DISCRETE CATS SEMINAR

discrete CATS seminar

Title: Recurrence Relations of Linear Tilings
 
Abstract: Let U ⊆ ℕ and let T be a linear tiling of a 1 x n board using 1 x i polyomino tiles where u is in U. We define T_U(n) to be the set of linear tilings of length n using tile sizes in U. Classically, when U = { 1, 2} these tilings are known as the Lucanomial tilings and they follow the Lucas sequence. Now, let U and V be distinct subsets of the natural numbers. In this talk, we will provide some general results and supportive examples of a large family of linear recurrence relations between the sequences T_U(n) and T_V(n) for fixed sets U and V. This is joint work with Christopher O'Neill and Pamela E. Harris.
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Zoom
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discrete CATS seminar

Title: Log-Concavity of Littlewood-Richardson Coefficients
 
Abstract:  Schur polynomials and Littlewood-Richardson numbers are classical objects arising in symmetric function theory, representation theory, and the cohomology of the Grassmannian. I will give a quick introduction to them, and then describe a new log-concavity property of the Littlewood-Richardson numbers. I will then explain the mechanism behind the log-concavity, Brändén and Huh's theory of Lorentzian polynomials.
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Zoom
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discrete CATS seminar

Title: The Lalanne--Kreweras Involution, Rowmotion, and Birational Liftings
 
Abstract: Our work ties together a few different actions studied in combinatorics.  First, The Lalanne–Kreweras involution (LK) on Dyck paths yields a bijective proof of the symmetry of two statistics: the number of valleys and the major index.  Panyushev studied an equivalent involution can be considered on the set of antichains of the type A root poset.  Second, we will discuss the action of rowmotion on the set of antichains of a poset.  This action, which sends an antichain A to the minimal elements of the complement of the order ideal generated by A, has received significant attention recently in dynamical algebraic combinatorics due to various phenomena (e.g. periodicity, cyclic sieving, homomesy) on certain "nice" posets including root posets.  The LK involution and rowmotion are connected in that they generate a dihedral action on the set of antichains of the type A root poset.  Furthermore, the periodicity of rowmotion on the type A root poset lifts to a generalization called "birational rowmotion" first studied by David Einstein and James Propp.  This motivated us to search for a birational lifting of the LK involution, where we discovered that the key properties of the LK involution are also satisfied in this generalization.  This is joint work with Sam Hopkins.
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Zoom
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discrete CATS seminar

Title: Shelling the m=1 amplituhedron
 
Abstract: The amplituhedron is a topological space related to the totally nonnegative Grassmannian that was inspired by high energy physics. In this talk we will apply combinatorics and poset topology to analyze a special case known as the m=1 amplituhedron. There is a poset of sign vectors which is the closure poset of the m=1 amplituhedron, and we show this poset has an EL shelling. This implies the m=1 amplituhedron is a ball. We also show this poset is strongly Sperner.
 
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Zoom
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discrete CATS seminar

Title: Generalized permutahedra: Minkowski linear functionals and Ehrhart positivity
 
Abstract: Generalized permutahedra form a combinatorially rich class of polytopes that appear naturally in various areas of mathematics. They include many interesting and significant classes of polytopes such as matroid polytopes. We study functions on generalized permutahedra that behave linearly with respect to dilation and taking Minkowski sums. We present classification results and discuss how these can be applied to prove positivity of the linear coefficient of the Ehrhart polynomial of generalized permutahedra. This is joint work with Mohan Ravichandran.
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Zoom
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discrete CATS seminar

Title: Permuto-associahedra as deformations of nested permutohedra 



Abstract: A classic problem connecting algebraic and geometric combinatorics is the realization problem: given a poset (with a reasonable structure), determine whether there exists a polytope whose face lattice is the poset. In the 1990s, Kapranov defined a poset, called the permuto-associhedron, as a hybrid between the face poset of the permutohedron and the associahedron, and he asked whether this poset is realizable. Shortly after his question was posed, Reiner and Ziegler provided a realization. In this talk, I will discuss a different construction we obtained as a deformations of nested permutohedra. This is joint work with Federico Castillo.

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Zoom
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discrete CATS seminar

Title: Pairwise Completability for 2-Simple Minded Collections.

Abstract:  Let Lambda be a basic, finite dimensional algebra over an arbitrary field, and let mod(Lambda) be the category of finitely generated right modules over Lambda. A 2-term simple minded collection is a special set of modules that generate the bounded derived category for mod(Lambda). In this talk we describe how 2-term simple minded collections are related to finite semidistributive lattices, and we show how to model 2-term simple minded collections for the preprojective algebra of type A. 

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Zoom
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discrete CATS seminar

Title: The Geometry and Combinatorics of Convex Union Representable Complexes  



Abstract: The study of convex neural codes seeks to classify the intersection and covering patterns of convex sets in Euclidean space. A specific instance of this is to classify "convex union representable" (CUR) complexes: the simplicial complexes that arise as the nerve of a collection of convex sets whose union is convex. In 2018 Chen, Frick, and Shiu showed that CUR complexes are always collapsible, and asked if the converse holds: is every collapsible complex also CUR? We will provide a negative answer to this question, and more generally describe the combinatorial consequences arising from the geometric representations of CUR complexes. This talk is based on joint work with Isabella Novik.

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Zoom
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discrete CATS seminar

TItle: Tropical Support Vector Machines
 
Abstract:  Support Vector Machines (SVMs) are one of the most popular supervised learning models to classify using a hyperplane in an Euclidean space. Similar to SVMs, tropical SVMs classify data points using a tropical hyperplane under the tropical metric with the max-plus algebra. In this talk, first we show generalization error bounds of tropical SVMs over the tropical projective space. While the generalization error bounds attained via VC dimensions in a distribution-free manner still depend on the dimension, we also show theoretically by extreme value statistics that the tropical SVMs for classifying data points from two Gaussian distributions as well as empirical data sets of different neuron types are fairly robust against the curse of dimensionality. Extreme value statistics also underlie the anomalous scaling behaviors of the tropical distance between random vectors with additional noise dimensions.  This is joint work with M. Takamori, H. Matsumoto and K. Miura.
 
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Zoom
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Discrete CATS seminar

Title: Determinantal formulas with major indices

Abstract: Krattenthaler and Thibon discovered a beautiful formula for the determinant of the matrix indexed by permutations whose entries are q^maj( u*v^{-1} ), where “maj” is the major index. Previous proofs of this identity have applied the theory of nonsymmetric functions or the representation theory of the Tits algebra to determine the eigenvalues of the matrix. I will present a new, more elementary proof of the determinantal formula. Then I will explain how we used this method to prove several conjectures by Krattenthaler for variations of the major index over signed permutations and colored permutations. This is based on joint work with Donald Robertson and Clifford Smyth.

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Zoom
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