Mathematics

Monthly Problem

The Mathematics department hosts a monthly problem competition. It is organized by Dr. Ohlinger. Problems are posted outside CCM121.

Mathematics Colloquium

November 2019
Dr. Caitlin Owens – DeSales University
“Color Me Proper”
Abstract: Vertex and edge coloring are popular topics of study in the field of graph theory.
A proper edge coloring of a graph is a coloring of the edges of the graph such that
two edges which share a vertex must be colored with different colors. The chromatic
index of a graph is the minimum number of colors required to properly edge color
a graph. In this talk, we will discuss some known results regarding the chromatic
index of graphs. Then we will discuss variations of the edge coloring problem. In
particular, we will replace the traditional proper edge coloring requirement with the
rule that every shortest path between each pair of vertices must be properly colored.
This type of edge coloring will not necessarily be a proper edge coloring of the entire
graph and so we will call such a coloring a very strong shortest path coloring. In the
remainder of this talk, we will discuss some results regarding very strong shortest
path colorings.

March 2019
Dr. Wing Hong Tony Wong – Kutztown University
“On John Conway’s Wizard Problem”
Abstract: John Conway, a famous mathematician, proposed the “wizard problem” in the 1960s.
Last night I sat behind two wizards on a bus and overheard the following:
•Blue Wizard: I have a positive integer number of children, whose ages are positive integers. The sum of their ages is the number of this bus, while the product is my own age.
•Red Wizard: How interesting! Perhaps if you told me your age and the number of your children, I could work out their individual ages?
•Blue Wizard: No, you could not.
•Red Wizard: Aha! At last, I know how old you are!
Apparently the Red Wizard had been trying to determine the Blue Wizard’s age for some time. Now, what was the number of the bus? There are a lot of interesting mathematics behind this riddle, and we are going to uncover some of them in this talk.

October 2018
Dr. Brian Kronenthal – Kutztown University
“Spying on cages, generalized quadrangles, and Moore!”
Abstract: You are a secret agent gathering intelligence on a meeting of enemy spies that is being held in a secure building. Your mission is to determine the number of spies in attendance. Unfortunately, the entrance is shielded from view, there are no windows, and you have no technology to let you observe the meeting. Fortunately, you have an informant inside; she describes two unusual observations:
• Every spy speaks to exactly 5 other spies.
• No “spy-triangle” is formed (for example, if James speaks to Sydney and Ethan, then Sydney and Ethan do not speak to each other).
Given this information, what is the minimum possible number of spies attending the meeting?
In this talk, we will study this and similarly posed problems. This will lead to a discussion of some special graphs called cages, and ultimately to incidence structures called generalized quadrangles. The properties of these objects, as well as their point-line incidence graphs, will be the primary focus of this talk, which will be accessible to undergraduates. All of the above terms will be defined and no particular mathematical background will be assumed.

April 2018
Dr. Eric Landquist – Kutztown University
“Blood, Sweat, and Tears: The Cable-Trench Problem and Some Applications”
Abstract: Suppose you need to connect each building on campus to the IT Center with its own underground internet cable. Trenches may be dug between any two buildings and any number of cables can be laid in a single trench. Due to bandwidth considerations, cables cannot be split. Where should you dig the trenches in order to minimize the total cost to dig (and fill) the trenches and also to purchase and lay the cables? This problem can be modeled as a graph-theoretic problem: vertices represent buildings (with a root vertex representing the IT Center), edges represent allowable trench routes between buildings, and edge weights are the Euclidean distances between buildings. Despite this seemingly simple model, finding the best solution for large instances of the Cable-Trench Problem is tantalizingly difficult (the sweat). In fact, the Cable-Trench Problem is NP-hard (the tears). In this talk, we will review related spanning tree problems and will discuss the Cable-Trench Problem, some of its natural generalizations, and some solution approaches. We will wrap up the talk with some important and surprising applications to medical imaging (the blood), radio astronomy, irrigation, car pooling, and possibly more. This talk will be suitable for students and faculty alike.

November 2017
Dr. Samantha Pezzimenti & Hannah Schwartz – Bryn Mawr College
“Introduction to Knot Theory”
Abstract: Since the 19^th century, mathematicians have studied knotted loops in space. Despite the concept of a knot being well-understood in artwork and culture, knot theory remains a rich field of topology with many unanswered questions. We will introduce some foundational ideas such as isotopy, invariants, and Seifert surfaces, in order to address some underlying questions in the field: When are two knots the same? When are they different? How can we distinguish them? This talk will be informal, interactive, and accessible.

April 2017
Dr. Michael A. Jones – Mathematical Reviews
“A Voting Theory Approach to Golf Scoring”
Abstract: The Professional Golfer’s Association (PGA) is the only professional sports league in the U.S. that changes the method of scoring depending on the event. Even without including match play or team play, PGA tournaments can be scored under stroke play or the modified Stableford scoring system; these two methods of scoring are equivalent to using voting vectors to tally an election. This equivalence is discussed and data from the 2004 Masters and International Tournaments are used to examine the effect of changing the scoring method on the results of the tournament. With as few as 3 candidates, elementary linear algebra and convexity can be used to show that changing how votes are tallied by a voting vector can result in up to 7 different election outcomes (ranking all 3 candidates and including ties) even if all of the voters do not change the way they vote! Sometimes, regardless of the voting vector the same outcome would have occurred, as in the 1992 US Presidential election. I relate this to the question: Can we design a scoring vector to defeat Tiger Woods? And answer it, retrospectively, for his record-breaking 1997 Masters performance.

October 2016
Dr. Eva Goedhart – Lebanon Valley College
“Minding My p’s and q’s: Using Continued Fractions to Solve Diophantine Equations”
Abstract: After a friendly introduction to continued fractions and Diophantine equations, we show how continued fractions can be used to solve a family of Diophantine equations.

March 2016
Dr. Hugh Denoncourt – CUNY
“Mathematical Oddities and Their Meanings”
Abstract: Many familiar operations have a simple interpretation. For example, Multiplication is repeated addition and exponentiation is repeated multiplication. Yet, even as early as high school algebra, we find ourselves raising numbers to negative and fractional values and somehow making sense of it. Similarly, we understand what it means to sum a collection of finite numbers. Yet, in calculus, we are tasked with making sense of assigning values to infinite sums. Is this expansion of familiar operations into unfamiliar territories just a game that mathematicians play? If so, are there rules or is it all pure invention? In this talk, we explore these questions through a tour of such oddities as complex numbers and divergent sums. We explore modern ideas that may help remove some of the mystery surrounding these objects.

November 2015
Dr. Tom Concannon – King’s College
“Understanding Gravity: Relativity, Black Holes, and Beyond”
Abstract: Newton originally conceived of a universal gravitational law in a explicit mathematical form, stating that the gravitational force is an attractive force between two massive objects proportional to the product of their masses and inversely proportional to the square of the distance between them. But Einstein improved on his definition and, in the process, unified space and time into one entity, space-time. This powerful formulation gave birth to radical new ideas about how we perceive motion in space and our passage through time as well as gave validity to the idea of black holes, strongly gravitationally bound systems from which not even light, the fastest thing in the universe, can escape. In this talk, we’ll explore the foundations of gravity, from Newton to Einstein and beyond, from its cosmological implications to its quantum ramifications via string theory, a possible unified description of all known physical laws.

April 2015
Dr. Stanley Ryan Huddy – Fairleigh Dickinson University
“An Introduction to Chaos Theory”
Abstract: If a butterfly flaps its wings in Brazil does this set off a tornado in Texas? Why are weather reports often incorrect? Join me as we answer these questions and more through a visually interactive introduction to chaos theory.

November 2014
Dr. Kathleen Ryan – DeSales University
“Non-Diplomatic Degrees!”
Abstract: A graph is a set of vertices together with a set of edges between some pairs of vertices.  We say that the degree of a vertex in a graph G is the number of edges incident to the vertex. If we create a list of integers consisting of the degree of each vertex in G, then the resulting list of integers is called the degree sequence of G.  Given a sequence of integers d=(d₁,d₂,…,d_n), we explore the answer to the following question:  When does there exist some graph whose degree sequence is d? We then explore a generalized version of this question for edge-colored graphs within certain graph families.