Todd Kemp

Current Course ("Winter" 2020)

• MATH 286: Stochastic Integration & Stochastic Differential Equations.  Course Webpage.

Teaching Experience

Here is an increasingly out-of-date teaching statement: Kemp-Teaching-2016.pdf

UC San Diego, 2010 - Present

• Stochastic Differential Equations: In Fall 2022 and 2018, I taught Math 286, an advanced graduate course on stochastic integration and stochastic differential equations. Topics included the stochastic integral with respect to a square integrable martingale integrator, the Ito / Doleans isometry, extension to local martingales and semi-martingales, quadratic variation and the Doob-Meyer decomposition, Ito's formula with applications to stochastic processes, the Feynman-Kac formula and other applications to PDE and harmonic analysis, Brownian local time, martingale representation theorems, and introduction to stochastic differential equations: strong existence and uniqueness for locally Lipschitz and sublinear growth drift and diffusion, weak solutions, and the Cameron-Marton-Girsanov theorem. We loosely followed the textbook "Introduction to Stochastic Integration" by Chung and Williams.
• Introduction to Random Matrix Theory: in Spring 2022, Fall 2013 and Fall 2011, I taught a topics graduate course (Math 289A or 247A). The topic was random matrix theory, aimed at students with a background in probability and analysis. Lecture notes are available here.
• Linear Algebra: In Winter 2022, Winter 2018 and Winter 2017, I taught Math 18, and in Fall 2012 I taught Math 20F (a previous version of the same course). There were 180 students in 2012, 310 students in 2017, and more than 400 (bewtween two lectures) in 2018. Topics included: Matrix algebra, Gaussian elimination, determinants; linear and affine subspaces, bases of Euclidean spaces; eigenvalues and eigenvectors, orthogonal matrices, orthogonal projections, and diagonalization of symmetric matrices. The course included a regular Matlab component emphasizing the computational aspects of the material. The textbook was "Linear Algebra and Its Applications", by David C. Lay.
• Probability Theory and Stochastic Processes: From Fall 2020 through Spring 2021, I taught Math 280ABC. Math 280 is the three-quarter graduate sequence in measure theoretic probability theory and stochastic processes. Topics covered through the whole sequence include the measure-theoretic foundations of probability theory, independence, the Law of Large Numbers, large deviations, convergence in distribution and associated metrics, Stein's method, the Central Limit Theorem, conditional expectation, martingales, Markov processes, and Brownian motion. I did not use a textbook; as a main source I used Bruce Driver's excellent notes Probability Tools with Examples. This was a fully remote asynchronous course, whose prerecorded lectures can be found on YouTube.
• Hidden Data in Random Matrices: In Winter 2020, I designed and taught a new topics undergraduate course, cross-listed with Data Science (DSC 155 & MATH 182). The topic was the theory of Principal Component Analysis (PCA). The course began with several weeks of requiste background material in linear algebra, probability, and statistics. I then introduced the main PCA algorithm, and studied its complexity. We then devled into the classical theory of random sample-covariance matrices, and the Marcenko-Pastur distribution for their eigenvalues. Finally, we introduec the spiked models pioneered by Baik, Ben Arous, and Peche in the early 2000s, and studied their affect on outlier eigenvalues and distortions (the "BBP Transition"). Accompanying the course was weekly homework and Python-based labs in Jupyter Notebooks, for the students to work through all the concept in reference to real high-dimensional data sets. They also produced a final project, using the methods of the class to analyze a high-dimensional data set of their own interest.
• Introduction to Probability Theory: in Fall 2019, Winter 2016, and Fall 2010, I taught Math 180A. I used "Elementary Probability for Applications" by Rick Durrett. Topics included probability spaces, combinatorial probability, conditional probability, random variables, discrete and continuous probability distributions, joint distributions, independence, variance and moments, the Laws of Large Numbers, and the Central Limit Theorem. The Fall 2019 version of the course was an experimental "Data Science flavored" course, which included a few extra topics (confidence intervals, moment generating function, Poisson process) and included an additional Python-based lab component (through Jupyter Notebooks) devoted to real-world data sets and their analysis using probabilistic ideas.
• Lie Groups: in Winter and Spring of 2019 and 2015, I taught Math 251B/C, a two quarter topics graduate course on Lie Groups. Topics included basics of Lie groups and group actions, Lie algebras and left-invariant vector fields, the exponential map and adjoint maps, normal subgroups and ideals, the Baker-Campbell-Hausdorff formula, the Lie correspondence, quotients and covering groups, the Lie-Cartan theorem, compact groups and maximal tori, the Weyl group, Cartan's torus theorem, the Weyl integration formula, basic representation theory, representation theory of semisimple Lie algebras, representative functions and characters for compact groups, and the Peter-Weyl theorem. I sampled from several sources, and I produced lecture notes (combined for this course and the preceding course on differential geometry) which can be downloaded here.
• Differential Geometry: in the Fall of 2018 and 2014, I taught Math 250A: the first graduate quarter course in differential geometry. Topics included: a review of topics from calculus and differential equations; topological manifolds; smooth structures, smooth maps; tangent spaces and the tangent bundle, differentials; vector fields, push forwards, Lie brackets; flows, Lie derivatives; the cotangent bundle, 1-forms, line integrals, and the Poincare lemma. The textbook was be Introduction to Smooth Manifolds by John M. Lee.
• Quantum Mechanics (for Mathematicians): In Spring 2017, I taught a topics graduate course, Math 247A. The topic was quantum mechanics, from a rigorous functional analytic viewpoint. Topics included a review of Hamiltonian mechanics, the key experiments of early 20th Century physics that necessitated quantum mechanics, the Borh-de Broglie "old quantum theory", the position and momentum operators (with careful domain analysis), the axioms of quantum mechanics, Schrodinger's equation, the quantized harmonic oscillator, the Heisenberg uncertainty principle, and a detailed analysis of the bound states of the Hydrogen atom (developing the necessary tools from the representation theory of SO(3) along the way). This was a very popular course, with almost 60 students attending and over 25 enrolled for a letter grade at the end (which required a final term paper and final exam to be completed). The textbook was Quantum Mechanics for Mathematicians by Brian Hall.
• Functional Analysis: In Winter 2017 and Fall 2016, I taught Math 241A and 241B, a two quarter sequence on functional analysis. In the first quarter, I taught roughly from chapters 1, 2, 3, 6, and 7 of John B. Conway's "A Course in Functional Analysis". Topics covered included basics of Hilberts spaces and Banach spaces, bounded operators on Hilbert spaces and Banach spaces, commutative C* algebras, and the Spectral Theorem for (commuting families of) normal operators on Hilbert spaces. The second course focused largely on unbounded operators, up to and including the spectral theorem for selfadjoint unbounded operators, and concluding with the Hille-Yosida theorem on strongly continuous contraction semigroups. Some of this material is in Chapter 10 of Conway, but I largely used outside sources (primarily notes from Bruce Driver and Leonard Gross).
• Foundations of Real Analysis II: in Spring 2016, I taught Math 140B, the second quarter of real analysis, as taught out of chapters 5-8 in "Principles of Mathematical Analysis" by Rudin; see 18.100B/C below. The topics covered included differentiation, the Riemann-Stieltjes integral, sequences and series of functions, power series, Fourier series, and special functions. Course notes can be found here.
• Foundations of Real Analysis I: in Winter 2016 and Winter 2014, I taught Math 140A, the first quarter of real analysis, as taught out of the first four chapters in "Principles of Mathematical Analysis" by Rudin; see 18.100B/C below. The topics covered included basic properties of the real numbers, complex numbers, metric spaces, sequences and series of real numbers, functions of a real variable and continuity.
• Vector Calculus: in Fall 2014, I taught Math 20E: vector calculus. This was a very large class (over 250 students). Topics included change of variable in multiple integrals, Jacobian, line integrals, Green's theorem, vector fields, gradient fields, divergence, curl, spherical/cylindrical coordinates, Taylor series in several variables, surface integrals, Stokes' theorem, Gauss' theorem, conservative fields. The textbook was Vector Calculus, 6th Edition by Jerrold Marsden and Anthony Tromba.
• Calculus for Science and Engineering II: in Spring 2014, I am taught Math 20B: integral calculus of one variable and its applications, with exponential, logarithmic, hyperbolic, and trigonometric functions. Topics also included methods of integration, infinite series, polar coordinates in the plane, and complex exponentials. The textbook was Calculus: Early Transcendentals, second edition, by Jon Rogawski; published by W. H. Freeman and Company; 2012.
• Complex Analysis: From Fall 2012 through Spring 2013, I taught Math 220ABC. Math 220 is the three-quarter graduate qual sequence in the subject. Topics covered through the whole sequence include Complex numbers and functions, Cauchy theorem and its applications, calculus of residues, expansions of analytic functions, analytic continuation, conformal mapping and Riemann mapping theorem, harmonic and subharmonic functions, the Dirichlet problem and principle, Caratheodory's Theorem, Hardy Spaces, Bergman Spaces, and boundary smoothness of conformal maps. The textbook used was "Functions of One Complex Variable, Volumes I and II" by John B. Conway.
• Stochastic Processes: In Spring, 2012, I taught Math 285, a graduate course in stochastic processes (without measure theory). Topics included Markov chains, hidden Markov models, martingales, Brownian motion, and Gaussian processes. The textbook loosely followed was "Introduction to Stochastic Processes, 2nd Edition" by G. Lawler. Lecture notes are available here.
• Multivariate Calculus: in Fall 2011, I taught Math 20C: multivariate calculus for scientists and engineers. Topics included: vector algebra, dot product, cross product, determinants in 2 and 3 dimensions. Parametrizations of lines and planes; quadric surfaces. Calculus of vector-valued functions; arclength, surface area, curvature. Multivariate functions; limits and continuity, partial derivatives, differentiability. Linear approximation. The chain rule. Optimization; Lagrange multipliers. Double and triple integrals. Polar, cylindrical, and spherical coordinates; change of variables.
• Introduction to Free Probability: in Spring 2011, I taught a topics graduate course, Math 247A, at UCSD. The class was an introduction to free probability, aimed at students in functional analysis, probability theory, and combinatorics. The textbook was "Lectures in the Combinatorics of Free Probability" by A. Nica and R. Speicher. Lecture notes are available here.

MIT and Cornell University, 2002 - 2010

• Real Analysis: in Spring 2010, I taught 18.100C at MIT; in Fall 2008, I taught 18.100B at MIT. These are standard introductory courses to real analysis, 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.)
• Vector Calculus: in Fall 2009, I taught 18.022 at MIT. This was a large class (about 90 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's theorem in one, two, and three dimensions.
• 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.

CURE: Collaborative Undergraduate Research Experience

Since 2013, I have run an NSF-funded undergraduate summer research program called CURE, for 4-5 undergraduate math majors and one math graduate student mentor. The CURE program is currently on hold due to the Covid-19 pandemic. I hope it will return in the near future.

Research Experience for Undergraduates