Philip Ording

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 changes that we see. 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.

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 changes that we see. 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, including artistic projects. This seminar is intended for students interested in advanced study in mathematics or science, preparing for careers in the health sciences or engineering, or simply wishing to broaden and enrich the life of the mind.

This seminar will study patterns in nature and design from the mathematical point of view. Examples will be primarily visual, including beadwork, braids, tilings, trees, waves, and crystals, among others. The workshop format of the class will give students the opportunity to discover, collaboratively, the structures that govern patterns. Students can expect to use both visual and logical reasoning to answer open-ended problems that involve hands-on experimentation and creative problem solving. By the end of the year, students will know how to reproduce a given pattern in one, two, or three dimensions; how to identify its symmetries; and how to compare it to related structures. For conference, there is a possibility of service-learning placements in community-based organizations, depending on availability. During the fall semester, students will meet with the instructor weekly for individual conferences. In the spring, we will meet weekly or every other week, depending on students’ needs and the progress of their conference projects. This course is recommended for any student interested in mathematics as the science of patterns, as well as those intending to study visual art or education. No particular math background is required.

The calculus of Newton and Leibniz was so successful that science forgave the logical shortcomings of its “fluxions” and “evanescent quantities.” In the 19th century, however, calculus evolved into the study of functions of a real variable—real analysis—which is a model of the foundational rigor that has come to define mathematics as a discipline. In the 20th century, the search for axioms of the real numbers uncovered subtle assumptions about spatial properties of the real line. These properties—such as continuity, separability, and dimension—do not depend on magnitude but on more general notions of position. The geometry of position, or topology as it is called today, is the study of exactly such properties. This yearlong seminar will begin with preliminaries of discrete mathematics, including symbolic logic, proof technique, and set theory. We will study these topics in the context of networks and surfaces, which are some of the most intuitive topological objects. This will be followed by an in-depth study of the real numbers, sequences and series, limits, continuity, the derivative, and the integral. To motivate our revision of these familiar calculus terms, the seminar will read and discuss important counterexamples, such as nowhere-differentiable continuous functions, rearrangements of infinite series, and the Cantor set. At the end of the year, we will return to topology. This will give us the opportunity to see how many of the geometric properties of curves, surfaces, and maps between them find a unified expression in terms of relations among point sets. Conference work will clarify seminar ideas and possibly their application to mathematical models in the natural sciences, computer science, or economics.

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.

This seminar will study patterns in nature and design from the mathematical point of view. Examples will be primarily visual, including beadwork, braids, tilings, trees, waves, and crystals, among others. The workshop format of the class will give students the opportunity to discover, collaboratively, the structures that govern patterns. Students can expect to use both visual and logical reasoning to answer open-ended problems that involve hands-on experimentation and creative problem-solving. By the end of the semester, students will know how to reproduce a given pattern in one, two, or three dimensions; how to identify its symmetries; and how to compare it to related structures. For conference, there is a possibility of service-learning placements in community-based organizations, depending on availability. No particular math background is required. This course is recommended for any students interested in mathematics as the science of patterns and strongly recommended for those studying education.

This lecture will present a formal analysis of ornament, using the mathematical principles of symmetry. Symmetric designs appear in material cultures from around the world and throughout history, from Bronze Age ceramics, 15th-century Islamic tiling, Latin American textiles, and Fijian bark cloth to the Arts and Crafts movement. Symmetry is a correspondence among the parts of a figure or object. Such a correspondence is often described in terms of an operation (“isometry,” in mathematical terms); for example, we will show that the symmetries of designs that repeat in one or two directions are comprised of just four types of operations: translation, rotation, reflection, and glide reflection. The collection of all possible symmetries of a figure comprises its “symmetry group,” and we will use this to classify finite and infinite ornamental designs. Many of the cultural artifacts that we study predate the mathematical theory of groups; in this sense, the lecture introduces the prehistory of modern mathematics. Museum visits and group conferences will offer students direct experience analyzing examples of visual structures in decorative art and design.