2014-2015 Mathematics Courses
Abstract Algebra: Theory and Applications
In pre-college mathematics courses, we learned the basic methodology and notions of algebra. We appointed letters of the alphabet to abstractly represent unknown or unspecified quantities. We discovered how to translate real-world (and often complicated) problems into simple equations whose solutions, if they could be found, held the key to greater understanding. But algebra does not end there. Abstract algebra examines sets of objects (numbers, matrices, polynomials, functions, ideas) and operations on these sets. The approach is typically axiomatic: One assumes a small number of basic properties, or axioms, and attempts to deduce all other properties of the mathematical system from these. Such abstraction allows us to study, simultaneously, all structures satisfying a given set of axioms and to recognize both their commonalties and their differences. Specific topics to be covered include groups, actions, isomorphism, symmetry, permutations, rings, fields, and applications of these algebraic structures to questions outside of mathematics.
Linear Algebra: The Mathematics of Matrices and Vector Spaces
An introduction to both the algebra and geometry of vector spaces and matrices, this course explores various mathematical concepts and tools used commonly in advanced mathematics, computer science, physics, chemistry, and economics. Developing systematic methods of solving systems of linear equations is the underlying theme, and applications of the theory will be emphasized. Topics of exploration include Gaussian elimination, determinants, linear transformations, linear independence, bases, eigenvectors, and eigenvalues. Applications will be pursued in the realms of game theory, optimization and linear programming, network/graph theory, and medical imaging, among others. Conference time will be allocated to exploring additional applications of linear algebra or exploring another branch of mathematics. This seminar is intended for students interested in advanced mathematics, computer science, the physical sciences, and/or economics. Prerequisite: prior study of Calculus or Discrete Mathematics.
Discrete Mathematics: A Gateway to Advanced Mathematics
Your voice will produce a mostly continuous sound signal when you read this sentence out loud. As it appears on the page, however, the previous sentence is composed of 79 distinct characters, including letters and a punctuation mark. Measuring patterns—whether continuous or discrete—is the raison d’être of mathematics, and different branches of mathematics have developed to address the two sorts of patterns. Thus, a course in calculus treats motion and other continuous rates of change. In contrast, discrete mathematics addresses problems of counting, order, computation, and logic. We will explore these topics and their implications for mathematical philosophy and computer science. The form of this seminar will be that of a (mathematical) writing workshop. We will work collaboratively to identify and reproduce the key formal elements of mathematical exposition and proof as they appear in both the discrete mathematics literature and each other’s writing. This seminar is a must for students interested in advanced mathematical study and highly recommended for students with an interest in computer science, law, logic, or philosophy.
Calculus II: Further Study of Motion and Change
This course continues the thread of mathematical inquiry following an initial study of the dual topics of differentiation and integration (see Calculus I course description). Topics to be explored in this course include the calculus of exponential and logarithmic functions, applications of integration theory to geometry, alternative coordinate systems, and power series representations of functions. For conference work, students may choose to undertake a deeper investigation of a single topic or application of the calculus or conduct a study in some other branch of mathematics. This seminar is intended for students interested in advanced study in mathematics or science, students preparing for careers in the health sciences or engineering, and any student wishing to broaden and enrich the life of the mind. The theory of limits, differentiation, and integration will be briefly reviewed at the beginning of the term. Prerequisite: One year of high-school calculus or one semester of college-level calculus. Permission of the instructor is required.
Calculus I: The Study of Motion and Change
The world is animated. Earth spins on its axis, as it rotates around the Sun; stock prices rise and fall; and an apple, acting solely in accordance with the laws of physics, falls onto the head of a modern day Newton. Calculus is the intriguing branch of mathematics whose primary goal is the understanding of the laws governing motion and change. The sum of the calculus—its methods, tools, and techniques—is often cited as one of the greatest intellectual achievements of humanity. Though just a few hundred years old, the calculus has become an indispensable research tool in both the natural and social sciences. Our study begins with the central concept of the limit and proceeds to explore the dual notions of differentiation and integration. Numerous applications of the theory will be examined. The pre-calculus topics of trigonometry and analytic geometry will be developed as the need arises. For conference work, students may choose to undertake a deeper investigation of a single topic or application of the calculus or conduct a study in some other branch of mathematics. This seminar is intended for students interested in advanced study in mathematics or science, students preparing for careers in the health sciences or engineering, and any student wishing to broaden and enrich the life of the mind. Prerequisite: One year each of high-school algebra and geometry. Permission of the instructor is required.
An Introduction to Statistical Methods and Analysis
An introduction to the concepts, techniques, and reasoning central to the understanding of data, this lecture course focuses on the fundamental ideas of statistical analysis used to gain insight into diverse areas of human interest. The use, abuse, and misuse of statistics will be the central focus of the course. Topics of exploration will include the core statistical topics in the areas of experimental study design, sampling theory, data analysis, and statistical inference. Applications will be drawn from current events, business, psychology, politics, medicine, and other areas of the natural and social sciences. Statistical software will be introduced and used extensively in this course, but no prior experience with the software is assumed. Conference work will involve working in a small group to conceive and execute a small-scale research study. This seminar is an invaluable course for anybody planning to pursue graduate work and/or research in the natural sciences or social sciences. No college-level mathematical knowledge is required.
Strange Universes: An Introduction to Non-Euclidean Geometries
If you draw two straight lines on a piece of paper, it’s not difficult to keep them from crossing. Imagine, however, that the lines extended in both directions off the page and without end. Do these hypothetical lines cross? Surprisingly, this mundane question goes to the heart of our modern conception of space. Your experience might suggest that the lines will cross unless they head off the edge of the page at exactly the same angle. In that case we call the lines parallel; and this is the answer Euclid asserts with his fifth (or “parallel”) postulate of the Elements. Roughly 2,000 years later, mathematicians came to the shocking realization that lines need not obey the parallel postulate. The resulting non-Euclidean geometries were so unexpected to the mathematicians who first conceived of them that one, János Bolyai, remarked, “Out of nothing I have created a strange new universe.” This course will explore the alternatives to Euclidean geometry that first appeared in the 18th century. These include hyperbolic, spherical, projective geometry, as well as more idiosyncratic geometries that we will devise together. Our exploration of these strange universes will be aided by visualizations that include drawing, computer graphics animation, and video game technology. Throughout, we will discuss the impact of the non-Euclidean revolution on astronomy, philosophy, and culture.
Beyond Perspective: Mathematics and Visual Art
For many, the experience of doing mathematics is dominated by formulas, order, and following rules. It might come as a surprise that some mathematicians (especially in so-called “pure mathematics”) view what they do as more of an art than a science. For example, Georg Cantor, a leading mathematician of the early modern era, claimed that the “essence of mathematics lies entirely in its freedom.” This course will explore similarities between mathematical and contemporary art practices. We will study a variety of ways that mathematics and art pose questions. We will also investigate the intersection of the two disciplines, including selected applications of mathematics to art-making (from the Renaissance on) and the presence of modern mathematical attitudes in contemporary art (from the historical avant-garde through the present). This course assumes no particular expertise with mathematics, studio art, or art history. Seminar readings, guest speakers, and a program of art viewings will establish a basis for investigating the relevance of fundamental mathematical concepts to contemporary art. These concepts will include axiom, proof, structure, and symmetry, among others. Conference work will involve more in-depth study of individual artists, art works, mathematical ideas, or student work in mathematics and/or art.