 # Summer Math Camp

### 2020 Simon School PhD Summer Math Camp

Training in mathematics and statistics is necessary for all course work in the Simon PhD Program. To help prepare you we offer three mini courses over the summer, before regular first-year PhD classes begin. The Summer Math program is covered under your full tuition scholarship.

These classes are strongly recommended and highly encouraged, however, they are not mandatory. Courses meet over 5 weeks, July 16 through August 21, 2020.

#### Topic 1: Linear Algebra

The goal of this course is to give an introduction to linear algebra. Topics include: Gaussian elimination, matrix operations, matrix inverses. Vector spaces and subspaces, linear independence, and the basis of a space. Row space and column space of a matrix, fundamental theorem of linear algebra, linear transformations. Orthogonal vectors and subspaces, orthogonal bases, and Gram-Schmidt method. Orthogonal projections, linear regression. Determinants: how to calculate them, properties, and applications. Calculating eigenvectors and eigenvalues, basic properties. Matrix diagonalization, application to difference equations and differential equations. Positive definite matrices, tests for positive definiteness, singular value decomposition. Classification of states, transience and recurrence, classes of states. Absorption, expected reward. Stationary and limiting distributions.

#### Topic 2: Optimization and Real Analysis

This course covers Optimization in Rn, Weierstrass Theorem, Unconstrained optimization, Lagrange Theorem and equality constraints, Kuhn-Tucker Theorem and Inequality constraints, Convexity, Parametric Monotonicity and Supermodularity. This course also introduces mathematical tools especially useful in economics, econometrics and finance. Topics include a basic topology of the real line, sequences and series, limits, continuity, differential and integral calculus.

#### Topic 3: Probability Theory

This course teaches Random Variable, Distribution, Independence; Transformations and Expectations; Common Families of Distributions; Multiple Random Variables, and Markov Chains.