- Use techniques of differentiation and integration (in one or more variables).
- Find local and absolute extrema of functions of one or more variables.
- Sketch graphs of functions of one variable and identify basic functions from their graphs (one or two variables).
- Solve systems of linear equations with the use of matrices and their inverses.
- Recognize and work with basic algebraic structures (groups, vector spaces, homomorphisms).
- Give direct proofs, proofs by contradiction, and proofs by induction.
- Formulate definitions and theorems.
- Write a simple computer program.
- Use mathematics to model and solve problems in other areas.
- Gain competency in other areas of mathematics (which ones and how many depends on the selected option).
- Demonstrate basic competency in both oral and written communication.
In the Mathematics MS program students choose one of two options: Option 1 (General Mathematics) and Option 2 (Applied Mathematics). The learning outcomes are somewhat different for each as seen below.
Graduating students in both options will:
- Have a broad exposure to advanced mathematics through electives chosen from a wide range of topics including abstract algebra, advanced calculus, geometry, differential equations, linear algebra, probability, number theory, and topology.
- Understand and devise proofs of mathematical theorems. This includes understanding the role of definitions, axioms, logic, and particular proof techniques such as proof by induction, proof by contradiction, etc.
- Be able to write a coherent, clear article on a mathematical theme, and to present this orally.
- Be able to search the mathematical literature to research a topic of interest.
- Understand and be able to apply basic results of complex analysis including: the relationship between complex analytic functions and harmonic functions, conformal mapping, and applications to Dirichlet problems; Cauchy’s integral formulas and their consequences including the fundamental theorem of algebra; series expansions, classification of singularities, and the application of residue calculus to definite integrals and sums.
Graduating students in Option 1 (General Math) will:
- Have a broad understanding at the graduate level of the content of the required courses of the option. This includes the theory of groups, rings and fields, topology, complex analysis, and real or functional analysis.
- Understand the basic theories of groups, rings and fields, including the structure of finite groups, polynomial rings and Galois theory.
- Understand how the main topological concepts (connectedness, compactness, products and separation properties) are introduced and used in abstract spaces where the topological structure is not derived from an underlying metric.
- Understand basic set theory including axiom of choice, basic topological properties of the real line; properties of real functions, sequences of real functions and various notions of convergence such as pointwise and uniform convergence.
- Understand the notions of outer-measure, measurability of sets, non-measurable sets, Riemann and Lebesgue integrability, convergence in measure, differentiation of functions, functions of bounded variation, absolutely continuous functions, basic properties of L^p spaces.
- Have an understanding of metric spaces, sequences, completeness; normed linear spaces, Banach spaces, classical sequence spaces, linear functionals and linear operators and their representations; the Hahn-Banach, Banach-Steinhaus and Open Mapping theorems; innerproduct and Hilbert spaces, orthonormal sets and sequences, representation of linear functionals on Hilbert spaces, Fourier series.
Graduating students in Option 2 (Applied Math) will:
- Have a broad understanding at the graduate level of the content of the required courses of the option. This includes numerical analysis, linear analysis, mathematical modeling and complex analysis.
- Be able to use a variety of mathematical tools (differential equations, linear algebra, etc) to formulate a mathematical model of real world problems. Understand the balance between the complexity of a model and its mathematical tractability. Understand the iterative process of modeling and the necessity to test a model against data.
- Be able to apply numerical methods to solve problems, such as large systems of linear equations, eigenvalue/eigenvector problems, and understand the theoretical underpinnings of these methods.
- Be able to solve partial differential equations numerically and be able to analyze the stability and convergence of these approximate solutions. This includes the understanding of the fundamental differences among parabolic, elliptic and hyperbolic partial differential equations, the Max/Min principle for certain elliptic partial differential equations and the method of characters for and second order hyperbolic partial differential equations.
- Understand metrics, norms, and inner products on important spaces of functions, including Banach spaces and Hilbert spaces and be able to use important applications including Fourier series and solutions of integral equations by contraction. They will be familiar with basic properties of linear operators, especially on Hilbert spaces, invertiblity and spectrum, and be able to apply these to solution of integral equations and differential equations.