Algebra & Discrete Mathematics Seminar

Jessica Striker (NDSU): A short history of rowmotion

Abstract: Rowmotion is an interesting action on order ideals of a poset that first appeared in a 1974 paper of Duchet and has been extensively studied and generalized in the past decade. In this talk, I'll discuss some important theorems on rowmotion and further recent developments.

Jessica Striker (NDSU): A short history of rowmotion, Part 2 - Recent developments

Abstract: Rowmotion is an interesting action on order ideals of a poset that first appeared in a 1974 paper of Duchet and has been extensively studied and generalized in the past decade. In this talk, I'll discuss some important theorems on rowmotion and further recent developments.

Cătălin Ciupercă (NDSU): Rational powers of ideals: properties and applications

Abstract: The study of asymptotic properties of ideals is a common theme in commutative algebra where algebraic and geometric properties of an ideal I are revealed by considering the asymptotic behavior of the power ideals I^n. In many instances, considering the asymptotic behavior of the integral closures of the power ideals instead of the regular powers has the advantage of eliminating some of the irrelevant information and simplifying the theoretical approach.

The rational powers of an ideal are a refinement of the integral closures of the power ideals. They capture more information associated with the ideal I while maintaining the theoretical advantage of working with multiplicative filtrations of integrally closed ideals. In an algebraic context, they were introduced by David Rees in the 70’s; in geometry, about the same time, they also appear (in some disguised form) in the work of Bernard Teissier about singularities.

In this talk we will give an exposition of the theory of rational powers with a focus on their behavior under derivations. We will also present several applications that justify the usefulness of this construction.

Tim Ryan (NDSU): The effective cone of cycles on a variety

Abstract: One of the motivating questions in algebraic geometry is to classify algebraic varieties, which are the shapes made by polynomials. In some sense, this problem has two parts -- first, given a variety, find all of the varieties isomorphic to it, and second, given two non-isomorphic varieties find a way to show they are not isomorphic. These parts of the problem overlap in an invariant called the effective cone (of dimension k cycles) on a variety. In particular, when k=1, the effective cone determines all varieties isomorphic to the given variety and for all k, it is an invariant that distinguishes (some) non-isomorphic varieties. In this talk, I will work up to the definition of the effective cone using minimal background (familiarity with polynomials and groups should suffice).

Tim Ryan (NDSU): Examples of effective cones

Abstract: Motivated by enumerative geometry (the branch of geometry that asks "How many objects X satisfy conditions Y?"), we will examine Hilbert schemes of points on the plane, which parameterize collections of points on the plane. In particular, we will compute (some of) the effective cones of cycles for the Hilbert scheme of two and three points on the plane and use this to answer enumerative questions.

Nate Krause (University of Minnesota): Topdrops: a bijective mapping on permutations.

Abstract: The topdrop map is a bijective mapping on permutations ascribed to John Conway and mentioned in a book by Martin Gardner in 1987. Unlike a related problem known as topswops, the topdrop map has not been the subject of any published work, despite exhibiting very interesting behavior. In this talk, I'll describe the map and explain some theorems about its orbit structure.

Cătălin Ciupercă (NDSU): Rational powers of ideals: properties and applications (Part 2)

Abstract: The study of asymptotic properties of ideals is a common theme in commutative algebra where algebraic and geometric properties of an ideal I are revealed by considering the asymptotic behavior of the power ideals I^n. In many instances, considering the asymptotic behavior of the integral closures of the power ideals instead of the regular powers has the advantage of eliminating some of the irrelevant information and simplifying the theoretical approach.

The rational powers of an ideal are a refinement of the integral closures of the power ideals. They capture more information associated with the ideal I while maintaining the theoretical advantage of working with multiplicative filtrations of integrally closed ideals. In an algebraic context, they were introduced by David Rees in the 70’s; in geometry, about the same time, they also appear (in some disguised form) in the work of Bernard Teissier about singularities.

In this talk we will give an exposition of the theory of rational powers with a focus on their behavior under derivations. We will also present several applications that justify the usefulness of this construction.

George Nasr (Augustana University): TBD

Abstract: TBD

Activity: Very short talks by participants

Abstract: This week for ADM seminar, we invite everyone to participate in giving very short (< 5 min) talks about something mathematical you've learned recently related to algebra or discrete mathematics. It could be something you learned at a conference or by reading a math paper. It could be an open problem you're interested in or some research you are working on. You are welcome to prepare a slide or two, write on the white board, or neither.

Janet Page (NDSU): Surfaces with Maximal Lines

Abstract: How many lines can lie on a smooth surface of degree d? In 1849, Cayley and Salmon proved their famous result that every smooth projective surface of degree 3 contains 27 lines. In higher degrees, we know that not all surfaces of degree d have the same number of lines (many have no lines at all), but Segre proved an upper bound: a smooth surface of degree d > 2 over the complex numbers has at most (d-2)(11d-6) lines. However, over fields of positive characteristic, there are smooth projective surfaces which break Segre’s upper bound. In this talk, I’ll give a new upper bound which holds over any field. In addition, we’ll fully classify those surfaces of degree d which contain the maximal number of lines possible. This talk is based on joint work with Tim Ryan and Karen Smith.

Activity: More very short talks by participants

Abstract: This week for ADM seminar, we invite everyone to participate in giving very short (< 5 min) talks about something mathematical you've learned recently related to algebra or discrete mathematics. It could be something you learned at a conference or by reading a math paper. It could be an open problem you're interested in or some research you are working on. You are welcome to prepare a slide or two, write on the white board, or neither. Both new topics and follow-up from prior seminars are welcome.

Jessica Striker (NDSU): Enumeration and dynamics on interval-closed sets

Abstract: Many nice formulas, such as binomial coefficients and Catalan numbers, have interpretations enumerating order ideals (downward-closed subsets) of certain posets (partially ordered sets). Many of these same posets have nice dynamical behavior under toggling actions such as rowmotion. In this paper, we study enumeration and toggle dynamics of interval-closed sets, a natural superset of order ideals. We find several results analogous to, though more complicated than, known results on order ideals. This talk is primarily based on a 2024 paper joint with Jennifer Elder, Nadia Lafreniere, Erin McNicholas, and Amanda Welch, and also work in progress with a subset of these authors and Sergi Elizalde and Joel Lewis.

Jessica Striker (NDSU): Pattern avoidance in alternating sign matrices: classical and key

Abstract: Alternating sign matrices (ASMs) are an interesting superset of permutations with connections to algebra, geometry, dynamics, and statistical physics. We study two notions of permutation pattern avoidance in alternating sign matrices, which we call classical- and key-avoidance. In the case of classical-avoidance, we completely characterize which avoidance classes of one pattern have an exponential upper bound and which have a factorial lower bound. This builds on a 2007 paper of Johansson and Linusson which enumerated alternating sign matrices that classically-avoid 132.

In the case of key-avoidance, we show ASMs that key-avoid 312 are in bijection with the gapless monotone triangles of [Ayyer, Cori, Gouyou-Beauchamps 2011], thus key-avoidance generalizes to all patterns the notion of 312-avoidance studied there. We enumerate by the Catalan numbers ASMs that key-avoid both 312 and 321. Finally, we enumerate ASMs with a given key avoiding 312 and 321 using a connection to Schubert polynomials, thereby deriving an interesting Catalan identity.

This talk is based on joint works with Mathilde Bouvel and Rebecca Smith (key-avoidance) and with these authors plus Eric Egge and Justin Troyka (classical-avoidance).

Ashleigh Adams (NDSU): Symmetry classes of plane partitions via webs

Abstract: Plane partitions are a combinatorial objects generalizing Young diagrams. They have beautiful combinatorial properties, such as product counting formulas for their symmetry classes. Recently, plane partitions have been shown by Gaetz, Pechenik, Pfannerer, Striker, and Swanson to be in bijection with certain U_q(sl_4)-webs lying on a hexagonal grid. U_q(sl_4)-webs can be viewed as combinatorial objects called hourglass plabic graphs and have deep representation theoretic connections to quantum invariant theory. In this talk, we discuss the connection between these seemingly very unrelated combinatorial objects. Moreover, we classify the symmetry classes of plane partitions via lattice words obtained from analyzing trip permutations of the corresponding hourglass plabic graphs.

Jesse Kim (University of Florida): Webs for the rook monoid

Abstract: Webs are a way of understanding the category of representations of the special linear group via the combinatorics of planar graphs. By applying Schur-Weyl duality, webs also provide insights into representations of the symmetric group. This talk will introduce a variant of webs based on a variant of Schur-Weyl duality introduced by Solomon in his study of the rook monoid. By restricting from the rook monoid to the symmetric group, we obtain bases for certain irreducible representations with interesting combinatorics.

Janet Page (NDSU): Rings of Differential Operators

Abstract: The collection of differential operators of a polynomial ring, called the Weyl algebra, has been studied in many fields across mathematics. Originally inspired by quantum mechanics, its study has also been useful to invariant theory, Lie theory, and more recently the study of singularities in algebraic geometry and commutative algebra. More generally, given any ring, its ring of differential operators tells us about the ring itself. When the ring has a natural associated geometric object, its ring of differential operators will tell us about the geometry of that object, including its singularities. In this talk, I'll introduce rings of differential operators and discuss some of their applications to commutative algebra. In addition, we'll discuss tools to compute rings of differential operators in some combinatorial settings.

Activity: Even more very short talks by participants

Abstract: This week for ADM seminar, we invite everyone to participate in giving short talks about something mathematical you've learned recently related to algebra or discrete mathematics. It could be something you learned at a conference or by reading a math paper. It could be an open problem you're interested in or some research you are working on. You are welcome to prepare a slide or two, write on the white board, or neither. Both new topics and follow-up from prior seminars are welcome.

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.