### Algebra & Discrete Mathematics Seminar

**Spring 2024 Schedule**

**Spring 2024 Location:**Minard 302**Time:**Tuesday 10:00 --10:50 am**Organizer:**Jessica Striker

##### 6 February 2024

**Ashleigh Adams **(NDSU): A Generalized Coinvariant Algebra

**Abstract**: The coinvariant algebra, a polynomial quotient ring whose ideal is S_n-invariant, has been well studied. In this talk we give a natural generalization of this ring motivated by the study of Stanley-Reisner rings, a family of rings that captures the structure of (finite, abstract) simplicial complexes. For the case when the Stanley-Reisner ideal is generated by minimally ordered square-free monomial ideals, we give an algorithm for writing down the Groebner basis. Using this recipe, we describe a combinatorial model for writing down the basis for the generalized coinvariant algebra and then we extend our intuition from these certain cases in order to give an explicit formula for the Hilbert series, in general. We will also give an application for the study of this particular ring.

##### 20 February 2024

**Colin Defant **(Harvard): A Smorgasbord of Ungar Moves

**Abstract**: Inspired by Ungar's solution to the famous slopes problem, I will introduce Ungar moves, which are operations that can be applied to elements of a finite lattice. I will discuss several problems and results concerning Ungar moves in the contexts of combinatorial dynamics, combinatorial probability, and combinatorial game theory.

##### 12 March 2024

**Esther Banaian **(Aarhus University): c-singleton Birkhoff polytopes and order polytopes of heaps

**Abstract**: For each Coxeter element c in the symmetric group, we define a pattern-avoiding Birkhoff subpolytope whose vertices are the c-singletons. We show that the normalized volume of our polytope is equal to the number of longest chains in a corresponding type A Cambrian lattice. Our work extends a result of Davis and Sagan which states that the normalized volume of the convex hull of the 132 and 312 avoiding permutation matrices is the number of longest chains in the Tamari lattice, a special case of a type A Cambrian lattice. Furthermore, we prove that each of our polytopes is unimodularly equivalent to the order polytope of the heap of the c-sorting word of the longest permutation. This gives an affirmative answer to a generalization of a question posed by Davis and Sagan. This talk is based on ongoing joint work with Sunita Chepuri, Emily Gunawan, and Jianping Pan.

##### 19 March 2024

**Torin Greenwood **(NDSU): Lattice walks in the Weyl chamber A_2

**Abstract**: A classical way to derive a formula for the Catalan numbers is to use the reflection principle on Dyck paths. How much does this generalize? We will survey existing results on lattice walks in higher dimensions, with different domains and stepsets. Then, we discuss how walks within Weyl chambers are a family of problems that naturally extend the reflection principle, as shown by Gessel and Zeilberger. Depending on the context of the problem, the number of walks can often be encoded within a rational generating function, making the problems amenable to techniques from analytic combinatorics in several variables. Ultimately, we find asymptotic results for weighted walks within the Weyl chamber A_2. Joint work with Samuel Simon.

##### 26 March 2024

**Tristan Larson **(NDSU): Asymptotics of bivariate algebraico-logarithmic generating functions

**Abstract**: We derive asymptotic formulae for the coefficients of bivariate generating functions with algebraic and logarithmic factors. Combinatorial enumeration problems can be attacked using generating functions derived via the symbolic method. Then, read-off results are used to determine the asymptotic growth of the combinatorial sequence. The asymptotic growth depends on the form of the generating function. Rational and algebraic generating functions are both well-studied. However, D-finite generating functions, such as those involving logarithms, have been studied far less. Logarithms appear when encoding cycles of combinatorial objects, and implicitly when objects can be broken into indecomposable parts.

##### 23 April 2024

**Janet Page **(NDSU): Extremal surfaces

**Abstract**: Perhaps one of the most famous classically known facts about lines on algebraic varieties is one proved by Cayley and Salmon in 1849--that every (smooth) cubic surface contains 27 lines. A hundred years later, Segre showed that a smooth surface of degree d > 2 over the complex numbers has at most (d-2)(11d-6) lines; when d = 3, we see that smooth cubic surfaces actually attain this upper bound. One might ask whether Segre's theorem holds when we look at surfaces over other fields. In this talk, I'll discuss some new results on a class of surfaces in positive characteristic which we’ve been studying for their “extremal” properties—including that they have more lines that Segre’s upper bound.

**Fall 2023 Schedule**

**Fall 2023 Location:**Minard 212**Time:**Tuesday 10:00 --10:50 am**Organizer:**Jessica Striker

##### 12 September 2023

**Jessica Striker **(NDSU): Hourglass plabic graphs and symmetrized six-vertex configurations

**Abstract**: In this talk, we introduce the title objects and explore their intriguing connections to tableaux dynamics, alternating sign matrices, and plane partitions. In a subsequent talk, we will we discuss the reason we defined these objects, namely, that they index a rotation-invariant SL4-web basis, a long-sought structure. This is joint work with Christian Gaetz, Oliver Pechenik, Stephan Pfannerer, and Joshua P. Swanson.

##### 19 September 2023

No seminar

##### 26 September 2023

**Jessica Striker **(NDSU): Hourglass plabic graphs and symmetrized six-vertex configurations, Part 2

**Abstract**: In this talk, we discuss the title objects and explore their intriguing connections to tableaux dynamics, alternating sign matrices, and plane partitions. We will also discuss the reason we defined these objects, namely, that they index a rotation-invariant SL4-web basis, a long-sought structure. This is joint work with Christian Gaetz, Oliver Pechenik, Stephan Pfannerer, and Joshua P. Swanson.

##### 10 October 2023

**Joshua P. Swanson **(University of Southern California): Type B q-Stirling numbers

**Abstract**: The Stirling numbers of the first and second kind are classical objects in enumerative combinatorics which count the number of permutations or set partitions with a given number of blocks or cycles, respectively. Carlitz and Gould introduced q-analogues of the Stirling numbers of the first and second kinds, which have been further studied by many authors including Gessel, Garsia, Remmel, Wilson, and others, particularly in relation to certain statistics on ordered set partitions. Separately, type B analogues of the Stirling numbers of the first and second kind arise from the study of the intersection lattice of the type B hyperplane arrangement. We combine the two directions and introduce new type B q-analogues of the Stirling numbers of the first and second kinds. We will discuss connections between these new q-analogues and generating functions identities, inversion and major index-style statistics on type B set partitions, and aspects of super coinvariant algebras which provided the original motivation for the definition. This is joint work with Bruce Sagan.

##### Thursday, 12 October 2023

**Stephan Pfannerer **(Technische Universitat Wien): Promotion and growth diagrams for r-fans of Dyck paths

**Abstract**: Using crystal graphs one can extend the notion of Schützenberger promotion to highest weight elements of weight zero. For the spin representation of type B_r these elements can be viewed as r-fans of Dyck paths. We construct an injection from the set of r-fans of Dyck paths of length n into the set of chord diagrams on [n] that intertwines promotion and rotation. This is done in two different ways: 1) as fillings of promotion–evacuation diagrams 2) in terms of Fomin growth diagrams This is joint work with Joseph Pappe, Anne Schilling and Mary Claire Simone.

**Location:** Minard 208

**Time: **11:00am

##### 7 November 2023

**Ben Adenbaum** (Dartmouth): Involutive Groups from Graphs

**Abstract:** We present a generalization of the toggle group, when thought of as a proper edge coloring of the Hasse diagram of the associated poset. Beyond general structure results, we focus on the case where the associated graph is a tree. This talk is based on joint work with Jonathan Bloom and Alexander Wilson.

##### 28 November 2022

**Tim Ryan** (NDSU): The Picard group

**Abstract:** The Picard group is a fundamental invariant of an algebraic variety. In this introductory talk, we will describe the Picard group starting with basic concepts. After defining it, we will explain a classical result, the Lefschetz hyperplane theorem, and a classical object, the Noether-Lefschetz locus of surfaces of degree d. These ideas will be central to next week’s colloquium and seminar by César Lozano Huerta.

Note: this talk is supplemental and is NOT required to understand either of next week’s talks, though I aim to make it helpful.

##### 5 December 2023

**César Lozano Huerta** (Universidad Nacional Autónoma de México - Oaxaca): The Noether-Lefschetz loci formed by determinantal surfaces in projective 3-space

**Abstract:** Solomon Lefschetz showed that the Picard group of a general surface in P3 of degree greater than three is ℤ. That is, the vast majority of surfaces in P3 have the smallest possible Picard group. The set of surfaces of degree greater than 3 on which this theorem fails is called the Noether-Lefschetz locus. This locus has infinite components and their dimensions are somehow mysterious.In this talk, I will calculate the dimension of infinite Noether-Lefschetz components that are simple in a sense, but still give us an idea of the complexity of the entire Noether-Lefschetz locus. This is joint work with Montserrat Vite and Manuel Leal.