## Spring 2019 Undergraduate Friday Seminars

**Speaker:** Dr. Sara Clifton

**Title:** Nice Rack: The Evolution of Deer Antlers and Other Mating Displays

**Abstract:** Species spanning the animal kingdom have evolved extravagant and costly ornaments to attract mating partners. Zahavi's handicap principle offers an elegant explanation for this: ornaments signal individual quality and must be costly to ensure honest signaling, making mate selection more efficient. Here, we incorporate the assumptions of the handicap principle into a mathematical model and show that they are sufficient to explain the heretofore puzzling observation of bimodally distributed ornament sizes in a variety of species.

**Speaker: **Prof. Zoi Rapti

**Title:** Can You be Both Central and Vacant? A Study of a Small Pond Network

**Abstract: **We will introduce a simple one-dimensional ordinary differential equation (Levin's equation) and concepts from network theory to analyze occupancy patterns in a small network of freshwater ponds. We will investigate various factors that determine whether a pond can be vacant or occupied by our organism (Daphnia pulex aka waterflea), which is prevalent in ponds and lakes of the Midwest. No knowledge of differential equations or network theory will be assumed: all background will be introduced in the talk.

**Speaker:** Prof. John P D’Angelo

**Title:** Functions of Operators

**Abstract:** We all know what we mean by the derivative operator D = d/dx. What might we mean by the square root of D? In other words, how do we take “half” of a derivative? More generally, how might we take g(D) for a more general function g. Starting from junior high school math (I am not joking!) we figure out the ideas that lead to a nice answer.

**Speaker: **Ravi Donepudi; PhD student

**Title: **When a Prime Number Ceases to be Prime

**Abstract: **Primes are commonly defined as those numbers whose only factors are 1 and themselves. This assumes that we only allow integers in their factorization. What happens if we allow fractions as factors or even irrational numbers? Will certain primes lose their status as "primes"? Will new "primes" be born to take their place? What does being prime even mean anymore? We will answer these and other questions which lead us to the exciting field of algebraic number theory.

**Speaker:** Agnès Beaudry; Assistant Professor at University of Colorado, Boulder

**Title:** Ants on Pants

**Abstract:** In this talk, I will give an introduction to manifolds and cobordism. What are manifolds? An ant living on a very large circle wouldn't know that it isn't living on the (flat) real line. In analogy, a d-manifold is a geometric object which, from an ant's perspective, looks flat like Euclidean space R^d, but which, from a bird's-eye view, can look curved or otherwise interesting, like the unit sphere in R^(d+1). What is cobordism? Think of a 2-dimensional surface that looks like a pair of empty pants. If the waist is the large circle which is the ant's universe, then the pants represent a transformation of the ant's world into a two circle universe. In analogy, a cobordism is a d+1 manifold with boundary which transforms one d-manifold into another. Two manifolds are cobordism equivalent if such a transformation exists. An interesting and difficult question is that of classifying manifolds. A raw classification in arbitrary dimensions is nearly impossible, and for this reason, mathematicians often settle for less precise answers. For example, can one classify manifolds up to cobordism equivalence? Come to my talk and find some answers to the ants on pants conundrum.

**Speaker:** Xiaomin Li; Undergraduate

**Title:** Beatty Sequences

**Abstract:** A Beatty sequence is a sequence of the form [a*n], where a is an irrational number and the bracket denotes the floor function. A remarkable result, called Beatty's Theorem, says that if a and b are irrational numbers such that 1/a+1/b=1, then the associated Beatty sequences "partition" the natural numbers. That is, every natural number belongs to exactly one of these two sequences. It is known that Beatty's Theorem does not extend directly to partitions into three or more sets, and finding appropriate analogs of Beatty's Theorem for such partitions is an interesting, and wide open, problem, which has applications to optimal scheduling questions. The goal of this project is to explore different constructions of partitions of integers into perturbed Beatty sequences and possible applications to optimal scheduling algorithms.

**Speaker:** Longzheng Chen; Undergraduate

**Title:** Introduction to Generating Functions

**Abstract:** n/a