Mathematics

Mathematics and Its Applications This course develops the mathematical and quantitative skills required of an effective citizen in our complex society. The emphasis is on the interpretation of material utilizing mathematics, as opposed to the development of simple numerical skills. Possible topics include the application of elementary algebra to common practical problems; exponential growth, with applications to financial and social issues; an introduction to probability and statistics; and the presentation and interpretation of graphically presented information. Instruction in the uses of a scientific calculator and of a computer to facilitate calculations is an integral part of the course.

Elementary Functions A transition from secondary school to college-level mathematics in both style and content, this course explores the elementary functions. Topics include polynomial, exponential, logarithmic, and trigonometric functions; graphing; inequalities; data analysis; and the use of a graphing calculator and/or computer. The course meets the College’s mathematics requirement and also prepares students for calculus. Prerequisite: Mathematics 101, or at least two years of high school mathematics and adequate performance on the mathematics placement exam.

Introduction to Statistics This course offers an introduction to statistical methods for the collection, organization, analysis, and interpretation of numerical data. Topics include probability, binomial and normal distributions, sampling, hypothesis testing, confidence limits, regression and correlation, and introductory analysis of variance. The course is oriented toward the increasingly important applications of statistics in the social sciences. Prerequisite: Mathematics 101.

Calculus I A course in differential and integral calculus in one variable. Topics include an introduction to limits and continuity, the derivative and its applications to max-min and related rate problems, the mean value theorem, the definite integral, and the Fundamental Theorem of Calculus. Prerequisite: Mathematics 109.

Calculus II This course is a continuation of Calculus I. Topics include techniques of integration, numerical integration, applications of the definite integral, Taylor approximations, infinite series, and an introduction to differential equations.

Linear Algebra This course deals with linear mathematics, including the geometry and algebra of linear equations, the mathematics of matrices, and vector spaces. The course provides an important foundation for the mathematical representation of phenomena in the social sciences and physical sciences as well as for more advanced analysis and algebra courses. Prerequisite: Mathematics 211 or permission of the instructor.

Vector Calculus This course deals with multivariable calculus and vector analysis. Topics include differentiation of vector functions, multiple integrals, line and surface integrals, vector fields, and the theorems of Stokes and Green. Applications to geometry and physics are considered as time permits. Prerequisites: Mathematics 211 and 220.

Complex Analysis This course in functions of one complex variable covers the Cauchy-Riemann equations, power series and analytic functions, the inverse and open mapping theorems, Cauchy’s Theorem, Cauchy’s Integral formula, isolated singularities and the calculus of residues, conformal mappings, and the Riemann Mapping Theorem. Prerequisite Math 221 or consent of the instructor.

Analysis I This course provides a firm foundation for calculus. Topics include a rigorous definition of the real numbers; Cauchy sequences; and definition of limit, along with proofs of the theorems of calculus, sequences of functions, uniform convergence, and continuity. Prerequisites: Mathematics 220 and 221 or permission of the instructor.

Analysis II This course is a continuation of Mathematics 312. Topics include series, the integral in one variable, Dirac sequences, Fourier series, improper integrals, and Fourier transforms. Prerequisite: Mathematics 312.

Modern Algebra I The fundamental structures of algebra play a unifying role in much of modern mathematics and its applications. This course is an introduction to some of the fundamental structures. Topics depend on the interests of students and may include groups, rings, fields, vector spaces, and Boolean algebras. Prerequisite: Mathematics 220.

Modern Algebra II This course is a continuation of Modern Algebra I. Topics include the theory of fields and Galois Theory and the theory of linear groups. Prerequisite Math 320, Modern Algebra I.

Number Theory An introduction to algebraic number theory, this course covers linear diophantine equations, congruences and Z/nZ, polynomials, the group of units of Z/nZ, quadratic reciprocity, quadratic number fields, and Fermat’s Last Theorem. Prerequisite: Mathematics 220 or permission of the instructor

Algebraic Geometry Algebraic geometry is the study of algebraic varieties, sets of zeroes of polynomials such as curves in the plane or curves and surfaces in space. This course is an introduction to such varieties in n-dimensional space. Such a study leads naturally to the study of a certain type of set of polynomials, namely an ideal. We establish a dictionary between an ideal and the variety consisting of the set of common zeros of all the polynomials in the ideal. For example, a curve might be the intersection of two surfaces; then each of these two surfaces has a corresponding ideal and these two ideals together generate the ideal of the curve in the intersection. We study these ideals from a theoretical and computational point of view.We describe the dictionary between polynomial ideals and affine algebraic sets. To be able to produce examples, we introduce the classification of ideals by means of Groebner bases. These give a constructive way to prove the Hilbert Basis Theorem, characterizing all algebraic varieties as intersections of a finite set of hypersurfaces. With these tools, we prove the Hilbert Nullstellensatz, and we establish the dictionary described earlier. Several applications are possible if time permits. Since the examples are computationally difficult, some time will be spent on Maple software and computer solutions and algorithms for computing Groebner bases. Prerequisite: Mathematics 220.

Statistics I This course provides the mathematical foundations underlying statistical inference. Topics include random variables, both discrete and continuous; basic sampling theory, including limit theorems; and an introduction to confidence intervals. Prerequisites: Mathematics 110 and 211, or permission of the instructor.

Statistics II This course is a continuation of Mathematics 330. Topics include estimation, tests of statistical hypotheses, chi-square tests, analysis of variance, regression, and applications. Case studies are examined as time permits. Prerequisite: Mathematics 330.

Differential Geometry I An introduction to the applications of calculus to geometry, this course is the basis for many theoretical physics courses. Topics include an abstract introduction to tangent spaces and differential forms; the Frenet Formulas for moving frames on curves in space; and the rudiments of the theory of surfaces, both embedded and abstract. Prerequisites: Mathematics 220 and 221, or permission of the instructor.

Differential Geometry II This course is a continuation of Mathematics 350. Topics include the shape operator of a surface, Gaussian and normal curvature, geodesics and principal curves, topology of surfaces, the covariant derivative, and the Gauss-Bonnet Theorem. Prerequisite: Mathematics 350.

Hyperbolic Geometry Hyperbolic geometry, sometimes called non-Euclidean geometry, was discovered independently by Gauss, Bolyai and Lobachevski in the 19th century, as a way of finally demonstrating that the parallel postulate of plane geometry is not a logical consequence of the other postulates. After the development of special relativity by Einstein, hyperbolic geometry found another use as one of several alternative models for the large-scale geometry of the universe. The philosophy of the course is to understand hyperbolic geometry via a close study of its symmetries. This will involve some of the basic concepts of abstract algebra and complex analysis (which will be explained as they are needed). Topology also enters the picture, since the vast majority of surfaces can be thought of as pasted-together hyperbolic polygons (in the same way that a cylindrical surface can be obtained by pasting together two opposite edges of a piece of paper). Thus, hyperbolic geometry serves as the meeting ground for many different kinds of mathematics. Prerequisites: Mathematics 220 and 221.

Topology I An introduction to topology—the study of properties preserved under continuous deformation. Topics include a brief introduction to set theory; open, closed, connected, and compact subsets of Euclidean space; and the classification of surfaces. Prerequisite: Mathematics 221 or permission of the instructor.

Topology II This course is a continuation of Mathematics 354. The main topic is the theory of knots, the study of which involves many different combinatorial, algebraic, and geometric techniques. In particular, the fundamental group is discussed in detail. Each student chooses a topic and produces a major paper. Prerequisite: Mathematics 354.

Ordinary Differential Equations This is an introductory course on ordinary differential equations. Topics include first-order equations, second order linear equations, harmonic oscillators, qualitative properties of solutions, power series methods, Laplace transforms, and existence and uniqueness theorems. Both the theory and applications are studied, including several problems of historical importance. Prerequisite: Mathematics 221 or permission of the instructor.

Partial Differential Equations This course offers an introduction to Fourier series and boundary value problems. Topics include the partial differential equations of physics, superposition of solutions, orthogonal sets of functions, Fourier series, Fourier integrals, boundary value problems, Bessel functions, Legendre polynomials, and uniqueness of solutions. Prerequisites: Mathematics 220 and 221 or permission of the instructor.

Mathematics Tutorial Under these course numbers, juniors and seniors design tutorials to meet their particular interests and programmatic needs. A student should see the prospective tutor to define an area of mutual interest to pursue either individually or in a small group. A student may register for no more than one tutorial in any semester.