### Algebra & Discrete Mathematics Seminar

**Spring 2022 Location:**Minard 404**Time:**Tuesday 10:00 --10:50 am**Organizer:**Jessica Striker

#### Spring 2022 Schedule

##### 3 May 2022

**Joseph Bernstein** (NDSU): Title TBA

**Abstract**: TBA

##### 26 April 2022

**Brandon Allen **(NDSU): Upper Bound on Chromatic Numbers of Coxeter Groups via Brink-Howlett Automaton

**Abstract****:** We first will give a historical perspective on computing chromatic numbers for Coxeter groups for non-exceptional types $A_n$, $B_n$, $C_n$, and $D_n$. The Brink-Howlett automaton turns an affine irreducible Coxeter into a finite state automaton. The main application of the Brink-Howlett automaton is to enumerate words of length $n$ for a given Coxeter group. Since we have a finite state automaton, and using the geometry of our groups, we can construct chromatic polynomials by the Tutte polynomial to give an upper bound on the chromatic number of the Coxeter group.

##### 19 April 2022

**Patricia Klein** (University of Minnesota): Geometric vertex decomposition and liaison

**Abstract: **Geometric vertex decomposition and liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties, the former primarily from an algebraic combinatorics perspective and the latter primarily from a commutative algebra/algebraic geometry perspective. In this talk, we will describe an explicit connection between these approaches. In particular, we describe how each geometrically vertex decomposable ideal is linked by a sequence of elementary G-biliaisons of height 1 to an ideal of indeterminates and, conversely, how every G-biliaison of a certain type gives rise to a geometric vertex decomposition. As a consequence, we can immediately conclude that several well-known families of ideals are in the **G**orenstein **l**iaison **c**lass of a **c**omplete **i**ntersection (glicci), including Schubert determinantal ideals, defining ideals of varieties of complexes, and defining ideals of graded lower bound cluster algebras. This connection also gives us a framework for implementing with relative ease Gorla, Migliore, and Nagel’s strategy of using liaison to establish Gröbner bases. This talk is based on joint work with Jenna Rajchgot.

##### 5 April 2022

**Joey Forsman** (NDSU): Two new closure operations on ideals

**Abstract**: We describe two closure operations on ideals which arise out of a ring operation called root closure. We describe these operations in the context of other well-known operations on ideas, rings. Several insights are gained due to the interplay between ideals and rings. In particular, the Rees algebra of an ideal is a central tool in the study of these operations.

##### 29 March 2022

**Torin Greenwood **(NDSU): Coloring the integers while avoiding monochromatic arithmetic progressions

**Abstract**: Consider coloring the positive integers either red or blue one at a time in order. How many integers W(k) do you need to color before you can guarantee that there are k equally-spaced integers all colored the same? Van der Waerden's classical result guarantees that W(k) exists for every k, but it remains challenging to find good bounds on the values of W(k). As a related question, we will look for colorings of {1, 2, ..., n} that minimize the total number of k-term monochromatic arithmetic progressions. We leverage a connection to coloring the continuous interval [0,1] that allows us to use tools from calculus. Our strategy will rely on identifying classes of colorings with permutations using mixed integer linear programming. Joint work with Jonathan Kariv and Noah Williams.

##### 22 March 2022

**Jessica Striker **(NDSU): Alternating sign matrix polytopes: Theme and variations

**Abstract**: Alternating sign matrices are {0, 1, −1}-matrices with deep connections to the symmetric group, enumerative combinatorics, algebraic geometry, and statistical physics. Their study from the geometric perspective of polytopes has been especially fruitful and includes many analogous properties to the classical Birkhoff polytope of doubly-stochastic matrices. In this talk, we’ll compare and contrast various properties of polytopes formed as convex hulls of alternating sign and related matrices, including their inequality descriptions, face lattices, and facet enumerations. This talk includes joint works with Dylan Heuer and Sara Solhjem.

#### Fall 2021 Schedule

**Fall 2021 Location:**Minard 404**Time:**Tuesday 10:00 --10:50 am**Organizer:**Cătălin Ciupercă

##### 7 December 2021

**Dennis Stanton **(University of Minnesota): Combinatorics of polynomials: orthogonal and type $R_I$

**Abstract**: I will review the combinatorial general theory of orthogonal polynomials via lattice paths. I will then compare these known results to those for the type $R_I$ orthogonal polynomials defined by Ismail and Masson in 1995. Explicit examples are given in the Askey scheme, along with new continued fractions. Open problems will be presented. This is joint work with Jang Soo Kim and Mourad Ismail.

**Note:** This talk will be on zoom. The link will be sent by email.

##### 26 October 2021

**Ben Noteboom **(NDSU): Decompositions of Symbolic Powers

**Abstract**: Symbolic powers of ideals have been a recent topic of study for commutative algebraists, particularly how they compare to regular powers. In this talk, we'll use tools from graph theory to find a decomposition of a certain class of symbolic powers, then use that decomposition to calculate an invariant of symbolic powers called the Waldschmidt constant.

##### 5 October 2021

**Jessica Striker **(NDSU): Web invariant polynomials

**Abstract**: We discuss classical results on bases for subspaces of invariant polynomials and their relation to symmetric group actions using combinatorial gadgets such as standard Young tableaux and matchings. We then generalize to very recent work involving increasing tableaux and webs. Both settings showcase interactions between combinatorics and algebra. This talk is based on joint work with Rebecca Patrias and Oliver Pechenik.