Todd Kemp

Current Course

18.022    Vector Calculus.

Teaching Experience

For a recent teaching statement, please see Kemp-Teaching-2008.pdf

• Vector Calculus: in Fall 2009, I will teach 18.022 at MIT. This is a large class (over 100 students). Topics include: Vector algebra, dot product, matrices, determinant. Functions of several variables, continuity, differentiability, derivative. Parametrized curves, arc length, curvature, torsion. Vector fields, gradient, curl, divergence. Multiple integrals, change of variables, line integrals, surface integrals. Manifolds with boundary, Stokes' theorem in one, two, and three dimensions.
• Analysis I: in Spring 2010, I will teach 18.100C at MIT; in Fall 2008, I taught 18.100B at MIT. These are standard introductory courses to real analaysis, at the level of "Principles of Mathematical Analysis" by W. Rudin. Topics include basic topology of metric spaces, continuity and differentiability, the Riemann-Stieltjes integral, and sequences and series of functions. (The contents of 18.100B and 18.100C are the same; 18.100C has an extra hour per week of face-time and extra written work for the students, who earn communication credit through the course.)
• Probability Theory: in Spring 2009, I taught 18.175 at MIT. This is the introductory graduate course in probability theory. Topics include laws of large numbers and central limit theorems for sums of independent random variables, introduction to large deviations, conditioning and martingales, and introduction to Brownian motion .
• Measure and Integration: in Spring 2008, I taught 18.125 at MIT. This is the introductory graduate course in measure theory, emphasizing Lebesgue measure, and including the basics of Banach and Hilbert spaces, and introduction to Fourier analysis.
• Stochastic Processes: in Fall 2007, I taught 18.177 at MIT. This was a graduate course in probability. After reviewing the basics of stochastic integration theory, we studied the Malliavin calculus (variational analysis on Wiener space) and its applications to PDE.
• Theoretical Calculus: in 2006/2007, I taught 18.014/18.024 at MIT; in 2005/2006, I taught Math 223/224 at Cornell University. Both are two-semester rigorous courses in calculus, linear algebra, and differential forms in one and many dimensional Euclidean space.
• Functional Analysis: in Spring 2004, I taught a topics graduate corse at Cornell University in functional analysis (compact operators and Schatten ideals).
• Calculus: in Summer 2002, I taught the standard second course in one-variable calculus at Cornell University.

Research Experience for Undergraduates

In Summer 2007, I ran an (NSF-funded) REU project at Cornell University. The topic was The Combinatorics of Free Probability. In studying *-moments of circular operators, we found a new and interesting family of convex posets associated to random strings and non-crossing partitions. The results are currently being compiled into two papers.

Calculus Textbook

I am working on a textbook, whose intended audience is the body of students in courses like 18.014/18.024 at MIT: a theoretical, mostly rigorous treatment of calculus in one and many variables, but more idea-centric than a standard "theorem, proof, corollary" course in real analysis. Here is a selection of course-notes I have created (they will eventually be incorporated into the book):

• R.pdf (Cantor's construction of the real numbers)
• Trig.pdf (rigorous development of the trigonometric functions)
• Chain.Rule.pdf (an almost-proof, and a proof, of the chain rule)
• Forms.pdf (differential forms on Euclidean space)