The student dedicated to becoming a mathematics teacher values the AQB MAT Program’s commitment to the discipline with its authentic research projects in mathematics (the academic research project) and in mathematics education (the classroom research project). MAT’s strong cohort model and small class size offer support to students over the course of the program and into the first years of their teaching career.

### Topics in Numerical Analysis 9201521 (3 Credits)

This course is intended to provide a practical introduction to computational problem solving. Topics covered include: the notion of well conditioned and poorly conditioned problems, with examples drawn from linear algebra; the concepts of forward and backward stability of an algorithm, with examples drawn from floating point arithmetic and linear-algebra; basic techniques for the numerical solution of linear and nonlinear equations, and for numerical optimization, with examples taken from linear algebra and linear programming; principles of numerical interpolation, differentiation and integration, with examples such as splines and quadrature schemes; an introduction to numerical methods for solving ordinary differential equations, with examples such as multistep, Runge-Kutta and collocation methods, along with a basic introduction of concepts such as convergence and linear stability; an introduction to basic matrix factorizations, such as the SVD; techniques for computing matrix factorizations, with examples such as the QR method for finding eigenvectors; Basic principles of the discrete/fast Fourier transform, with applications to signal processing, data compression and the solution of differential equations.

### Linear Algebra 9201562 (3 Credits)

This course examines Linear spaces, subspaces.  Linear dependence, linear independence; span, basis, dimension, isomorphism.  Quotient spaces.  Linear functional, dual spaces.  Linear mappings, null space, range, fundamental theorem of linear algebra.  Underdetermined systems of linear equations.  Composition, inverse, transpose of linear maps, algebra of linear maps.  Similarity transformations.  Matrices, matrix Multiplication, Matrix Inverse, Matrix Representation of Linear Maps, Determinant, Laplace expansion, Cramer's rule.  Eigenvalue problem, eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem. Diagonalization.

### Real Analysis 9201561 (3 Credits)

This course presents a rigorous treatment of fundamental concepts in analysis. Emphasis is placed on careful reasoning and proofs. Topics covered include the completeness and order properties of real numbers; limits and continuity; conditions for integrability and differentiability; infinite sequences and series. Basic notions of topology and measure are also introduced.

### Geometry 9201522 (3 Credits)

The course has the following goals:

• To learn new mathematics through inquiry, using open-ended problems, hands- on explorations, written and small group dialogs, and non-test based assessments.
• To develop an experiential understanding of the geometric properties of surfaces in 2 and 3-dimensional space, and use that understanding to devise their own mathematical arguments and proofs.
• To introduce students to many topics from geometry such as curve theory, spherical and hyperbolic geometry, detail of surfaces, curvature, torsion , and analytic geometry.

### Special Topics 9201563 (3 Credits)

Selected topics in mathematics chosen by the instructor.