Erin Carmody

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. Advanced algebra examines sets of objects (numbers, matrices, polynomials, functions, ideas) and operations on those 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 those. 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. The pace and level of discussion is aimed at students who have experience reading and writing proofs.

Our existence lies in a perpetual state of change. An apple falls from a tree; clouds move across expansive farmland, blocking out the sun for days; meanwhile, satellites zip around the Earth, transmitting and receiving signals to our cell phones. The calculus was invented to develop a language to accurately describe and study the change that we see. The ancient Greeks began a detailed study of change but were scared to wrestle with the infinite; so it was not until the 17th century that Isaac Newton and Gottfried Leibniz, among others, tamed the infinite and gave birth to this extremely successful branch of mathematics. 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 topics of differentiation and integration. Numerous applications of the theory will be examined. 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.

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, infinite series, 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 of some other mathematically-related topic. This seminar is intended for students interested in advanced study in mathematics or science, for those preparing for careers in the health sciences or engineering, or for any simply wishing to broaden and enrich the life of the mind.

Rarely is a quantity of interest—tomorrow’s temperature, unemployment rates across Europe, the cost of a spring-break flight to Fort Lauderdale—a simple function of just one primary variable. Reality, for better or worse, is mathematically multivariable. This course introduces an array of topics and tools used in the mathematical analysis of multivariable functions. The intertwined theories of vectors, matrices, and differential equations and their applications will be the central themes of exploration in this yearlong course. Specific topics to be covered include the algebra and geometry of vectors in two, three, and higher dimensions; dot and cross products and their applications; equations of lines and planes in higher dimensions; solutions to systems of linear equations, using Gaussian elimination; theory and applications of determinants, inverses, and eigenvectors; volumes of three-dimensional solids via integration; spherical and cylindrical coordinate systems; and methods of visualizing and constructing solutions to differential equations of various types. Conference work will involve an investigation of some mathematically-themed subject of the student’s choosing.

One of the remarkable ironies of modern mathematics is that the success of its methodology has exposed its own limitations. In particular, the advances in mathematical foundations and logic of the early 20th century precipitated Kurt Gödel's incompleteness theorems—which establish that, for any effective axiomatic system of mathematics, there are mathematical truths that mathematics cannot prove. Gödel's proof is remarkable for both its philosophical implications and its very ingenuity. To prepare our study of the proof, the seminar will review basic logic, set theory, elementary number theory and the standard techniques of mathematical proof. Having completed a close reading of Gödel's proof, we will then explore the relationship between proof and understanding in more recent mathematical practice. Students will have an opportunity to strengthen their mathematical reading and writing abilities while engaging contemporary mathematical issues concerning the progress of the discipline, the role of computers in proof, and best practices in mathematical exposition. This seminar is recommended not only for the mathematically inclined but also for students interested in computer science, law, or philosophy.