### Mathematics Colloquium

#### Spring 2023

###### Location and Time: Minard 306 at 3:00 PM (Refreshments at 2:30 in Minard 404)

***Special Colloquia or Tri-College Colloquia venues and times may vary, please consult the individual listing.**

##### Tuesday, January 31

###### Miklos Bona, University of Florida

*A method to prove that the solution of some enumeration problems is a non-rational generating function.*

Abstract: The solution of an enumeration problem is very often a generating function $F$. Some problems are too difficult for us to find the explicit form of $F$. In this talk, we will introduce a method that leads to negative results that are rare in this part of combinatorics. Pattern avoiding permutations will serve as our main examples of the use of the method, but other uses will be mentioned as well. We will discuss a 22-year-old conjecture of Zeilberger and Noonan. When our method applies, it shows that $F$ is not a rational function, which provides at least some explanation of the fact that the original enumeration problem is difficult.

The talk will be accessible to graduate students.

##### Tuesday, February 21

*Special Tri-College Colloquium at MSUM - please note the different location & time

###### Maria Alfonseca, North Dakota State University

*Uniqueness questions in convex geometry regarding the Euclidean ball and bodies of revolution*

**Location & Time: **

**Reception:** 2:30-3pm Maclean 0268 "Presidents" ITV Conference Room at MSUM**Talk:** 3:00-4pm Maclean 0274 at MSUM

Abstract: The Euclidean ball B is distinguished among convex bodies by the fact that many geometrical quantities (such as width, areas or perimeters of central sections, areas or perimeters of projections, etc) which generally depend on the direction are constant for B. Thus, many uniqueness problems have been posed by asking: "If a convex body K has property P (which B has), must K necessarily be a Euclidean ball?" In the same spirit, uniqueness questions can also be posed for bodies of revolution, since all their axial sections are symmetric with respect to a line, and all their sections perpendicular to the axis are Euclidean balls. In this talk we will survey some uniqueness results for the Euclidean ball, and prove some new results for bodies of revolution.

##### Tuesday, February 28

###### Rebecca R. G., George Mason University

*Canonical forms of neural ideals*

Abstract: Place cells are neurons in the hippocampus that fire when an animal is in a particular location. A neural ideal captures the firing pattern of a set of place cells, turning questions from neuroscience and coding theory into algebraic problems. In Curto, Itskov, et al. 2013, the authors gave an algorithm for computing the canonical form of a neural ideal, a unique set of pseudomonomial generators (products of $x_i$ and $1-x_j$) corresponding to the neural code. In this talk I will give a simple criterion for determining whether a neural ideal is in canonical form, as well as an improved algorithm for computing the canonical form of a neural ideal. This work is joint with Hugh Geller.

##### Tuesday, March 7

###### Jim Coykendall, Clemson University

*From polynomials to power series: the complete disaster*

Abstract: Two of the most fundamental and central constructions in commutative algebra are polynomials and power series. From the point of view of commutative rings with identity, one can think of these constructions from a purely algebraic perspective, but if one sprinkles a little topology in the mix, the power series ring R[[x]] is the completion of the polynomial ring R[x] at the ideal (x). To be sure, there are some similarities between R[x] and R[[x]] (for instance R[x] is Noetherian if and only if R[[x]] is Noetherian), but there are many, many striking differences (with sometimes surprising consequences). In this talk (which will be “general audience algebra” and should be accessible to graduate students), we will explore some of these similarities and differences from a number of perspectives (factorization, dimension-theoretic, and extension stability). We will give some interesting open problems, and if we are lucky, we might get to some new results hot off the presses!

##### Tuesday, March 28

###### Janet Page, North Dakota State University

*Gorenstein rings and the chicken McNugget problem*

Abstract: If chicken McNuggets are only sold in packs of 6, 9 and 20, then it’s impossible to order exactly 7, 8, 10, 11, or 13 nuggets. What is the largest number of nuggets that it’s impossible to order? It turns out that this question is deeply connected to a property in commutative algebra called Gorenstein-ness. In this talk, I’ll introduce and discuss Gorenstein rings and their algebraic and geometric properties. Then, we’ll delve deeper into some combinatorial examples coming from rings defined by finite posets and I’ll discuss some recent results.

##### Tuesday, April 18

###### Kateryna Tatarko, University of Waterloo

*Title: Reverse isoperimetric problems under curvature constraints*

Abstract: The well-known classical isoperimetric problem states that among all convex bodies of fixed surface area in $\mathbb{R}^n$, the Euclidean ball has the largest volume. In this talk, we will discuss the question of reversing the classical isoperimetric inequality for convex bodies. In particular, we explore a class of $\lambda$-convex bodies, i.e., convex bodies with curvature at each point of their boundary bounded below by some positive parameter $\lambda$.

##### Thursday, April 20

###### Vladyslav Yaskin, University of Alberta

*Title: On the distance between the projection of the center of mass of a convex body and the center of mass of its projection.*

Abstract: We show that there is a constant $D \approx 0.2016$ such that for every $n$, every convex body $K\subset \mathbb R^n$, and every hyperplane $H\subset \mathbb R^n$, the distance between the projection of the centroid of $K$ onto $H$ and the centroid of the projection of $K$ onto $H$ is at most $D$ times the width of $K$ in the direction of the segment connecting the two points. The constant $D$ is asymptotically sharp. Joint work with K. Tatarko and S. Myroshnychenko.

##### Tuesday, May 2

*Special Tri-College Colloquium at NDSU

###### Damiano Fulghesu, Minnesota State University - Moorhead

*Title: The Integral Chow Ring of the Space of Rational Stable Maps.*

Abstract: In this talk, we give a presentation for the integral Chow ring of the stack of rational stable maps whose image is an irreducible curve of odd degree, as a quotient of a polynomial ring. We describe an efficient set of generators for the ideal of relations, and compute them in generating series form.