Our core program for students electing an Area of Concentration (AOC) in Mathematics includes three semesters of calculus, linear algebra, differential equations, two semesters of modern abstract algebra, and two semesters of real analysis and complex analysis. In addition, students are encouraged to take courses in topology, discrete mathematics, graph theory and number theory as well as computer science and the other sciences.

Beyond these core components, there is a great deal of flexibility involving coursework for advanced students. Past advanced courses, tutorials and independent study projects have included algebraic geometry, algebraic topology, combinatorial optimization, differential geometry, differential topology, foundations of mathematics, Fourier analysis, functional analysis, Galois theory, representation theory of finite groups, group theory, mathematical biology, measure theory, model theory, partial differential equations, probability, projective geometry and topics in mathematical physics.

For detailed requirements, check out our General Catalog and the Mathematics Academic Learning Compact.

**Here’s a list of recent course offerings in Mathematics:**

**Patterns**

This interdisciplinary course is intended as a general education class primarily for first year incoming students that will expose them to a variety of disciplines taught here at New College. It will be offered as a single term class in the fall of 2006. The title is chosen to reflect the central theme of patterns that appear in such areas as history, religion, sociology, music, mathematics, literature, psychology, physics, biology, chemistry, art etc… A guest lecturer from the faculty will be invited each week to give a seminar on patterns as they appear in his/her field of study. Each invited speaker will be asked to prepare an assignment for the students due the following week. Prerequisites: None.

**Puzzles, Proofs, and Paradoxes**

This course could be considered a “logic and proof” laboratory. We will look at a variety of puzzles and paradoxes, practicing problem solving and logic skills. We will also discuss a little of the history of mathematics, getting to know some of the outstanding mathematicians of the past, and learn to appreciate their contributions. Prerequisites: Enjoyment of math.

**Calculus I**

Calculus is a means for calculating the rate of change of a quantity which varies with time and the total accumulation of the quantity whose rate of change varies with time. Although calculus is only about three centuries old, calculus ideas are the basis for most modern applications of mathematics, especially those underlying our technology. The development of the calculus is one of the great intellectual achievements of Western civilization. A balance will be struck between presenting calculus as a collection of techniques for computation, and as a handful of difficult but very powerful concepts. Wherever possible, we will motivate the ideas as ways of answering questions about real world problems. Prerequisites: Complete the math placement exam.

**Calculus II**

This course takes up where Calculus I leaves off. The topics covered include integration techniques, sequences, series, Taylor series, complex numbers, areas and volumes. This course is recommended for students pursuing interests in the physical sciences, applied mathematics and economics. Prerequisite: Calculus I and instructor's permission.

**Calculus III**

This class is a continuation of Calculus I and II. We will cover the calculus in n-dimensional Euclidean space. The topics covered during the course of the semester include the fundamental constructions of the calculus of multivariable functions (vector fields, gradients, line integrals, surface integrals etc) and the associated fundamental results (Green's Theorems, Gauss' Theorem, Stokes' Theorem, etc). The course will focus on application and computation and will include an introduction to differential equations. Prerequisite: Calculus II.

**Calculus with Theory I**

This course is the first in a two semester sequence designed as a rigorous introduction to the calculus. This class targets students that want a deep understanding of the theoretical under-pinnings of calculus and the ability to reprove the classical theorems of calculus. This course will cover considerably more detail than a regular calculus course and includes an introduction to writing proofs. The first semester will cover differential calculus with an in-depth look at limits, continuity, and differentiability as well as applications such as optimization and linear approximation. We will complete the course by rigorously developing the Riemann integral and proving the fundamental theorem of calculus. Prerequisites: Permission of Instructor.

**Calculus with Theory II**

This is a continuation of Calculus with Theory 1. This course will continue with techniques of integration, logarithms and exponential functions, infinite sequences and sums and power series. If time remains the course will touch on Fourier series. Prerequisites: Calculus With Theory 1.

**Discrete Math**

Discrete Math is a collection of tools needed by mathematicians, computer scientists, economists, and anybody applying math to model and solve real world problems. All the tools pertain to finite objects, and use finite methods in the solution - Calculus is not required! The approach taken in this course is primarily mathematical, although many algorithms and techniques with a wide range of applications will be discussed. We will focus on disciplined thinking and symbolic computation, learn to appreciate well know proofs, and practice to discover and formulate our own. The topics covered will include logic, elementary set theory, algorithms, graph theory, trees, combinatorics and elementary probability, and some algebra and theory of computation. Prerequisites: None, other than the ability to think in a disciplined way and to enjoy math.

**Introduction to Number Theory with Applications to Cryptography**

In this course, we will introduce axiomatically the basic ideas and tools of classical number theory. Students will be exposed to elements of logic, mathematical induction, divisibility properties of the integers, modular arithmetic, congruences, quadratic reciprocity and elements of Abstract Algebra. Throughout the course, we will study modern applications of number theory in primality testing and cryptography. The class will be an excellent preparation for students interested in the Abstract Algebra sequence. Prerequisite: Interest in math and permission of instructor

**Graph Theory**

This course will provide an introduction to the theory of graphs. Following a short introduction of the basic definitions, we will study further: coloring of graphs, circuits and cycles, counting, labeling of graphs, drawing of graphs, measurement of closeness to planarity and finally applications. Many open problems will be introduced and discussed. Students will be encouraged to work on open/research problems. Prerequisite: Permission of instructor

**Graphs, Networks and Algorithms**

In this course we investigate applied problems in which networks and graphs appear naturally. Topics which we will investigate over the course of the semester include: combinatorial optimization, Markov chains on graphs, simulation and Markov Chain Monte Carlo methods, difference operators on discrete structures, inverse problems for networks, and an introduction to computational complexity. Prerequisites: Calculus I and II.

**Linear Algebra**

This course is an introduction to the theory of vector spaces and linear transformations and to their representation by means of matrices. The topics that will be covered are: matrices and linear systems of equations, algebra of matrices, determinants, vector spaces, linear transformations, eigenvalues and eigenvectors, matrix diagonalization, and inner product spaces. Prerequisites: Calculus or the consent of instructor.

**Computational Ordinary Differential Equations**

This course will focus on differential equations and computational methods using Matlab/Maple. It is intended for Mathematics and Science students who are going to apply these techniques in their coursework. Reflecting the shift in emphasis from traditional methods to new computer-based methods, we will focus on the mathematical modeling of real-world phenomena as the goal and constant motivation for the study of differential equations. Topics covered include some material on complex numbers and matrix algebra, first and second order linear and non-linear systems with applications, introductory numerical methods, and Laplace transform techniques. Prerequisites: Calculus I and II.

**Probability**

The course will consist of two parts. In the first part, we will begin by studying discrete spaces and simple games of chance. We will introduce and study the basic notions of probability including random variables, distribution, expectation, and variance. We will study continuous distributions as they relate to approximations of various discrete objects. In the second part of the course we will use our knowledge of simple games of chance to construct discrete models of simple physical systems. The models and the ideas behind their construction have found applications in many different areas (Physics, Chemistry, Biology, Economics, etc.). Time permitting; we will study several such examples in detail. Prerequisite: Calculus.

**Discrete Dynamical Modeling**

An important problem in science is to predict the behavior of systems that change in time. Such systems are called dynamical systems. This course introduces students to a set of mathematical methods used to model dynamical systems. It focuses on discrete dynamical models in which time is viewed as a sequence of steps. Students will learn how to translate real world problem into mathematical equations and they also learn how to use mathematical and computational methods to analyze the problem and make prediction. Mathematical concepts on steady states, cycles and chaos will be introduced. Concrete examples will be drawn from biology when possible. Prerequisite: Calculus

**Mathematical Modeling I**

Mathematical modeling plays a central role in understanding of complex systems that are changing in time. Such systems are called dynamical systems. This course is designed to introduce students to the elements of dynamical systems. Both continuous and discrete systems will be covered. In the course of the term, students will come to understand how mathematical models are formulated, and how their short and long term behaviors can be uncovered through a combination of analysis and computer simulation. Qualitative, quantitative and graphical techniques will be used to analyze and understand mathematical models and to compare theoretical predictions with available data. Mathematical concepts of steady states, cycles and chaos will be introduced. Examples will be given from physics, biology, chemistry and economics. Prerequisites: Calculus and differential equations (or the approval of instructor).

**Introduction to Numerical Methods**

This is a survey course of the basic numerical methods which are used to solve practical scientific problems. Important concepts such as accuracy, stability, and efficiency and convergence are discussed. The course provides an introduction to MATLAB, an interactive program for numerical linear algebra. Objectives of the course: Develop numerical methods for approximately solving problems from continuous mathematics on the computer. Examine the accuracy, stability, and failure modes of these methods. Implement these methods in a computer language MATLAB. Prerequisites: Calculus and Differential Equations.

**Mathematical Biology**

This course introduces the study of nonlinear interactions in biology and medicine. We consider physical problems which are well modeled by systems of coupled ordinary differential equations and develop techniques to obtain qualitative information about such systems. Mathematical concepts on nonlinear dynamics and chaos, qualitative and quantitative mathematical techniques as local and global stability theory, bifurcation analysis, phase plane analysis, and numerical simulation will be introduced. Concrete and detailed examples will be drawn from molecular, cellular and population biology and mammalian physiology. Prerequisite: Calculus, Differential Equations. Programming experience preferred.

**Advanced Linear Algebra**

Linear algebra is a critical mathematical tool in all of the sciences. Therefore, an in-depth knowledge of linear algebra is useful not only to mathematicians, but also to any scientist using mathematics. Topics to be covered include a review of basic linear algebra, the Moore-Penrose Pseudoinverse, singular value decompositions, generalizations of matrix equations, projections and inner products, least squares problems, Jordan canonical form, linear differential equations and the matrix exponential, and difference equations. Prerequisite: Linear Algebra or permission of the instructor.

**Algebraic Graph Theory**

In algebraic graph theory one expresses properties of graphs in algebraic terms and then deduces theorems about them. First we will tackle the applications of linear algebra and matrix theory to the study of graphs; algebraic constructions such as adjacency matrix and the incidence matrix and their applications will be discussed in depth. We will also study the theory of chromatic polynomials, a subject that has strong links with the Ainteraction models studied in theoretical physics, and the theory of knots. The last part of the course will deal with symmetry and regularity properties where important connections with other branches of algebraic combinatorics and group theory will be explored. Prerequisites: Linear Algebra or consent of instructor.

**Abstract Algebra I**

Abstract Algebra generalizes the idea of solving equations to mathematical objects other than numbers. At its core is the axiomatic method, which consists of making a small number of initial assumptions and deducing powerful theorems from them. These theorems can then be applied in a wide variety of mathematical contexts where the assumptions are valid. Topics that will be covered are introduction to the axiomatic method, sets and equivalence relations, groups, subgroups, homomorphisms, factor groups. Also, rings and fields, rings of polynomials, homomorphisms, factor rings, and ideals. Prerequisites: Linear Algebra.

**Abstract Algebra II**

In the second term of the “algebra” sequence we begin by studying more advanced topics in group theory including group actions, the use of group theory in counting, symmetry groups and the Sylow Theorems. We continue with the study of factorization domains, polynomial rings and field extension and conclude with the beautiful and powerful “Galois Theory”, which determines what polynomials are solvable by radicals. Prerequisites: Abstract Algebra I and Linear Algebra.

**Real Analysis I**

Real Analysis is a core course of the mathematics curriculum. The material for the course centers on the fundamental notions of the calculus – complete with proofs. Topics include an axiomatic development of the real numbers, sequences of real numbers, topology of the real line, continuous functions, differentiable functions, a construction of the Riemann integral, a proof of the fundamental theorem of calculus, Euclidean spaces and metric spaces and various additional topics. Prerequisite: A year of calculus and exposure to the notion of proof.

**Real Analysis II**

This class is a continuation of Real Analysis 1. Topics covered vary from year to year but usually include extensions of the material covered in Real Analysis 1 to the multivariable case (including the Implicit Function Theorem and the Inverse Function Theorem for Euclidean space), topics in Fourier analysis, topics in ODEs, and a construction of the Lebesgue integral. Prerequisites: Linear Algebra and Real Analysis I.

**Complex Analysis**

Complex numbers were introduced in the study of the roots of polynomial equations and have found applications in nearly every branch of modern mathematics. This course will develop the notion of a function of a complex variable and the corresponding calculus. The theorems and applications to be discussed are some of the most beautiful results of modern mathematics. Topics for the course include analytic functions, complex integration and the Cauchy integral formula, series representations, residues, the Dirichlet problem, and conformal mappings. Prerequisites: Real Analysis I or permission of instructor.

**Partial Differential Equations**

This course is designed to prepare students for advanced work in geometry and mathematical physics by developing the knowledge of partial differential equations common to both topics. Topics covered during the semester include: Laplace equations, wave equations, heat equations, Hamilton-Jacobi equations, Fourier theory, and the theory of distributions. Prerequisites: Calculus III and Ordinary Differential Equations.

**Computational Fluid Mechanics**

This course is an interdisciplinary introduction to Computational Fluid Mechanics. The course focuses on physical and mathematical foundations of computational fluid mechanics with emphasis on applications. We will consider solution methods for model equations and the Euler and the Navier-Stokes equations; the finite volume formulation of the equations; classification of partial differential equations and solution techniques; truncation errors, stability, conservation, and monotonicity. The main programming language is Matlab. Prerequisites: Calculus III, Ordinary Differential Equations.

**Computational Partial Differential Equations**

This course will focus on applied partial differential equations and their computational methods. It is intended for math and science students who apply these techniques in their work. Topics we will consider include, but are not limited to heat, wave, and Laplace equation, harmonic functions, Fourier series expansions, separation of variables, spherical and cylindrical Bessel functions, and Legendre polynomials. For each topic we will study numerical and computer algebra approaches with Matlab and Maple. Prerequisites: Calculus III, Ordinary Differential Equations.

**Differential Geometry**

Differential Geometry can be considered as a continuation of the concepts of arc length and surface area, together with questions about shortest paths (geodesics). This course will restrict itself to the geometry of curves and surfaces, covering the local theory of curves, geodesics, the Gauss map, first and second fundamental forms, parallel transport, and the Gauss-Bonnet theorem. Prerequisites: Calculus III and Linear Algebra or permission of the instructor.

**Topology**

General topology investigates the fundamental concepts necessary to develop function theory, and hence, calculus and analysis, on abstract spaces. The subject developed from analysis, geometry, and set theory, and the material reflects this, often reducing ideas down to set theory. For example, functions are defined in terms of sets. This course on general topology will start with set theory, and include a discussion of the axiom of choice. The course will then move on to the concept of open sets, the fundamental unit in analysis. The final target of the class is to determine just what structure on a space is necessary for the space to have a metric. The material assumes no knowledge of advanced mathematics but does require some sophistication in proof-writing. Prerequisites: Some experience with writing proofs.

**Topics in Algebra**

This is an advanced course intended for students who have completed the Abstract Algebra sequence. The content of the course varies from year to year. Past topics have included: Character Theory, Galois Theory, Representation Theory of Finite Groups, Representation Theory of Lie Groups and Lie Algebras. Prerequisites: Permission of Instructor

**Topics in Analysis**

This is an advanced course intended for students who have completed the Real Analysis sequence. The content of the course varies from year to year. Past topics have included: Measure Theory, Probability, Functional Analysis, Partial Differential Equations. Prerequisites: Permission of Instructor

**Topics in Geometry and Topology**

This is an advanced course intended for students who have completed either the Real Analysis or Abstract Algebra sequence. The content of the course varies from year to year. Past topics have included: Algebraic Topology, Differential Topology, Groups and Geometry, Hyperbolic Geometry. Prerequisites: Permission of Instructor

**Topics in Number Theory and Cryptography**

The Theory of Numbers is that branch of mathematics which deals with properties of counting numbers, the most primitive of our mathematical creations. This content of the course varies from year to year, and ranges over elementary to advanced material . Past topics have included: Introduction to Classical Number Theory, Coding and Cryptography, and Analytic Number Theory. Prerequisites: Permission of Instructor

**Science on the Computer**

In this course we will learn how to use the computer algebra system (Maple) and the scientific programming package (Matlab) to solve real world problems. To give just a sample of topics, we will consider least squares data fitting for Dow Jones index, regression analysis, scaling, maximizing profit from sales data, Kirchhoff laws and RLC circuits, projectile motion, Monte Carlo simulations, phase plane portraits, competition of species, predator-pray models, nonlinear diode, fractal patterns. In addition, any other topics of interest to students could be included in class material or developed as an individual project. Prerequisites: None, but Calculus I recommended.

**Mathematics Seminar**

Math Seminar has been a traditional forum for students interested in mathematics. The purpose of this seminar is to cover many interesting or advanced topics in mathematics that cannot be titled under one subject. Students enrolled in this seminar are expected to present several lectures prepared under supervision of the math faculty. Prerequisites: None

**Tutorials in Mathematics**

The faculty offer regular tutorials on many mathematical topics. Past tutorials have included Analytic Number Theory, Algebraic Combinatorics, Homology, Lie Theory, Galois Theory, Godel's Theorems, Fourier Analysis, Stable Marriage Problems, Coding and Cryptography. Prerequisites: Permission of instructor.

For a complete list of courses, click here.

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