Abstracts

Pouya Layeghi (University of Minnesota)  

• Title: "Homotopy Path Algebras and Koszulity"

• Abstract: Homotopy path algebras (HPA) were introduced by David Favero and Jesse Huang in their paper "Homotopy Path Algebras". They are the path algebras of quivers with special relations. In this talk, I will give an introduction to homotopy path algebras and their origins, which involves some of the origins of toric homological mirror symmetry first stated in Bondal's Oberwolfach report. Then, I will talk about Koszulity of the path algebras and its interactions with derived categories of quivers (path algebras) based on another paper written by Bondal. Finally, I will try to explain when an HPA is Koszul which involves some topological and combinatorial tools. 

Yuxuan Sun (University of Minnesota)

• Title: "Crystal Chute Moves on Pipe Dreams"

• Abstract: Schubert polynomials represent a basis for the cohomology of the complete flag variety and thus play a central role in geometry and combinatorics. In this context, Schubert polynomials are generating functions over rc-graphs, equivalently reduced pipe dreams. By restricting the chute moves of Bergeron and Billey, we define a crystal structure on the monomials of a Schubert polynomial. As a consequence, we provide a method for decomposing Schubert polynomials as sums of key polynomials, complementing work of Assaf and Schilling via reduced factorizations with cutoff. (Joint work with Sarah Gold and Elizabeth Milićević.)

Lilly Webster (University of Minnesota)

• Title: "A Brief Introduction to Cluster Algebras"

• Abstract: If you spend any amount of time around combinatorialists here at Minnesota, you will surely hear someone mention cluster algebras. First defined by Fomin and Zelevinsky in 2000 as a tool in Lie Theory, cluster algebras have since found applications in diverse areas of mathematics and have gone on to form an interesting topic of study in their own right. In this talk, I’ll define cluster algebras and then we’ll explore some of my favorite examples.  You don’t need any background, but you should bring something to write with!

Jesse Kim (UC San Diego)

• Title: "Combinatorial spiders and their webs"

• Abstract: Combinatorial spiders and webs are a diagrammatic way of studying invariant spaces in the representation theory of Lie groups via certain planar graphs. We will give an introduction to these objects in this talk, particularly the webs for $SL(2)$ and $SL(3)$. We will explain how algebraic relations in the representation theory side correspond to combinatorial rules, and how to compute using these combinatorial rules. This is a pretalk for Friday's talk.

Robbie Angarone (University of Minnesota)

• Title: "Totally nonnegative matrices and real-rooted polynomials in combinatorics"

• Abstract: This talk concerns matrices with real entries whose minors are nonnegative and polynomials with real coefficients whose roots are real. And, well... those don't sound like particularly combinatorial topics, right? Wrong! In this talk, I aim to convince you that totally nonnegative matrices and real-rooted polynomials are fundamentally combinatorial phenomena. I'll do this by introducing the Gessel-Lindstrom-Viennot lemma and the Jacobi-Trudi identity, two of our field's greatest hits.

Suki Dasher (University of Minnesota)

• Title: "An Introduction to Lie (Super)algebras, Quantum Groups, and Combinatorial Representation Theory"

• Abstract: Besides being inherently beautiful, Lie theory is a foundational subject related to many areas of mathematics. This talk will present one such instance. The first part of the talk will be an accessible introduction to Lie groups, Lie algebras, and some highlights of their theory. With this foundation, I will define and present examples of Lie superalgebras, which are generalizations of Lie algebras with a Z/2Z-grading. Finally, I will give an idea of how Lie (super)algebras give rise to algebraic objects called quantum groups, and how a combinatorial study of their representations can produce familiar special functions such as Schubert polynomials.

Emily Gullerud (University of Minnesota)

• Title: "Simplicial complexes for group cohomology"

• Abstract: We can construct various simplicial complexes which when endowed with a group action by G give insight into the structure of G. In this talk, we will explore a few of these complexes, primarily under the action of alternating groups. We will particularly be interested in complexes where certain fixed point spaces are contractible, as this will lead to a tantalizing application to group (co)homology: the ability decompose the cohomology of a group completely in terms of the cohomology of its subgroups.

Anastasia Nathanson (University of Minnesota)

• Title: "Matroids, posets, and their simplicial complexes"

• Abstract: A matroid is a combinatorial structure that abstracts the idea of independence, which has gotten a lot of attention in the mathematical world in recent years. We will use graphs to define a matroid and its flats and discuss a simplicial complex defined using these. This is what we call the “normal” story. We will then talk about the conormal story, a new development that uses information from both the matroid and its dual and see how it relates to the “normal” story. This is a pre-talk for Friday’s seminar. 

Kayla Wright (University of Minnesota)

• Title: "Path Algebras and Module Categories" 

• Abstract:  Have you ever thought to yourself... 🤔 why is the combinatorics group at UMN obsessed with quivers? 🤔 Maybe you have seen that UMN combinatorialists are equally obsessed with tableaux and you've seen that the reason for this is because these cute pictures actually dictate lots of representation theory of groups! It turns out that the obsession with  🏹 quivers 🏹  is also very justified. This is because quivers allow us to understand most of the representation theory of algebras! In this talk, we will give a crash course in representation theory for finite dimensional algebras. In particular, we will see examples of algebras that you can create from quivers and look at modules for them. I hope to illustrate an example of how you can see all of the modules for an algebra using a geometric model. This is a pre-talk for Friday’s seminar.

Theo Douvroupoulos (University of Massachusets, Amherst)

• Title: "Recursions and Proofs in Coxeter-Catalan combinatorics: A précis"

• Abstract: The noncrossing partition lattice NC(W) associated to a finite Coxeter group W, and its sibling, the W-cluster complex Y(W), have become central objects in Coxeter-Catalan combinatorics during the last 25 years. I am going to give brief descriptions of these objects and of two natural polynomials associated to them that will play a prominent role in my talk on Friday: the zeta polynomial of NC(W) and the face-enumerator of Y(W).

These polynomials are given via remarkable product formulas that are determined by natural invariants of W and its associated hyperplane arrangement and the main goal of Friday's talk will be to give proofs of these formulas that do not rely on the classification of Coxeter groups. In the pre-talk, I will define these arrangement-theoretic invariants, give some motivation for them, and if time permits sketch a special case of the proof that highlights one of the main technical tools: a Laplacian operator associated to W.

Anastasia Nathanson (University of Minnesota)  

• Title: "Permutation Action on the Chow Ring of a Matroid: a bande d'annonce"

• Abstract: Noticing symmetries of a geometric lattice, we will enhance a result of Adiprasito, Huh, and Katz about the Chow ring of a matroid defined by that geometric lattice. To do this, we will use the monomial basis defined by Feichtner and Yuzvinski when they defined the Chow ring itself, intuition developed by symmetries of a lattice, and fun worksheets. 

ULeonid Rybnikov (Harvard) 

• Title: "Spatial polygons and Representations of SL(2)" 

• Abstract: Fix a triangulation of a (flat) n-gon and consider all spatial n-gons in the 3-dimensional Euclidean space with the fixed integer side lengths $l_1,\ldots,l_n$ and (arbitrary) integer lengths of the diagonals forming the given triangulation, modulo isometries of the Euclidean space. It turns out that the set of all such spatial polygons is a disjoint union of tori, with the number of the connected components depending on the integers $l_i$ -- but not on the triangulation. I will explain how to make this statement obvious using the Representation Theory of SL(2). I will also discuss some Combinatorics standing behind this fact. 

Trevor Karn (University of Minnesota) 

• Title: "Combinatorial formulas for paving matroid Kazhdan--Lusztig polynomials: à l’envers"

• Abstract: Representation theory provides tools to prove remarkable combinatorial identities. In this talk, we outline the combinatorics of skew Young tableaux and the Littlewood-Richardson coefficients to provide a foundation for the results presented in the UMN Combinatorics Seminar on Friday, Feb 17. All tableaux will be drawn in English notation. 

Moriah Elkin (University of Minnesota)

• Title: "Twists of Grassmannian Cluster Variables"

• Abstract: he Grassmannian $\textrm{Gr}(k,n)$ is the space of $k$-dimensional subspaces of an $n$-dimensional space; we may embed it into $\left(\binom{n}{k}-1\right)$-dimensional projective space via the Plücker embedding. Scott showed that the coordinate ring of the Grassmannian is a cluster algebra -- it has distinguished generators called cluster variables, which include the Pl{\"u}cker coordinates, that are produced recursively -- and Postnikov described a way to encode this structure in a plabic graph. Each Plücker coordinate has a corresponding set of dimers, or ways to match vertices in the graph. We can assign a weight to each dimer so that summing the weights of a set of dimers gives a cluster variable, called the "twist" of the Plücker coordinate it came from. However, Plücker coordinates are not the only cluster variables: for $k=3$ and $n\geq 6$ some cluster variables are quadratic expressions in the coordinates, and for $n \geq 8$ some are cubic. We present a concise description of the sets of dimers corresponding to many of these non-Plücker cluster variables, via connectivity conditions on double and triple dimers, that allows us to efficiently compute their twists.

Sean Griffin (UC Davis) 

• Title: "Springer fibers: avec tableaux"

• Abstract: Springer fibers are a family of varieties that have remarkable connections to combinatorics and representation theory. We'll explore some of these connections, including a bijection between irreducible components of a Springer fiber and standard Young tableaux. We'll take a hands-on approach and start from scratch using simple linear algebra. This is the pre-talk for Friday's talk on "Springer fibers and the Delta Conjecture."

Mahrud Sayrafi (University of Minnesota) 

• Title: "The Polytope, the Toric Variety, and the Line Bundle: un conte fan-tastique"

• Abstract:  There are many ways to construct toric varieties: convex polytopes, polyhedral fans, even moduli of quiver representations! I'll describe the polytope construction for projective toric varieties and, by way of examples, show how cohomology of line bundles (a thing algebraic geometers care about, I hear) reduces to simple manipulations of polytopes.

Yibo Gao (University of Michigan) 

• Title: "Establishing the Sperner property of partially ordered sets"

• Abstract:   A ranked poset is Sperner if the size of its largest antichain does not exceed the size of its largest rank. Sperner showed in 1928 that the Boolean lattice is Sperner. In this talk, we explore two methods that establish the Sperner property of a ranked poset: sl2-representations and normalized flows, both of which establish Sperner's result instantly. The first method allows us to show that the weak Bruhat order of the symmetric group is Sperner and the second method allows us to show that the absolute order on the symmetric group is Sperner. 

Sheila Sundaram (Pierrepont School)  

• Title: "Homotopy and Homology of order complexes of posets"

• Abtract: This will be an expository talk on some techniques for computing homotopy type 

• and homology representations of order complexes of posets.  I hope to say something about Whitney homology, the Hopf trace formula,  recursive techniques and a little bit of discrete Morse theory.

Dorian Smith (University of Minnesota)

• Title: "Sandpile Groups of Cones Over Trees"

• Abstract: Let $G=(V,E)$ be a loopless undirected, connected graph. The sandpile group $K(G)$ is the finite abelian group isomorphic to the cokernel of the reduced graph Laplacian of $G$. Here we prove the exact group structure of $K(G)$ for two types of graphs $G$: a path with a cone vertex attached, commonly referred to as an n-fan, and that of a star with a cone vertex attached. The motivation is that we hope that these two families will be extreme cases, giving bounds on the structure of the sandpile groups for all other cones over trees on a fixed number of vertices. We also prove some additional properties of cones over trees.

Tianyuan Xu (Haverford College)

• Title:"Fully commutative Kazhdan–Lustig cells"

• Abstract: The Kazhdan–Lusztig (KL) cells of a Coxeter group W are certain special subsets of W constructed via the KL basis of the Iwahori–Hecke algebra associated to W. In this talk, we recall this construction and summarize some recent results on KL cells consisting of so-called fully commutative elements. We will emphasize the role played in these results by Viennot’s heaps, which are certain posets associated with fully commutative elements.

Son Nguyen (University of Minnesota) 

• Title: "Bender-Knuth Involutions on Linear Extensions of Posets: groupe de cactus"

• Abstract:  We study the group $BK_P$ generated by Bender-Knuth involutions on linear extensions of a poset P, an analog of the Berenstein-Kirillov group on column-strict tableaux. We explore the group relations, with an emphasis on identifying posets P for which the cactus relations hold in  $BK_P$.

Sylvester Zhang (University of Minnesota)

• Title: "Expansion Formulas for Surface Cluster Algebras"

• Abstract: A crash course on cluster algebras from surfaces, with emphasis on combinatorial constructions of T-paths (Schiffler) and Snake graphs (Schiffler-Musiker) and a SL_2 matrix formula (Musiker-Williams). This is a pre talk for Friday seminar and is not meant to be an introduction to the general theory of cluster algebras. 

Kaelyn Willingham (University of Minnesota)

• Title: "How Deep Learning Algorithms can help answer Combinatorial Questions"

• Abstract: The use of computers has become widespread in mathematics, beginning with the proof of the famous Four Color Theorem in the 1970's. In recent years, the development of machine learning algorithms has provided new tools for scientists to add to their scientific toolkit. This raises a natural question: could mathematicians also benefit from utilizing said tools? Some recent papers in combinatorics attempt to answer this question. By surveying two such papers, this talk will introduce the core principles of deep learning and examine the use of deep learning algorithms in building conjectures & counterexamples to problems in graph theory & polytope theory. In particular, we'll see how machine learning processes can predict properties of lattice polytopes, such as volume, dual volume, & reflexivity, with near-100% accuracy.

Marta Pavelka (University of Miami)

• Title: "Topology and Geometry in Combinatorics"

• Abstract: We review and explore definitions of cell and simplicial complexes and some of their interesting properties, like Shellability, Constructibility, and Cohen–Macaulayness. One of many applications of the mentioned complexes is to triangulations of manifolds and pseudomanifolds. How many triangulations of d-manifolds with a fixed number of d-simplices are there? We will see the motivation for this question and prove that there are 'too many' surfaces with a given number of triangles. 

Katie Bruegge (University of Kentucky)

• Title: "Lattice Polytopes and their Properties"

• Abstract: Polytopes are geometric objects that generalize polygons in the plane and polyhedra in 3-dimensional space. Of particular interest in geometric combinatorics are lattice polytopes, whose vertices have integer coordinates. In this talk, we will introduce lattice polytopes and several properties of interest in their study. We’ll also discuss some known results regarding these properties and give a survey of some families of polytopes that have interesting definitions, structure, or applications. 

Harper Niergarth (University of Minnesota)

• Title: "Spectral Faux Trees"

• Abstract: A classic problem in mathematics is whether one can "hear the shape of a drum". For graphs, one could phrase this in terms of the spectrum (the set of eigenvalues) of a matrix associated to a graph. In this talk, we will consider the existence and construction of graphs that are not trees but share their spectrum with a tree with respect to a matrix. We will consider several commonly studied matrices and see that the answer to whether one "can hear the shape of a tree" depends greatly on the matrix being considered. 

Patty Commins (University of Minnesota)

• Title: "Group Actions on Partially Ordered Sets"

• Abstract: Many combinatorially appreciated partially ordered sets (posets) come equipped with natural group actions. In this talk, we'll explore how enumerative and topological properties of posets can be generalized to explain how the group acts on spaces associated to the poset. We'll work mainly through examples, focusing especially on the boolean lattice and the lattice of set partitions. This talk will serve as a pretalk for Friday's combinatorics seminar. 

Kayla Wright (University of Minnesota)

• Title:  "What the Hecc is a Cluster Algebra? 🤯"

• Abstract: If you are interested in combinatorics, you've probably heard of the phrase 👻 cluster algebra 👻 before. As an astute young grad student, maybe you were even brave enough to google the phrase. But the Wikipedia article is a bit EEK! 

In my talk, I am hoping to show y'all that cluster algebras are not EEK at all, but very concrete, combinatorial objects that are quite tangible to work with. We will be defining cluster algebras from triangulations of polygons and if time permits, I will present my favorite result of my advisor's... the cluster expansion formula using snake graphs by 😍 Gregg Musiker 😍, 🐍 Ralf Schiffler 🐍, and 🐐 Lauren Williams 🐐. 

Carolyn Stephen (University of Minnesota)

• Title:  "Cluster Algebras and Root Systems"

• Abstract:  We will explore some remarkable connections between cluster algebras and root systems by considering triangulations of regular polygons. Investigating these connections will also lead to a more intuitive understanding of cluster algebras than the standard algebraic description.

Stephen Lacina (University of Oregon)

• Title:  "Lexicographic Shellability of Posets"

• Abstract: We will review the notion of shellability of finite simplicial complexes and its topological consequences. Then we will review the technique of lexicographic shelling of poset order complexes due to Björner and Wachs. We will focus on EL-labelings and give several examples which will feature in my talk on maximal chain descent orders in the Combinatorics Seminar on Nov. 18.

Lina Liu (University of Minnesota)

• Title:  "An Introduction to Exact k-Typed Parking Functions"

• Abstract: Honk honk!!! In this talk, we will introduce parking functions, a combinatorial object that was first introduced by Konheim and Weiss in 1966. We will start by contrasting how these classical parking functions compare with the exact k-typed parking function, introduced by Mark Dukes in 2021. The exact k-typed parking functions are tuples describing the parking of M cars on a street with M parking spots. We will also examine the uniqueness of each parking configuration, and classify them to see how different components of this parking arrangement interact. Note: No drivers' licence needed!