Mathematics

The major in mathematics is designed to enable the student to enter the marketplace (industrial or educational) or to pursue further studies in mathematics or allied fields at the graduate level. Interested students should meet with an advisor in the Department of Mathematics as early as possible for assistance in formulating an appropriate course of study.

A student majoring in mathematics cannot declare a second major in statistics.

Program Learning Goals

Upon completion of the two-year sequence in calculus, students will be able to:

1. Differentiate and integrate a wide variety of algebraic and transcendental functions;

2. Apply such knowledge to a variety of verbal problems in economics, physics, and related rates;

3. Develop the Taylor series expansion for functions and compute the error terms occasioned by truncation of the series to a finite number of terms;

4. Use geometric vectors to prove theorems;

5. Deal with functions and surfaces (areas, volumes) in 3-dimensional space;

6. Use other (than Cartesian) coordinate systems, especially polar coordinates, in the study of graphs and, by change of variable, to facilitate certain integrations;

7. Follow subtle lines of reasoning, detect breaches of logic and validity, write sustained logical arguments;

8. List several approaches to the real number system, such as Dedekind cuts, the Bolzano–Weierstrass property, the nested-interval property, the existence of suprema and infima of bounded sets, the convergence of Cauchy sequences.

Upon completion of our courses in analysis beyond calculus, students will be able to:

1. Point out the analogies—the interplay and interconnections—between corresponding real-valued functions of a real variable and complex-valued functions of a complex variable;

2. Highlight some of the properties that follow from analyticity of functions on various domains;

3. Perform computations with complex numbers, evaluate contour integrals, evolve Laurent series of functions;

4. Show how metric spaces endowed with Euclidean and non-Euclidean metrics are particular examples of topological spaces;

5. Present properties of metrizable and nonmetrizable topological spaces as generalizations of properties that originate in the set of real numbers;

6. Explicate properties of connectedness and compactness in topological spaces.

Upon completion of our courses in algebra, students will be able to:

1. Trace the construction of the integral domain of rational integers and the fields of rational and complex numbers by successive refinements of, and additions to, the properties of a set;

2. Show how abstract initial conditions can be used to derive facts and features of a variety of algebraic structures;

3. Apply abstract algebra, which had its origins and motivation in number theory, back to number theory, to elucidate number-theoretic properties by placing them in a general (abstract) setting;

4. Prove theorems about groups, rings, fields, and other algebraic structures;

5. Account for the advantages of abstract formulations in mathematics;

6. Define the dimension of a vector space in terms of the (unique) number of vectors in a basis, accomplish basis-to-basis transformations, compute characteristic values and vectors, and enumerate some of the profound connections among the invertibility of matrices, systems of linear equations, determinants, linear independence, spanning sets and bases, rank, orthogonality.

Upon completion of our courses in geometry, students will be able to:

1. Discourse with authority on the impact and role of initial assumptions (postulates) on the structure of a geometrical system, mainly with reference to Lobachevskian and Riemannian geometry;

2. Cite facts (theorems) of Euclidean geometry that depend on the parallel postulate and hence are absent in neutral geometry;

3. Provide examples of finite and infinite incidence geometries and their isomorphisms;

4. Trace some of the history of geometry, especially as it concerns attempts to prove Euclid’s parallel axiom as a consequence of the other axioms;

5. Speak on difficulties encountered in endeavoring to establish the physical validity of a geometric theory – which the actual geometry of the universe is, given the homogeneity of space with respect to the parallel postulate; and of course

6. Compose mathematically correct proofs of geometric statements.

Upon completion of our other classes, students will be able to:

1. Solve differential equations using series expansions, Laplace transforms, and other standard techniques [differential equations];

2. Enunciate properties and applications of Eulerian, Hamiltonian, connected, cyclic, acyclic, planar, traversable, and other types of graphs [graph theory];

3. Approach combinatorics problems from two points of view which, when united, lead to solutions of problems in combinatorics using permutations, combinations, partitions, mathematical induction [combinatorics];

4. Trace the historical development of mathematics from antiquity to the present, including contributions to that cumulative subject from various cultures and countries [history of mathematics];

5. Stipulate properties and characteristics of whole numbers – divisibility, the division algorithm, Diophantine equations, unique factorization, the integers modulo n, Fermat’s theorem, Euler’s theorem, representation in different bases [theory of numbers];

6. Write computer programs in a high-level programming language to solve mathematical problems and verify their correctness, and invoke techniques of object-oriented programming to represent objects and their behaviors in code [algorithms, computers, and programming class].