# Mathematics

Albright’s mathematics faculty are committed to giving students the tools and knowledge needed to achieve their goals. Faculty are available for extra help during office hours or individual appointments. They employ the latest technology—such as various iPad applications, Matlab, Excel with Visual Basic, Maple, Mathematica, and TI-89 and TI-200 graphing calculators—combined with traditional and experiential learning techniques to prepare students for success after graduation. Small class sizes benefit students from their first calculus course all the way through Abstract Algebra and Real Analysis. The most advanced classes offer even more attention, which helps students to understand complex subjects.

**Mathematics Mission Statement:**

Mathematics offers an important way of viewing, analyzing and interpreting the world. It allows us to observe patterns, develop conjectures and make predictions. We strive to nurture a mathematical point of view in our students at all levels whether they are pursuing a liberal arts education in the humanities, in need of mathematical skills to complement other fields of study or wish to study mathematics as a core discipline.

We also aim to provide a thorough undergraduate training in mathematics for those students who wish to pursue graduate study in mathematics, teach mathematics in the secondary school systems, or work in various fields of business and industry.

The mathematics department will attain these goals by:

- Providing an up-to-date curriculum that is both broad in diversity of subject matter and rigorous in content.
- Expand students’ mathematical reasoning, problem solving and communication abilities.
- Promote connections with other disciplines through the co-concentration program.
- Develop students’ use and appreciation of technologies relevant to the study of mathematics.
- Provide a curriculum of general education mathematics courses with substantive skill development in quantitative and abstract reasoning.

Albright Mathematics students learn to:

- become confident in their ability to do mathematics and statistics,
- become mathematical problem solvers,
- learn to communicate, reason, and function mathematically in a mathematical, statistical, and technological learning community,
- apply both qualitative and quantitative methods,
- demonstrate oral communication skills,
- demonstrate written mathematical communication skills.

**Christopher J. Catone, Ph.D., Chair**

Associate Professor of Mathematics; Department Chair

610-921-7850

ccatone@albright.edu

**David B. Nawrocki, Ph.D. '87**

Associate Professor of Mathematics

610-921-7853

dnawrocki@albright.edu

**Brittany Shelton, Ph.D.**

Associate Professor of Mathematics

610-921-7851

bshelton@albright.edu

### Major in Mathematics

MAT 131, 132, 233

MAT 250

MAT 320

MAT 325

MAT 431

Six elective mathematics courses at the 300-400 level

MAT 491

PHY 201 (satisfies the General Studies Foundations Natural Science Requirement)

Students interested in pursuing graduate study in mathematics are strongly encouraged to take MAT 310, 334, 360, 435, 438, 440, and CSC141.

### Secondary Mathematics Education

Mathematics Majors preparing for a career in education take Math courses and a series of Education and other courses specified by the Education Department to meet Pennsylvania Department of Education regulations. As early as possible in their college experience, candidates for teacher certification in English should consult the Requirements section of the Education website and the chair of Education regarding specific course requirements. The Mathematics Education certification is a grades 7-12 program.

### Combined Major in Mathematics

For Classes before the Class of 2023

MAT 131, 132, 233

MAT 250

Three elective mathematics courses at the 300-400 level

MAT 491

Beginning with the Class of 2023

MAT 131, 132, 233

MAT 250

MAT 320

Two elective mathematics courses at the 300-400 level

MAT 491

Students interested in the actuarial profession should take MAT 131, 132, 233, 250, 310, 320, 360, 491, CSC 141, ECON 105, 207, 307, and should co-major in economics, accounting, or business. Exam P and Exam FM should be taken before graduation.

Courses

**MAT 102 – Topics in Mathematics **This course provides a general survey of mathematical topics that are useful in a variety of fields. Topics include: set theory, logical operators, topics in number theory, graph theory, Euler circuits, Hamiltonian circuits and search algorithms, voting methods, and topics in interest rate theory.

**MAT 110 – Elementary Statistics **This course gives students a general overview of modern statistics. Topics include: organization of data; probability and probability distributions; measures of central tendency and variability; normal distributions; sampling; hypothesis testing; correlation and regression.

**MAT 120 – Pre-Calculus Mathematics **This is a review of algebra and trigonometry intended to be taken before 131 or 125 by those students whose background in algebra and trigonometry is insufficient. The major emphasis is on the concept of functions. Elementary analytic geometry is discussed, along with algebra, composition of functions, inverse functions, trigonometry, and logarithmic and exponential functions.

**MAT 125 – Calculus with Business/Economics Applications **Designed as a one-semester course for concentrators in business administration or economics, topics such as linear functions and models; matrices and matrix algebra; linear systems; functions and graphs; derivatives and integrals; and extremization are included. Partial differentiation also is introduced.

**MAT 131 – Calculus and Analytic Geometry I **This course involves fundamental concepts of functions of one variable. Topics include: limits, continuity, differentiation, derivative applications, curve sketching, related rates, and maxima-minima problems. Introduction to indefinite and definite integration including the fundamental theorems, and numerical approximation techniques are also covered.

**MAT 132 – Calculus and Analytic Geometry II **This course is a continuation of MAT 131. Topics include transcendental functions, applications of integration, including volume, surface area, arc length, and work. Also covered are integration techniques, indeterminate forms, improper integrals, sequences and series, and Taylor’s theorem.

Prerequisite: MAT 131 with a C- or better.

**MAT 233 – Calculus and Analytic Geometry III **This course is a continuation of MAT 132. Topics include polar coordinates, parametric representation, vectors, analysis of functions of two or more variables, multiple integration, line and surface integrals, the divergence and Stokes’ Theorems.

Prerequisite: MAT 132

**MAT 250 – Foundation of Mathematics **This is an introduction to abstract mathematics. Topics include symbolic logic, methods of proof (direct, contradiction, and induction), set theory, relations, functions, countable and uncountable sets.

Prerequisite: MAT 132 or permission of the department

**MAT 300 – Discrete Mathematics **In this course students will study counting and finite structures. Topics from combinatorics include: permutations and combinations, binomial and multinomial coefficients, recursion, generating functions, and the twelvefold way. Topics from graph theory include graphs, trees, Eulerian and Hamiltonian cycles, planarity, and coloring. Additional topics may include but are not limited to Catalan numbers, Sterling numbers, partitions, Ramsey theory, Latin squares, design theory, and coding theory. Proof techniques, especially proof by induction, will be used throughout the course.

Prerequisite: MAT250 or permission of the department

**MAT 310 – Probability and Mathematical Statistics **This course introduces probability and mathematical statistics at the level presupposing knowledge of calculus. Descriptive and inferential statistics are included, along with hypothesis testing, estimation and analysis of variance.

Prerequisite: MAT132 & 250 or permission of the department

**MAT 320 – Linear Algebra **This is an introduction to matrix algebra, linear equations, linear independence, determinants, vector spaces, linear transformations, eigenvalues, eigenvectors, and diagonalization.

Prerequisites: MAT 132 & 250 or permission of the department

**MAT 325 – Abstract Algebra **This is an introduction to groups, rings, and fields, ideals, and polynomial rings.

Prerequisite: MAT 250 or permission of the department

**MAT 334 – Differential Equations **This course is a study of solution techniques for ordinary differential equations. Topics include the principal types of equations of first and second order, linear equations with constant coefficient, higher order linear and non-linear equations, series solutions, operational methods, systems of equations and modeling problems. Runge- Kutta and other numerical approximation methods may be covered.

Prerequisite: MAT 233 & 320 or permission of the department

**MAT 340 – Geometry **This course begins with a study of the most important ideas of Euclidean plane geometry, but also considers the historical significance of Euclid’s original postulates. Special attention is given to the notion of parallelism of lines and the resulting non-Euclidean geometries when the axiom of parallelism is altered. Differential geometry of curves and surfaces is also covered.

Prerequisites: MAT 233 & 250 or permission of the department

**MAT 360 – Numerical Analysis **This is a study of numerical methods used in interpolation, differentiation and integration, solutions of equations and systems of equations, solutions of differential equations, fitting of empirical data and error estimation. Some computer or calculator programming will be employed and a basic level of programming is assumed. Applications are made to the sciences and engineering.

Prerequisite: MAT 233 & 250 or permission of the department

**MAT 431 – Real Analysis **This course is designed to take a rigorous look at definitions, theorems and concepts taken from the foundational calculus courses. Rigorous treatment is given to topics such as continuity, mean-value theorems, analysis of functions of several variables, extremization and limits. Other topics include sequences, series, the Heine-Borel covering theorem and the Riemann Integral.

Prerequisites: MAT 233 & 250 or permission of the department

**MAT 435 – Partial Differential Equations **Topics include: Orthogonal functions; Sturm-Liouville system; initial and boundary value problems; Fourier series; higher transcendental functions; separation-of-variables method; and other methods of solution of equations of mathematical physics.

Prerequisite: MAT 233, 250 & 334 or permission of the department

**MAT 438 – Complex Analysis **This is an introduction to the theory of functions of a complex variable, including derivatives, integrals, Cauchy’s theorem, power series, theory of residues, and conformal mappings.

Prerequisite: MAT 233 & 250 or permission of the department

**MAT 440 – Introduction to Topology **This course introduces definitions and properties of topological spaces, metric spaces, continuity, homeomorphisms, separation axioms, compactness, connectedness, and fundamental group.

Prerequisites: MAT 233 & 250 or permission of the department

**MAT 480 – Advanced Topics in Mathematics **Designed to cover topics of interest that are not covered in other courses.

Prerequisites: Permission of the department

**MAT 491 – Senior Seminar **A seminar in topics selected by the course instructor in which independent learning is stressed. The student will present both oral and written reports on the topics covered. Each student will select an additional topic with the approval of the instructor. The student is expected to present both oral and written reports on their topic of choice. This seminar is to be taken in the fall of the student’s senior year.

### What Can I do With a Major in Mathematics?

### Monthly Problem

The Mathematics department hosts a monthly problem competition. It is organized by Dr. Shelton. 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=(d

_{1},d

_{2},…,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.

### Undergraduate Research

Students in the Mathematics department have opportunity to pursue scholarship outside of class. This is usually done via independent study, the ACRE program, or the honors program.

**Recent projects include:**

- Stephen Brittain – “Fixed and Periodic Points of Multiples of the Sum of the Digits Function”
- Katherine Betz – “Go Directly To Jail: The Mathematics Behind Family Game Night”
- Sophie Bass – “Applying Queueing Theory to Real World Situations”
- Tyler VanBlargen – “SafeCracker 40 and its Unique Solution”
- Matt Smith – “Minimal Surfaces, Soaps Films, and the Weierstrass-Enneper Representation”
- Hilary Hanford – “Comparing the effectiveness of individual tutoring and online videos”
- Danielle Smiley – “A Comparison of Riemann and Lebesgue Integration”

### The Ray Mest Senior Award

This award is given to a graduating senior. Recent awardees are listed below.

2021 – Stephen Brittain

2019 – Katherine Betz

2017 – Mat Costa

2014 – Hillary Hanford

2014 – Matt Smith

2013 – Dustin Hill

2012 – Joshua Blank