Teaching

I hadn't planned on going into mathematics when I started as an undergraduate, but after dabbling in some upper-level math courses I found the subject too beautiful to pass by. Perhaps because it took me a while to discover the aesthetic of mathematics for myself, one of my favorite parts of my job is being able to share an appreciation of this subject with students — especially students who have never enjoyed mathematics.

For research projects I've explored with undergraduates, please see my research page.

Current courses

± Applied Linear Algebra (Calculus & Mathematica), Sections F83 & Z83, UIUC, fall 2009: This is a Calculus & Mathematica version of the typical 415 course offered by the department. The class focuses on the singular value decomposition of a matrix and what it tells us about the geometric action of matrices. Since the SVD encodes much of classical matrix theory, this really is the `holy grail' of linear algebra.
Students follow the Mathematica notebooks which allow them to explore linear algebra and its applications through the lens of SVD. Topics range from interpretting SVD factors, to basic applications of the SVD, to understanding why the SVD theory works. Course content is managed through the ClassComm system, so there is no course webpage. A syllabus will be avaiable soon.

Previous courses

± Elementary Number Theory, Section X13, UIUC, Spring 2009: This is a basic introductory course to number theory with a focus on `elementary' methods. After covering basic topics like divisibility, primality, and the Fundamental Theorem of Arithmetic, we'll then introduce modular arithmetic, study basic arithmetic functions and their properties, learn about quadratic reciprocity, and investigate primitive roots for various moduli. If time permits, we'll also cover some of the current applications of elementary number theory, particularly the RSA cryptosystem.

This course includes a variety of methodologies to promote active and creative student engagement in the class. For instance, the primary course management tool is its own Wiki, located at http://math453spring2009.wikidot.com. Throughout the term students are generating and participating in discussions of course content through the Wiki's own forum. Students will be responsible for preparing their own Wiki pages and in-class presentations on number theoretic topics of their choice; these will occur at the end of the term.
± Calculus II, Section BL1, UIUC, Spring 2009: This is a second course in calculus, beginning with techniques in integration, moving through infinite series, and concluding with parametric equations and coordinate changes. The course meets en masse for lecture twice per week and breaks into six smaller discussion sections twice per week. As the instructor for this section, I am in charge of managing all details for the course, from giving the lectures to coordinating the dicussion sections to writing tests, quizzes and homework assignments.
± Elementary Number Theory, Section D13, UIUC, Fall 2008: This is a basic introductory course to number theory with a focus on `elementary' methods. After covering basic topics like divisibility, primality, and the Fundamental Theorem of Arithmetic, we'll then introduce modular arithmetic, study basic arithmetic functions and their properties, learn about quadratic reciprocity, and investigate primitive roots for various moduli. If time permits, we'll also cover some of the current applications of elementary number theory, particularly the RSA cryptosystem.

This course includes a variety of methodologies to promote active and creative student engagement in the class. For instance, the primary course management tool is its own Wiki, located at http://math453fall2008.wikidot.com. Throughout the term students are generating and participating in discussions of course content through the Wiki's own forum. Students will be responsible for preparing their own Wiki pages and in-class presentations on number theoretic topics of their choice; these will occur at the end of the term.
± Applied Linear Algebra (Calculus & Mathematica), Sections E83 & Z83, UIUC, Spring 2007: This is a Calculus & Mathematica version of the typical 415 course offered by the department. The class focuses on the singular value decomposition of a matrix and what it tells us about the geometric action of matrices. Since the SVD encodes much of classical matrix theory, this really is the `holy grail' of linear algebra.
Students follow the Mathematica notebooks which allow them to explore linear algebra and its applications through the lens of SVD. Topics range from interpretting SVD factors, to basic applications of the SVD, to understanding why the SVD theory works. Course content is managed through the ClassComm system, so there is no course webpage. A syllabus is avaiable here.
± Introductory Matrix Theory, Sections P1 & Q2, UIUC, Fall 2007: This course is basic introduction to the ideas of linear algebra, with an emphasis on real examples. The course necessarily covers a great deal of theory, and so serves the double-purpose of being something of an introduction to abstract mathematical thought. The course is designed by the Math Department at UIUC, and we follow their syllabus.
± Introductory Matrix Theory (Calculus & Mathematica), Section T8, UIUC, Fall 2007: This is a Calculus & Mathematica version of the typical 225 course offered by the department. Instead of the usual `systems of equations' approach to linear algebra, the class focuses on the singular value decomposition of a matrix and what it tells us about the geometric action of matrices. Since the SVD encodes much of classical matrix theory, this really is the `holy grail' of linear algebra.
Students follow Mathematica notebooks which allow them to explore linear algebra and its applications through the lens of SVD. Topics range from interpretting SVD factors to basic applications of the SVD. Course content is managed through the ClassComm system, so there is no course webpage. A syllabus is avaiable here.
± Math 51 Stanford University, Winter 2007 and Fall 2006: As the uber-TA for Math 51 in the Winter of 2007, I designed and mainted the course webpage for this edition of Math 51 (450+ enrollment). Additional responsibilities included writing and posting homework and exam solutions as well as various administrative tasks.

As the teaching assistant for the Accelerated Calculus for Engineers (ACE) section of Math 51 in the Fall of 2006, I led an extended discussion section twice a week for students enrolled in the Math 51 ACE. This page contains the few handouts I gave out to the class or solutions I posted for interested students.
± Stanford Engineering Summer Academy -- Mathematics Module Stanford University, Summer 2006: The Stanford Summer Engineering Academy (SSEA) is designed to help attract and maintain `a diverse student body to the School of Engineering' and is a `rigorous introduction to [Stanford's] engineering, math, and physical sciences programs.' The mathematics module prepares students for either Math 51 (linear algebra and multivariable calculus) or Math 41/42 (single variable calculus) by introducing material they will see during the regular term. I designed and taught the module as well as coordinated discussion sections with course assistants.
± Matrix Theory and its Applications Stanford University, Summer 2006: As the name suggests, this course is an introduction to linear algebra with a special emphasis on presenting applications. I designed and taught the course and used Mathematica as a tool for visualizing concepts or presenting real-world applications. You can find these Mathematica calculations on the course webpage under Mathematica Examples.
± Etale cohomology reading group Stanford University, Winter and Spring 2006.: This was a collection of grad students interested in learning a little more about Etale cohomology. It was informal, but we put up a webpage to keep track of what we covered. Use material at your own risk!
± Introduction to Calculus Stanford University, Winter 2006 (and 2005): This class is the first in a three part series which introduces students to single variable calculus; the emphasis in this part of the series is differentiation and some of its applications. I designed and taught the course.

MatLab exercises

Rob Easton and I wrote some MatLab exercises for the multivariable calculus/linear algebra series at Stanford. Each of these gives a `real world' application of the ideas which were covered in the classes.

± Assignment 1: Math 51: This assignment introduces students to MatLab and has them compute some problems dealing with Markov processes.
± Assignment 2: Math 51 (An alternate version with modified final exercise): Students explore Google's PageRank algorithm in this assignment.
± Assignment 3: Math 51 (An alternate version with modified bonus problem): This application has students investigating least squares and other polynomial fitting functions in MatLab.
± Assignment 1: Math 52: This problem set deals with Fourier transformations and finishes with an application to image processing. It also uses the following image.
± Assignment 2: Math 52: An introduction to complex numbers and integrals in the complex plane.
± Assignment 3: Math 52: Building on Assignment 2, this problem set has students develop a program that finds poles of meromorphic functions in the complex plane.