In 1988, at the initiative of Dr. Anne Hudson, the then *Department of Mathematics and Computer Science* at Armstrong State College began a near-weekly luncheon colloquium. Students and faculty would gather in the luxurious confines of Hawes 203 for hot dogs, spaghetti, taco salad, etc., and an enjoyable talk on some topic in mathematics or computer science. In 2003 this luncheon-colloquium series was named in honor of Anne and Sigmund Hudson.

Today, the colloquium is sponsored by the Department of Mathematics and takes place on **Wednesdays at 12:15pm **in University Hall, room 157 (unless otherwise noted). For a donation of a dollar—$2 for faculty and other non-students—you can enjoy a delicious snack and drink, invigorating conversation with students and faculty members, and a lecture, demonstration, or other event arranged by faculty, students and/or visitors. Please come.

Please contact Dr. Tricia Brown if you are interested in giving a presentation. Also, please send your email address to Dr. Brown if you would like to be added to the mailing list. If you're interested in helping with lunch preparation, please contact Dr. Brown. Her email is Patricia.Brown@armstrong.edu

## Fall 2017

### November 15 - James Brawner

*Elliptagons*

Abstract: Suppose a cow is in a field outside of a tall barn with a square base. The cow is tied to a rope, which is longer than the perimeter of the barn and stretches around the outside of the barn. What is the area of the region that the cow can graze?

### November 1 - Sungkon Chang

*Generalization of Pythagorean Triples and Heronian Triples*

Abstract: Three positive integers that satisfy the Pythagorean theorem are called Pythagorean triples. The method of generating Pythagorean triples is an inspiration to problems in algebraic geometry and advanced number theory, and its algebraic generalization called Fermat's Last Theorem inspired the birth of Commutative Algebra. In this talk we return to its geometric origin, namely right triangles, and consider its geometric generalization. Among its generalization Pythagorean triples possess a unique property, which I call a direct parametrization property. An algebraic demonstration of this property always felt charming to me, and in this talk I shall present my discovery of a great number of "generalized Pythagorean triples" that have a direct parametrization property. If three positive integers form the side lengths of a triangle whose area is integer valued, the three integers are called Heronian triples. They are less popular than Pythagorean triples, and the method of generating Heronian triples is less efficient than that of Pythagorean triples. While studying "generalized Pythagorean triples" I discovered a curious connection between Heronian triples and Pythagorean triples, which turned out to be known in the literature. In this talk I shall introduce a generalization of this connection to all triangles with integer side lengths, and an open problem on the distribution of integer-side triangles.

### October 18 - Dawit Denu

*Trace of a Point on a Rolling Circle Along a Smooth Curve*

Abstract: In this talk, I will present different types of well known plane curves such as cycloid, epitrochoid, hypotrochoid and others, together with their parametric equations. All these curves are generated by rolling a circle of radius r along a line or another circle. Then, I will show how to find the parametric equation of a curve traced by a point attached to a circle that rolls along any smooth curve. I will also talk about some interesting plane curves obtained by the trace of a point on a wheel mounted on a wheel mounted on another wheel, each turning at a different rate. Finally I will discuss about the symmetry of these curves and their relation with the frequencies in the Fourier series expansion of the function representing the curves.

### October 4 - Tricia Brown

*Minimal Free Resolutions of Domino Tilings*

Abstract: Tilings of closed regions of the plane have been of interest to mathematicians for years - especially in terms of enumeration and existence proofs. In this talk, we want to take tilings into the realm of commutative algebra. Beginning with the set of domino tilings of a 2 by n rectangle, we'll look at the monomial ideal generated by these tilings and demonstrate how the elements of the minimal free resolution of the ideal correspond to layered domino tilings. Then, we give enumerative results which connect the entries in the Betti tables with Fibonacci numbers.

## Spring 2017

### April 7 - James Brawner, Sungkon Chang, Jack Wagner

*Reflections on the 77th Annual Putnam Mathematical Competition*

Abstract: In December 3, 2016, an intrepid team of Armstrong students spent most of their Saturday working on a dozen frighteningly challenging mathematical problems. Why? Just another installment of the notoriously difficult William Lowell Putnam Mathematical Competition. Members of the team and faculty members will discuss solutions of their favorite problems from the most recent competition.

### March 22 - Jared Schlieper

*The Intersection of Football, Mobile Games, and Mathematics*

Abstract: Mobile games like Madden Mobile have sets that you need to complete as a way to gain better players or items. A question typically asked is "how many duplicates will I have before I complete a certain set?". This is the classic coupon collector's problem. We consider the history of the problem as well as several versions. We will consider how to approximate the distribution of coupons from a set in the Madden Mobile game and apply the coupon collector's problem solution.

### March 1 - Tricia Brown

*Counting k-omino towers*

Abstract: Path ideals are algebraic objects generated by all paths of a fixed length in a graph G. Associated with a path ideal is a unique minimal free resolution which in turn provides a topological invariant known as Betti number. Informally, we can understand the kth Betti number as the number of k-dimensional "holes" in the simplicial complex identified from the path ideal. In this talk, we will describe three constructions on trees which allow us to give a much simpler, combinatorial description of the Betti numbers.

### February 15 - Duc Huynh

*The Equation p=x²+qy² and Restriction on Class Groups*

Abstract: Let N ∈ ℤ be a positive integer. We wish to study the equation p=x²+qy² for rational primes p,q such that |C(ℤ(√-q))| ≥ N. We will look at various attacks on the problems and applications toward construction of elliptic curves and class fields.

### February 1 - Michael Tiemeyer

*What do you do for a living? I lead horses to water.*

Abstract: "Self Regulation refers to our ability to direct our behavior and control our impulses so that we meet certain standards, achieve certain goals, or reach certain ideals. Self regulation involves being able to set goals, monitor one's behavior to ensure that it is in line with those goals, and having the willpower to persist until goals are reached." In this talk, we discuss self-regulated learning, how we may encourage it in our students, and a method that students can learn to interact with technical texts.