplex. I will introduce these notions and investigate when logarithmic Voronoi cells are polytopes and when they are non-polytopal convex sets. I will also talk about logarithmic Voronoi cells for the finite model consisting of all empirical distributions of a fixed sample size. These cells are polytopes dual to the logarithmic root polytopes of Lie type A, and I will characterize their faces. |

May 3rd | Vasu Tewari University of Hawai'i |
## An algebra of Klyachko, and Macdonald's reduced word identityMotivated by the striking resemblance between an identity of Klyachko from 1985 in the context of the permutahedral variety and Macdonald's reduced word identity from 1991, I will discuss what we call the q-Klyachko algebra. My focus will largely be on an intriguing q-analog of Postnikov's mixed Eulerian numbers that arises naturally in this algebra, and I will interpret this new family both combinatorially and probabilistically. I will conclude with a simultaneous generalization of the aforementioned identities of Klyachko and Macdonald, and discuss some applications. Joint work with Philippe Nadeau (CNRS & Université Lyon 1). |