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.

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. 

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.

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.

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.

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.

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.