Bard College at Simon's Rock: the Early College
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# Mathematics Courses

Explore the mathematics courses offered at Simon’s Rock.

### Algebra Workshop

Mathematics 099 | Staff | 1 p/f credit

Intended for students who need extra preparation before enrolling in a college-level mathematics course, Algebra Workshop offers an in-depth treatment of the foundational skills needed for later mathematics and science courses. Students gain fluency in problem-solving through understanding rather than memorization, as they extend the properties of operations with real numbers into algebra. Topics include fractions and percents, exponents and roots, factoring, solving equations and inequalities, graphing, linear and quadratic functions and their applications.

This course is graded on a Pass/Fail basis and does not meet the Mathematics requirement for the AA degree.

### Mathematics and Its Applications

Mathematics 101 | All Math Faculty | 3 credits

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.

Prerequisites: Adequate performance on the mathematics placement exam or completion of Math 099. This course is generally offered every semester.

### Elementary Functions

Mathematics 109 | All Math Faculty | 3 credits

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. This course is generally offered every semester.

### Introduction to Statistics

Mathematics 110 | All Math Faculty | 3 credits

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: Adequate performance on the mathematics placement exam. This course is generally offered every semester.

### Calculus I

Mathematics 210 | All Math Faculty | 3 credits

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 or adequate performance on the mathematics placement exam. This course is generally offered every semester.

### Calculus II

Mathematics 211 | All Math Faculty | 3 credits

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.

Prerequisite: Mathematics 210. This course is generally offered every semester.

### Linear Algebra

Mathematics 220 | All Math Faculty | 3 credits

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. This course is generally offered once a year.

### Vector Calculus

Mathematics 221 | All Math Faculty | 3 credits

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 or permission of instructor. This course is generally offered once a year.

### Complex Analysis

Mathematics 310 | Dunbar, Knox | 4 credits

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: Mathematics 221 or permission of the instructor. This course is offered when there is sufficient student interest.

### Analysis I

Mathematics 312 | Knox, Landi, Putz | 4 credits

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. This course is generally offered once every two years.

### Analysis II

Mathematics 313 | Knox, Landi | 4 credits

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. This course is generally offered once every two years.

### Modern Algebra I

Mathematics 320 | Dunbar, Knox, Landi | 4 credits

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. This course is generally offered once every two years.

### Modern Algebra II

Mathematics 321 | Dunbar, Knox | 4 credits

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: Mathematics 320. This course is generally offered once every two years.

### Number Theory

Mathematics 324T | Dunbar, Knox | 4 credits

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. This course is offered when there is sufficient student interest.

### Algebraic Geometry

Mathematics 326T | Dunbar, Knox | 4 credits

This course is an introduction to commutative algebra and algebraic geometry. Commutative algebra topics include algebras, ideals, Noetherian rings, tensor products, localization, direct limits, the Hilbert basis theorem, and Hilbert's Nullstellensatz. Algebraic geometry topics include affine algebraic varieties, finite maps and the principal ideal theorem, projective varieties and Bezout's theorem, Grassmannians, tangent spaces to algebraic varieties, dimension theory, curves, divisors, and the Riemann-Roch theorem.

Prerequisite: Mathematics 320. This course is generally offered as a tutorial.

### Statistics I

Mathematics 330 | Dunbar | 4 credits

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.

Prerequisite: Mathematics 221 or permission of instructor. This course is generally offered once every two years.

### Statistics II

Mathematics 331 | Dunbar | 4 credits

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 220 and Mathematics 330. This course is generally offered once every two years.

### Differential Geometry I

Mathematics 350T | Knox | 4 credits

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. This course is generally offered as a tutorial.

### Differential Geometry II

Mathematics 351T | Knox | 4 credits

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. This course is generally offered as a tutorial.

### Hyperbolic Geometry

Mathematics 352T | Dunbar | 4 credits

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. This course is generally offered as a tutorial.

### Topology I

Mathematics 354 | Dunbar, Knox, Landi | 4 credits

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. This course is generally offered once every three or four years.

### Topology II

Mathematics 355 | Dunbar, Knox | 4 credits

This course is a continuation of Math 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: Math 354. This course is generally offered once every year.

### Ordinary Differential Equations

Mathematics 364 | Knox, Landi | 4 credits

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. This course is generally offered once a year.

### Partial Differential Equations

Mathematics 365T | Landi | 4 credits

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. This course is generally offered as a tutorial.

### Dynamical Systems and Transversality

Mathematics 366 | Dunbar | 4 credits

Dynamical systems are mathematical models for changes in behavior over time . They can be either continuous (using differential equations) or discrete (using functions from a state-space into itself). This course focuses on the former, using qualitative methods to predict long-term behavior. For many differential equations, there is no "closed form" for the solution; also, the differential equation itself may be only roughly known, its coefficients depending on inexact physical measurements. As we develop these qualitative methods, we bring in a few concepts from other branches of mathematics, including topology and real analysis, which are explained when the need arises.

Prerequisite: Mathematics 364.

### Machine Learning and Neural Nets

Mathematics 370 | Landi | 4 credits

This course develops the foundations of machine learning, an area of broad application of mathematics and computer science to technology, the health sciences, internet applications, and science, engineering, and business in general. Students will learn current approaches to both supervised and unsupervised learning, as well as how to apply their knowledge to develop a solution to a problem amenable to machine learning techniques. Students will also make written and oral presentations of their work.

Prerequisites: Students are expected to have some mathematical and/or programming maturity. Therefore, a combination of one of the following math classes and one of the following computer science classes will be required for admission into the class: Mathematics 220; Mathematics 330; Computer Science 234; Computer Science 252.

### Mathematics Tutorial

Mathematics 300/400 | Staff | 4 credits

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.