## Fall 2019

**Math 22: Linear Algebra**

## Previous courses

This course is an introduction to graduate level analysis. Divided roughly in half, the first part of the course covers abstract measure theory. This part starts with the introduction of measurable sets and measures on spaces. With these one can define the Lebesque integral which has a number of advantages over the Riemann integral. The course then continues with \( L^p \) spaces which are very useful in functional analysis. This part ends with the proof of the Radon-Nikodym theorem.

The second half of this course covers complex analysis. It starts with the definition of holomorphic functions and then covers the most important theorems in complex analysis, Cauchy's theorem, the Residue theorem and Laurent series expansions.

This course is a sequel to Math 3 and provides an introduction to Taylor series and functions of several variables. The first third of the course is devoted to approximation of functions by Taylor polynomials and representing functions by Taylor series. The second third of the course introduces vector-valued functions. It begins with the study of vector geometry, equations of lines and planes, and space curves. The last third of the course is devoted to studying differential calculus of functions of several variables.

Real analysis is a formal, mathematical study of the basic objects of calculus. In the first part we treat real numbers, functions, sequences and limits and in the second part differentiation and integration. We will revisit this material from calculus, taking a more abstract and mathematically sophisticated approach. In addition to learning real analysis, students will improve their skills in reading mathematics and in writing proofs. The course finishes with an outlook on Fourier series.

This course provides an introduction to fundamental algebraic structures. The majority of the course will consist of the study of groups, rings and fields. Additionally, to better understand groups from a geometric point of view, we will look at their representation as a Cayley graph. Finally, the aim of the hands-on project will be to find a solution to the 2x2 Rubik's cube and to understand its connection with abstract algebra.

#### Math 13: Multivariable Calculus

This course is a sequel to Math 8 and provides an introduction to calculus of vector-valued functions. The course starts with iterated, double, triple, and surface integrals including change of coordinates. The remainder of the course is devoted to vector fields, line integrals, Green’s theorem, curl and divergence, and Stokes’ theorem.

This course presents the fundamental concepts and applications of linear algebra. Topics include bases, subspaces, dimension, determinants, characteristic polynomials, eigenvalues, eigenvectors, and especially matrix representations of linear transformations and change of basis. As an application we will study Markov chains, Google's PageRank algorithm and the JPEG image compression. In this course the students will develop the ability to perform meaningful computations and to write accurate proofs.

#### Math 3: Calculus

This course aims to prepare the students to use calculus in many other disciplines without losing sight of its theoretical value. This course is an introduction to single variable calculus and is intended for students who are planning to go on to Math 8. Topics include, but are not limited to limits, continuity, derivatives, definite and indefinite integrals over the real line. The jewel of this course is the Fundamental Theorem of Calculus and it will be used to develop some techniques of integration.

By associating a group to one of its Cayley graphs, the properties of a group can be studied from a geometric point of view. The group itself acts on the Cayley graph via isometries which is reflected in the symmetries of the graph. The inherent beauty of a group can thus be visualized in its Cayley graph making these graphs a fascinating object of study. Geometric group theory examines the connection between geometric and algebraic invariants of a group. In order to obtain interesting invariants one usually restricts oneself to finitely generated groups and takes invariants from large scale geometry.

Geometric group theory closely interacts with low-dimensional topology, hyperbolic geometry and differential geometry and has numerous applications to problems in classical fields, like combinatorial group theory, graph theory and differential topology.