Professor, UC San Diego
Department of Mathematics
University of California, San Diego
La Jolla, CA 92093-0112
Phone: (858) 534-3985
Fax: (858) 534-5273
Current Courses (Fall 2019)
- Math 180A: Introduction to Probability (for Data Science). Course Webpage.
Here is a recent teaching statement: Kemp-Teaching-2016.pdf
UC San Diego, 2010 - Present
- Lie Groups: in Winter and Spring of 2015 and 2019, 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 2014 and 2018, 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.
- Stochastic Differential Equations: In Fall 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 follows the textbook "Introduction to Stochastic Integration" by Chung and Williams.
- Linear Algebra: In Fall 2012, I taught Math 20F, and in Winter 2017 and Winter 2018, I taught Math 18 -- the same course, but now revamped to hold a new primary
pre-calculus place in the curriculum (which was my successful proposal). 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.
- 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 Fall 2016 and Winter 2017, 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.
- Introduction to Probability Theory: in 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.
- 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.
- Introduction to Random Matrix Theory: in Fall 2011 and Fall 2013, I taught a topics graduate course, Math 247A.
The topic was random matrix theory, aimed at students with a background in probability and analysis. Lecture notes are available
- Complex Analysis: In Fall 2012 and Winter and 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
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
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
Under the aegis of my NSF CAREER Award, I run a summer research program each year for 4-5 undergraduate math majors.
Information about the program in general and current and past projects can be found here.
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.