This course is an introduction to abstract mathematics through the lens of discrete mathematics, a field centered on the study of mathematical objects such as sets, functions, and graphs. We will learn many techniques that allow one to rigorously prove mathematical facts, and we will apply those techniques to the study of discrete mathematics. Emphasis is placed on logical, abstract thinking and clear and precise mathematical writing.
Spring 2018 (Dartmouth College)
Math 60 is an honors version of Math 20. It covers the material of Math 20 but in more depth and from a much more theoretical perspective, with an emphasis on formal proofs. Several additional topics (including generating functions, branching processes, random walks) are added.
Summer 2017 (Dartmouth College)
Our capacity to fathom the world around us hinges on our ability to understand quantities which are inherently unpredictable. Therefore, in order to gain more accurate mathematical models of the natural world we must incorporate probability into the mix. This course will serve as an introduction to the foundations of probability theory. Topics covered will include some of the following: discrete and continuous random variables, random vectors, multivariate distributions, expectations; independence, conditioning, conditional distributions and expectations; strong law of large numbers and the central limit theorem; random walks and Markov chains.
Summer 2017 (Dartmouth College)
This course will provide an introduction to fundamental algebraic structures, and may include significant applications. The majority of the course will consist of an introduction to the basic algebraic structures of groups and rings. Additional work will consist either of the development of further algebraic structures or applications of the previously developed theory to areas such as coding theory or crystallography.
Spring 2017 (Dartmouth College)
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.
Winter 2017 (Dartmouth College)
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.
Fall 2016 (Dartmouth College)
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.
Spring 2016 (Dartmouth College)
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.
Winter 2016 (Dartmouth College)
This course starts by covering the symbolic method of Flajolet and Sedgewick, the asymptotic analysis of generating functions, and selected applications to combinatorial problems.
Fall 2015 (Dartmouth College)
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.
2011-2015 (Unisverity of Florida)
Notes from a few of the courses that I took as a graduate student are available.