In this course we'll learn about Language Theory as a tool for solving combinatorial problems. The course will cover many classical topics---finite automata, regular automata, and context-free grammars---as well as a selection of more advanced topics, including transducers, Ogden's Lemma, and decidability problems. Along the way, we'll present many applications of language theory to combinatorial problems.
Combinatorics is the study of structures in mathematics. This course starts by developing basic set theory and counting principles, and proceeds to the study of graphs, permutations, lattice walks, and more. Other topics that include proof techniques (induction, etc.), probability, and generating functions.
This course is a sequel to Math 8 and provides an introduction to calculus of vector-valued functions. Topics include differentiation and integration of parametrically defined functions with interpretations of velocity, acceleration, arc length and curvature. Other topics include iterated, double, triple and surface integrals including change of coordinates. The remainder of the course is devoted to vector fields, line integrals, Green's theorem, curl and divergence, and Stokes' theorem.
This course presents the fundamental concepts and applications of linear algebra with emphasis on Euclidean space. Significant goals of the course are that the student develop the ability to perform meaningful computations and to write accurate proofs. Topics include bases, subspaces, dimension, determinants, characteristic polynomials, eigenvalues, eigenvectors, and especially matrix representations of linear transformations and change of basis. Applications may be drawn from areas such as optimization, statistics, biology, physics, and signal processing.
This course starts by covering the symbolic method of Flajolet and Sedgewick, the asymptotic analysis of generating functions, and selected applications to combinatorial problems.
This course can be viewed as equivalent to Math 13, but is designed especially for first-year students who have successfully completed a BC calculus curriculum in secondary school. In particular, as part of its syllabus it includes most of the multivariable calculus material present in MATH 8. Topics include vector geometry, equations of lines and planes, and space curves (velocity, acceleration, arclength), limits and continuity, partial derivatives, tangent planes and differentials, the Chain Rule, directional derivatives and applications, and optimization problems. It continues with multiple integration, vector fields, line integrals, and finishes with a study of Green's and Stokes' theorem.
Notes from a few of the courses that I took as a graduate student are available.