Selected Topics in Mathematics (MASTER)

Strengthen logical and abstract thinking and understanding of mathematical concepts and their economic interpretations and applications.

Acquire sufficient knowledge of analysis, linear algebra and optimization for further studies and independent work with quantitative methods.

Develop the ability to apply mathematical methods to formulate, model and solve complex problems in economics, actuarial sciences and finance.

1. Linear algebra
1.1. Matrices and determinants
1.2. Systems of linear equations, Gaussian elimination
1.3. Vector spaces and subspaces, linear independence, basis, dimension
1.4. Linear transformations, nullspace and image
1.5. Eigenvalues and eigenvectors, matrix diagonalization
1.6. Vector spaces with scalar product
1.7. Orthonormal basis, Gram-Schmidt orthogonalization
1.8. Symmetric matrices, quadratic forms, definiteness, least squares method

2. Linear optimization
2.1. Linear program, unbounded and infeasible program
2.2. Simplex method and two-phase simplex method
2.3. Dual program, complementary slackness

3. Function analysis
3.1. Derivative, partial derivative, gradient, Hessian matrix, vector functions
3.2. Total differential, linear and polynomial approximation
3.3. Indefinite integral, definite integral, improper integral
3.4. Multiple integral, polar and spherical coordinates, numerical integration
3.5. Implicit functions, derivative of an implicit function
3.6. Metric spaces, Cauchy sequences, fixed points theorems

4. Differential equations
4.1. Differential equations with separable variables
4.2. Homogeneous and nonhomogeneous linear differential equations
4.3. Autonomous differential equations and equilibrium solutions
4.4. Numerical solution of differential equations
4.5. Classification of second-order linear partial differential equations
4.6. Partial differential equations with separable variables

5. Non-linear optimization
5.1. Open, closed, bounded and compact sets
5.2. Local and global extrema, first and second order conditions
5.3. Constrained extrema, substitution, Lagrangian and Lagrange multipliers
5.4. Convexity and convex problems, Karush-Kuhn-Tucker conditions