The fundamental theorem of algebra and factoring polynomials 4. Vector spaces 5. Span and bases 6. Linear maps 7. Eigenvalues and eigenvectors 8. Permutations and the determinant 9. Inner product spaces Change of bases The spectral theorem for normal linear maps MATH F. Agent-Based Modeling with NetLogo. This course uses mathematical techniques to study the behavior of real-world systems. The focus is on agent-based models ABMs.
ABMs are especially useful for understanding dynamics in systems that are affected by individual agents within the system, and the variations, interactions, decisions, and adaptations of those agents. Students learn the guiding principles of ABM modeling and how to build models in NetLogo, then have the opportunity to create and analyze their own models. Sophomore-Junior Seminar. With varying subject matter, this writing-attentive seminar addresses both the oral and written communication of mathematics.
The seminar focuses on understanding why rigor is necessary and what constitutes effective communication of mathematical ideas to different audiences. Students practice peer editing and peer reviewing, and learn how to write effective grant and thesis proposals. Prerequisite s : MATH s Real Analysis.
An introduction to the foundations of mathematical analysis, this course presents a rigorous treatment of fundamental concepts such as limits, continuity, differentiation, and integration.
Majors and Minors
Elements of the topology of the real numbers are also covered. Prerequisite s : MATH , , and s Topics in Analysis. This course is a continuation of MATH The content varies semester to semester. Possible topics include sequences and series of functions, Fourier series, metric spaces, measure theory, theories of integration, and functions of several variables. Partial differential equations PDEs arise in several fields of study in the natural and social sciences.
This course provides an introduction to the theory and applications of partial differential equations. Students learn solution techniques and qualitative analysis for linear partial differential equations including initial and boundary value problems for second-order linear partial differential equations. Applied and Computational Linear Algebra. This course explores techniques for solving huge linear systems, covering both the theory behind the techniques and the computation.
The course reviews and further develops concepts from MATH and uses them to efficiently solve problems across natural and social sciences. Problems are drawn from numerical analysis, mathematical biology, data analysis and machine learning, imaging and signal processing, chemistry, physics, economics, computer science, engineering, and other disciplines.
Complex Analysis. This course extends the concepts of calculus to deal with functions whose variables and values are complex numbers. Instead of producing new complications, this leads to a theory that is not only more aesthetically pleasing, but is also more powerful. The course should be valuable to those interested in pure mathematics as well as those who need additional computational tools for physics or engineering. Topics include the geometry of complex numbers, differentiation and integration, representation of functions by integrals and power series, and the calculus of residues.
Abstract Algebra I. An introduction to basic algebraic structures common throughout mathematics. These may include the integers and their arithmetic, modular arithmetic, rings, polynomial rings, ideals, quotient rings, fields, and groups. Prerequisite s : MATH and s This course studies geometric concepts in Euclidean and non-Euclidean geometries. Topics include isometries, arc lengths, curvature of curves and surfaces, and tessellations, especially frieze and wallpaper patterns. The notion of "closeness" underlies many important mathematical concepts, such as limits and continuity. Topology is the careful study of what this notion means in abstract spaces, leading to a thorough understanding of continuous mappings and the properties of spaces that they preserve.
Topics may include metric spaces, topological spaces, continuity, compactness, connectedness, homotopy theory, fixed-point theorems, and applications of these topics in areas such as geographic information systems, robotic, and game theory. Optimal Control Theory with Economic Applications. Optimal control theory unifies numerous economic problems related to the creation and use of physical capital. This course introduces optimal control theory as a tool for dynamic optimization and applies that theory to a variety of classic economic problems involving capital.
Among the economic problems examined are optimal use of a renewable resource, optimal use of a nonrenewable resource, and optimal economic growth when growth begets pollution. The course includes formally proving Pontryagin's maximum principle, which characterizes dynamic optima, in the special case context of common economic problems. Prerequisite s : MATH , , and one economics course. Topics in Computational Mathematics.
Computational mathematics considers how computing can aid the study of mathematics. Computers may be used to perform numerical calculations involved in modeling, provide graphical understanding in analysis, or complete symbolic calculations in algebra. Mathematical study of computer algorithms can lead to more efficient algorithms and actual implementation of algorithms helps students deepen their understanding of mathematics while gaining appreciation for computers' capabilities as well as deficiencies. MATH A. Numerical Analysis. This course studies the best ways to perform calculations that have already been developed in other mathematics courses.
Mathematics (MATH) Courses
For instance, if a computer is to be used to approximate the value of an integral, one must understand both how quickly an algorithm can produce a result and how trustworthy that result is. While students implement algorithms on computers, the focus of the course is the mathematics behind the algorithms. Topics may include interpolation techniques, approximation of functions, solving equations, differentiation and integration, solution of differential equations, iterative solutions of linear systems, and eigenvalues and eigenvectors.
MATH B. Graph Algorithms. How can we create a network of cables between houses with minimum cost? Under what circumstances can a mail delivery van traverse all the streets in the neighborhood it serves without repeating any street?
Linear Algebra - As an Introduction to Abstract Mathematics (A Free Textbook)
Graph theory is the branch of mathematics that provides the framework to answer such questions. Topics may include definitions and properties of graphs and trees, Euler and Hamiltonian circuits, shortest paths, minimal spanning trees, network flows, and graph coloring. Some of the class meetings are devoted to learning to program in Maple. Students then write computer programs to provide solutions to questions such as the ones mentioned before. Dynamical Systems and Computer Science. The study of long-term behaviors of feedback processes, the field of dynamical systems is best understood from both theoretical and computational viewpoints, as each informs the other.
Students explore attracting and repelling cycles and witness the complicated dynamics and chaos a simple quadratic function can exhibit. Real and complex functions are considered. Simultaneously, students learn sound computer science fundamentals by writing Visual Basic programs that illustrate the theory of dynamical systems. In particular, students plot both orbit and bifurcation diagrams, Julia sets, and the Mandelbrot set.
The course explores both dynamical systems and computer science in depth, thus requiring four meetings per week. Independent Study. Students, in consultation with a faculty advisor, individually design and plan a course of study or research not offered in the curriculum. Course work includes a reflective component, evaluation, and completion of an agreed-upon product. Students may register for no more than one independent study per semester. This course may not be used to fulfill requirements for the mathematics major or minor in mathematics.
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- The Photoreading Whole Mind System.
- Philosophy of mathematics.
- Development of abstract mathematical reasoning: the case of algebra?
Special Topics. Content varies from semester to semester. Possible topics include chaotic dynamical systems, number theory, mathematical logic, measure theory, algebraic topology, combinatorics, and graph theory. Abstract Algebra II. Advanced topics in group theory, ring theory, and field theory are covered. Applications include geometric constructions, crystallography, and algebraic coding theory. Senior Thesis.
Prior to entrance into MATH , students must submit a proposal for the work they intend to undertake toward completion of a thesis. Open to all majors upon approval of the proposal. Required of candidates for honors. Students register for MATH in the fall semester.
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Students register for MATH in the winter semester. Senior Seminar.
Prior to entrance into MATH , students must submit a proposal for the section of senior seminar they wish to undertake. While the subject matter varies, the writing-attentive seminar addresses an advanced topic in mathematics. The development of the topic draws on students' previous course work and helps consolidate their earlier learning.
Students are active participants, presenting material to one another in both oral and written form, and conducting individual research on related questions. Instructor permission is required.
MATH C. Numerical Linear Algebra.