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Teaching

At UPenn

Spring 2015:

Math 104: Calculus I (SAIL):
Brief review of High School calculus, applications of integrals, transcendental functions, methods of integration, integration applications, sequences, infinite series, Taylor's theorem. Use of symbolic manipulation and graphics software in calculus.
On SAIL: Structured Active In-class Learning, an alternative to pure lecturing in which students do group-work in class under the supervision of the instructor and several teaching assistants, with content delivery mostly outside of class. See Teaching a SAIL Class

Math 260: Honors Calculus:
Honors version of Math 240. Linear algebra: vectors, matrices, systems of linear equations, eigenvalues and eigenvectors. Series solutions of ordinary differential equations, Laplace transformations and systems of ordinary differential equations. Use of symbolic manipulation and graphics software.

Math 603: Graduate Abstract Algebra II:
Module theory: Tensor products, flat and projective modules, introduction to homological algebra, Nakayama's lemma. Field theory: separable and normal extensions, cyclic extensions, fundamental theorem of Galois theory, solvability of equations.

Fall 2014:

Math 114: Calculus II:
Functions of several variables, vector-valued functions, partial derviatives and applications, double and triple integrals, conic sections, polar coordinates, vectors, Vector calculus: functions of several variables, vector fields, line and surface integrals, Green's, Stokes' and divergence theorems. Applications to physical sciences. Use of symbolic manipulation and graphics software in calculus.

Math 170: Ideas in Mathematics:
Topics from among the following: logic, sets, calculus, probability, history and philosophy of mathematics, game theory, geometry, and their relevance to contemporary science and society.

Math 602: Graduate Abstract Algebra I:
Group theory: permutation groups, symmetry groups, linear algebraic groups, Jordan-Holder and Sylow theorems, finite abelian groups, solvable and nilpotent groups, p-groups, group extensions. Ring theory: Prime and maximal ideals, localization, Hilber basis theorem, integral extensions, Dedekind domains, primary decomposition, rings associated with affine varieties, semisimple rings, Wedderburn's theorem. Linear algebra: Diagonalization and canonical form of matrices, bliniear forms, quotient spaces, dual spaces, tensor products, exact sequences, exterior and symmetric algebras.

Spring 2014:

Math 501: Geometry:
The course moves from a study of extrinsic geometry (curves and surfaces in n-space) to the intrinsic geometry of manifolds. After a section on tensor algebra, we study manifolds and intrinsic geometry, including metrics, connections, and the Riemann curvature tensor. Other topics as time permits.

Math 601: Algebraic Topology:
Covering spaces and fundamental groups, van Kampen's theorem and classification of surfaces. Basics of homology and cohomology, singular and cellular; isomorphism with de Rham cohomology, Brouwer fixed point theorem, CW complexes, cup and cap products, Poincare duality, Kunneth and universal coefficient theorems, Alexander duality, Lefschetz fixed point theorem.

Fall 2013:

Math 500: Topology:
Point set topology: metric spaces and topological spaces, compactness, connectedness, continuity, extension theorems, separation axioms, quotient spaces, topologies on function spaces, Tychonoff theorem. Fundamental groups and covering spaces, and related topics.

Summer 2012:

Math 114: Calculus II:
See above.

Spring 2012:

Math 203: Proving Things: Algebra:
This course focuses on the creative side of mathematics, with an emphasis on discovery, reasoning, proofs and effective communication, while at the same time studying arithmetic, algebra, linear algebra, groups, rings and fields. Small class sizes permit an informal, discussion-type atmosphere, and often the entire class works together on a given problem. Homework is intended to be thought-provoking, rather than skill-sharpening.

Fall 2011:

Math 240: Calculus III: Linear algebra:
vectors, matrices, systems of linear equations, eigenvalues and eigenvectors. Series solutions of ordinary differential equations, Laplace transforms and systems of ordinary differential equations. Use of symbolic manipulation and graphics software.

Spring 2011:

Math 104: Calculus I:
See above.

Fall 2010:

Math 104: Calculus I:
See above.

At Harvard

Spring 2009:

Math 118r: Dynamical Systems:
Introduction to dynamical systems theory with a view toward applications. Topics include existence and uniqueness theorems for flows, qualitative study of equilibria and attractors, iterated maps, and bifurcation theory.

Spring 2007:

Math 23b: Linear Algebra and Real Analysis II:
A rigorous, integrated treatment of linear algebra and multivariable calculus. Topics: Riemann and Lebesgue integration, determinants, change of variables, volume of manifolds, differential forms, and exterior derivative. Stokes' theorem is presented both in the language of vector analysis (div, grad, and curl) and in the language of differential forms.

Fall 2006:

Math 25a: Honors Linear Algebra and Real Analysis I:
A rigorous treatment of linear algebra. Topics include: Construction of number systems; fields, vector spaces and linear transformations; eigenvalues and eigenvectors, determinants and inner products. Metric spaces, compactness and connectedness.

At the Math Circle

I am also interested in student-driven pedagogical methods. I have been involved, first as a student and later as a teacher, in the Boston-Area Math Circle, an extracurricular program in mathematics for students in primary and secondary school. Unlike traditional math classes, the Math Circle classes are driven almost entirely by the participation and direction of the students, with the instructors mainly providing questions rather than answers.

Here is a pdf of an article that appeared in Focus Magazine featuring a new solution to a familiar problem by the Math Circle students, to which I contributed.

The picture depicts Math Circle students; I'm standing third from the left. The students on the floor form an example of an integer-sided triangle.

Returning to the Math Circle as an instructor while I was in college, I taught semester-long courses on Paul Sally's Four-Numbers game, group theory and the Rubik's cube, the Cayley Klein geometries, and the building the real numbers. Here are notes on the Four-Numbers game 1 2 3 4 5 and the real numbers 1 2 3 4

I try to adapt some of the approaches to teaching I learned during my time in the Math Circle to my current teaching.



Current Teaching

Math 104     Math 260