# MAT: Mathematics

### MAT 118 - C: Mathematical Thinking

Development of quantitative thinking and problem solving abilities through a selection of mathematical topics: logic and reasoning; numbers, functions, and modeling; combinatorics and probability; growth and change. Other topics may include geometry, statistics, game theory, and graph theory. Through their engagement in problem solving, students develop an appreciation of the intellectual scope of mathematics and its connections with other disciplines.

*Prerequisite:* C or better in MAP 103 or level 2+ or higher on the mathematics placement examination

(Prerequisite must be met within one year of beginning this course.)

3 credits

### MAT 122 - C: Overview of Calculus with Applications

The basics of calculus in a self-contained, one-semester course. Properties and applications of polynomial, exponential, and logarithmic functions. Derivatives: slopes, rates of change, optimization, integrals, area, cumulative change, and average. The fundamental theorem of calculus. Emphasis on modeling examples from economics. Students who subsequently wish to enroll in MAT 125 or 131 will be required to take MAT 130 as a pre- or corequisite to either course or to score level 4 on the mathematics placement examination before taking either course.

*Prerequisite:* C or better in MAP 103 or level 3 on the mathematics placement exam

(Prerequisite must be met within one year prior to beginning the course.)

3 credits

### MAT 123 - C: Introduction to Calculus

Comprehensive preparation for the regular calculus sequences, with introduction to derivatives. Careful development of rational, exponential, logarithmic, and trigonometric functions, and their applications. Asymptotics and limits. Linear approximations, slope and derivatives, detailed curve sketching. General modeling examples.

*Prerequisite:* C or better in MAP 103 or level 3 on the mathematics placement exam

(Prerequisite must be met within one year prior to beginning the course.)

3 credits

### MAT 125 - C: Calculus A

Differential calculus, emphasizing conceptual understanding, computations and applications, for students who have the necessary background from 12th-year high school mathematics. Differentiation of elementary algebraic, trigonometric, exponential and logarithmic functions; graphing; modeling; and maximization. May not be taken for credit in addition to MAT 131 or 141 or AMS 151.

*Prerequisite:* C or higher in MAT 123; or level 4 on the mathematics placement examination; or MAT 122 and coregistration in MAT 130

3 credits

### MAT 126 - C: Calculus B

A continuation of MAT 125, covering integral calculus: the fundamental theorem, symbolic and numeric methods of integration, area under a curve, volume, applications such as work and probability, complex numbers. May not be taken for credit in addition to MAT 142 or AMS 161.

*Prerequisite:* C or higher in MAT 125 or 131 or 141 or AMS 151 or level 6 on the mathematics placement examination

3 credits

### MAT 127 - C: Calculus C

A continuation of MAT 126, covering: improper integrals and l'Hospital's rule, sequences, series, Taylor series, differential equations and modeling. May not be taken for credit in addition to MAT 132 or 142 or AMS 161.

*Prerequisite:* C or higher in MAT 126 or level 8 on the mathematics placement examination

3 credits

### MAT 129: Introduction to Integration

This course covers the material that is in MAT131 and not MAT125: The Riemann integral and the Fundamental Theorem of Calculus. This course serves as a bridge for students who took MAT125 but are ready to handle the faster pace of MAT132. Students must have a solid understanding of differentiation, which is covered by MAT125. Not for credit in addition to any of MAT126, MAT131, MAT141, or AMS151. Offered in winter session only.

*Prerequisites:* B or better in MAT 125 or Math placement level 6

1 credit

### MAT 130: Functions

Inverse functions, exponential and logarithmic functions, radian measure of angles and trigonometric functions. Open to prospective students in engineering, physical sciences, and mathematics who need to bridge the gap between MAT 122 and 125 or MAT 131 or AMS 151. May not be taken for credit in addition to MAT 123.

*Advisory Prerequisite:* C or higher in MAT 122

*Advisory Corequisite:* MAT 125 or 131 or AMS 151

1 credit

### MAT 131 - C: Calculus I

The differential calculus and integral calculus, emphasizing conceptual understanding, computations and applications, for students who have the necessary background from 12th-year high school mathematics. Differentiation of elementary algebraic; trigonometric, exponential, and logarithmic functions; graphing; modelling and maximization; the Riemann integral; and the fundamental theorem. May not be taken for credit in addition to MAT 125 or 141 or AMS 151.

*Prerequisite:* B or higher in MAT 123, or level 5 on the mathematics placement examination, or B or higher in MAT 122 and coregistration in MAT 130

4 credits

### MAT 132 - C: Calculus II

A continuation of MAT 131, covering symbolic and numeric methods of integration; area under a curve; volume; applications such as work and probability; improper integrals and l'Hospital's rule; complex numbers; sequences; series; Taylor series; differential equations; and modelling. May not be taken for credit in addition to MAT 127 or 142 or AMS 161.

*Prerequisite:* C or higher in AMS 151 or MAT 131 or 141, or level 7 on the mathematics placement examination

4 credits

### MAT 141 - C: Honors Calculus I

The topics of MAT 131 treated with additional attention to the underlying theory as a means of understanding why the processes of calculus work. May not be taken for credit in addition to MAT 125 or 131.

*Prerequisite:* Level 5 on the mathematics placement examination; priority given to students in the University's honors programs

4 credits

### MAT 142: Honors Calculus II

A continuation of MAT 141 in the same spirit, covering the topics of MAT 132. May not be taken for credit in addition to MAT 126 or 127 or 132 or AMS 161.

*Prerequisite:* C or higher in MAT 141, or B or higher in MAT 131 or AMS 151, or level 7 on the mathematics placement examination

4 credits

### MAT 160: Mathematical Problems and Games

Intended for students interested in sharpening their problem-solving skills and in developing their ability to express mathematical ideas.

1 credit, S/U grading

### MAT 171 - C: Accelerated Single-Variable Calculus

A single semester, honors-level, course which reviews the material in MAT 131 in a few weeks, then concentrates on the topics covered in MAT 132, with additional attention paid to the underlying theory. Primarily intended for students who have had calculus in high school. May not be taken for credit in addition to MAT 126 or 127 or 132 or 142 or AMS 161.

*Prerequisites:* Level 5 on the AB Calculus AP exam, Level 3 on the BC Calculus exam, A or A- in MAT 131 or AMS 151, MAT 141, or level 7 on the mathematics placement exam. Priority given to students in the University's honors programs.

4 credits

### MAT 200: Logic, Language and Proof

A basic course in the logic of mathematics, the construction of proofs and the writing of proofs. The mathematical content is primarily set theory, combinatorics and Euclidean geometry. There is considerable focus on writing.

*Prerequisites:* C or higher in MAT 203, 205 or AMS 261, and in MAT 211 or AMS 210; or A- or higher in MAT 125, 131, 141 or AMS 151; or B- or higher average in MAT 125/126/127, 131/132, 141/142, 171 or AMS 151/161; or permission of instructor

3 credits

### MAT 203: Calculus III with Applications

Vector algebra in two and three dimensions, multivariate differential and integral calculus, optimization, vector calculus including the theorems of Green, Gauss, and Stokes. Applications to economics, engineering, and all sciences, with emphasis on numerical and graphical solutions; use of graphing calculators or computers. May not be taken for credit in addition to AMS 261 or MAT 205.

*Prerequisite:* C or higher in MAT 127 or 132 or 142 or AMS 161 or level 9 on the mathematics placement examination

4 credits

### MAT 205: Calculus III

Vector algebra, multivariate differential and integral calculus, divergence and curl, line and surface integrals, theorems of Green, Gauss, and Stokes. More theoretical than MAT 203 with applications to the physical sciences. Not for credit in addition to AMS 261 or MAT 203.

*Prerequisite:* C or higher in MAT 127 or 132 or 142 or AMS 161 or level 9 on the mathematics placement examination

*Pre- or Corequisite:* MAT 211 or AMS 210

4 credits

### MAT 211: Introduction to Linear Algebra

Introduction to the theory of linear algebra with some applications; vectors, vector spaces, bases and dimension, applications to geometry, linear transformations and rank, eigenvalues and eigenvectors, determinants and inner products. May not be taken for credit in addition to AMS 210.

*Prerequisite:* C or higher in AMS 151 or MAT 131 or 141 or coregistration in MAT 126 or level 7 on the mathematics placement examination

3 credits

### MAT 260: Problem Solving in Mathematics

Students actively solve challenging problems in plane geometry, basic number theory, and calculus, and write precise arguments. Relevant preparation for problem-solving is provided in the course.

*Prerequisite:* MAT 203 or 205 or 211 or AMS 261, or B or higher in MAT 127 or 132 or 142 or AMS 151

1 credit

### MAT 303: Calculus IV with Applications

Homogeneous and inhomogeneous linear differential equations; systems of linear differential equations; series solutions; Laplace transforms; Fourier series. Applications to economics, engineering, and all sciences with emphasis on numerical and graphical solutions; use of computers. May not be taken for credit in addition to AMS 361 or MAT 305.

*Prerequisite:* C or higher in MAT 127 or 132 or 142 or AMS 161 or level 9 on the mathematics placement examination

4 credits

### MAT 305: Calculus IV

Linear versus nonlinear equations and their numerical solutions, existence and uniqueness, Duhamel's principle for linear equations, series solutions, systems. Fourier series; comparison of separation of variables with integral formulas. More theoretical than MAT 303. Applications to the physical sciences. May not be taken for credit in addition to MAT 303 or AMS 361.

*Prerequisite:* C or higher in MAT 127 or 132 or 142 or AMS 161 or level 9 on the mathematics placement examination

4 credits

### MAT 307: Multivariable Calculus with Linear Algebra

Introduction to linear algebra: vectors, matrices, systems of linear equations, bases and dimension, dot product, determinants. Multivariate differential and integral calculus, divergence and curl, line and surface integrals, theorems of Green, Gauss, and Stokes. More theoretical and intensive than MAT 203, this course is primarily intended for math majors. Together with MAT 308, it forms a 2-semester sequence covering the same material as the 3-semester sequence of MAT 205, MAT 211 and MAT 305. May not be taken for credit in addition to MAT 203, MAT 205 or AMS 261.

*Prerequisite:* MAT 127 or MAT 132

4 credits

### MAT 308: Differential Equations with Linear Algebra

Linear algebra: determinants, eigenvalues and eigenvectors, diagonalization. Differential equations; existence and uniqueness of solutions. First- and second-order equations; linear versus nonlinear equations. Systems of linear equations. Laplace transform. Applications to physics. More theoretical and intensive than MAT 303, this course is primarily intended for math majors. Together with MAT 307, it forms a 2-semester sequence covering the same material as the 3-semester sequence of MAT 205, MAT 211 and MAT 305. May not be taken for credit in addition to MAT 303, MAT 305 or AMS 361.

*Prerequisite:* MAT 307 or MAT 205 and MAT 211

4 credits

### MAT 310: Linear Algebra

Finite dimensional vector spaces, linear maps, dual spaces, bilinear functions, inner products. Additional topics such as canonical forms, multilinear algebra, numerical linear algebra.

*Prerequisites:* C or higher in MAT 211 or 305 or AMS 210; C or higher in MAT 200 or permission of instructor

4 credits

### MAT 311: Number Theory

Congruences, quadratic residues, quadratic forms, continued fractions, Diophantine equations, number- theoretical functions, and properties of prime numbers.

*Prerequisites:* C or higher in MAT 312 or 313 or 318; C or higher in MAT 200 or permission of instructor

3 credits

### MAT 312: Applied Algebra

Topics in algebra: groups, informal set theory, relations, homomorphisms. Applications: error correcting codes, Burnside's theorem, computational complexity, Chinese remainder theorem. This course is offered as both AMS 351 and MAT 312.

*Prerequisite:* C or higher in AMS 210 or MAT 211

*Advisory Prerequiste:* MAT 200 or CSE 113

3 credits

### MAT 313: Abstract Algebra

Groups and rings together with their homomorphisms and quotient structures. Unique factorization, polynomials, and fields.

*Prerequisites:* C or higher in MAT 310 or 312 or 318; C or higher in MAT 200 or permission of instructor

3 credits

### MAT 316: Invitation to Modern Mathematics

Mathematical reasoning and the process of mathematical research. The power and range of modern mathematics are discussed in detail through a few key theorems in algebra, analysis, geometry, and topology together with some applications.

*Prerequisites:* MAT 211 or AMS 210; MAT 203 or 205 or AMS 261; MAT 200 or permission of instructor

3 credits

### MAT 319: Foundations of Analysis

A careful study of the theory underlying topics in one-variable calculus, with an emphasis on those topics arising in high school calculus. The real number system. Limits of functions and sequences. Differentiations, integration, and the fundamental theorem. Infinite series.

*Prerequisites:* C or higher in MAT 200 or permission of instructor; C or higher in one of the following: MAT 203, 205, 211, AMS 261, or A- or higher in MAT 127, 132, 142, or AMS 161

4 credits

### MAT 320: Introduction to Analysis

A careful study of the theory underlying calculus. The real number system. Basic properties of functions of one real variable. Differentiation, integration, and the inverse theorem. Infinite sequences of functions and uniform convergence. Infinite series.

*Prerequisites:* C or higher in MAT 200 or permission of instructor; C or higher in one of the following: MAT 203, 205, 211, AMS 261, or A- or higher in MAT 127, 132, 142, or AMS 161

4 credits

### MAT 322: Analysis in Several Dimensions

Continuity, differentiation, and integration in Euclidean n-space. Differentiable maps. Implicit and inverse function theorems. Differential forms and the general Stokes's theorem.

*Prerequisites:* C or higher in MAT 203, 205, or AMS 261; C or higher in MAT 211 or AMS 210; B or higher in MAT 320

*Advisory Pre- or Corequisite:* MAT 310

3 credits

### MAT 324: Real Analysis

Metric spaces, including compactness, connectedness, completeness, and continuity. Introduction to Lebesgue integration. Aspects of Fourier series, function spaces, Hilbert spaces, Banach spaces.

*Prerequisites:* C or higher in MAT 203 or 205 or AMS 261; B or higher in MAT 320

3 credits

### MAT 331: Computer-Assisted Mathematical Problem Solving

Exploration of the use of the computer as a tool to gain insight into complex mathematical problems through a project-oriented approach. Students learn both the relevant mathematical concepts and ways that the computer can be used (and sometimes misused) to understand them. The particular problems may vary by semester; past topics have included cryptography, fractals and recursion, modeling the flight of a glider, curve fitting, the Brachistochrone, and computer graphics. No previous experience with computers is required.

*Prerequisite:* C or higher in MAT 203 or 205 or AMS 261

3 credits

### MAT 336 - H: History of Mathematics

A survey of the history of mathematics from the beginnings through the 19th century, with special attention to primary sources and to the interactions between culture and mathematics. Emphasis on topics germane to the high school curriculum. Mesopotamian, Egyptian, and Greek mathematics; non-European mathematics; early Renaissance mathematics; the birth and flowering of calculus; the beginnings of probability theory; and the origin of non-euclidean gemetrics and the modern concept of number.

*Prerequisite:* MAT 200 or AMS 301

3 credits

### MAT 341: Applied Real Analysis

Partial differential equations of mathematical physics: the heat, wave, and Laplace equations. Solutions by techniques such as separation of variables using orthogonal functions (e.g., Fourier series, Bessel functions, Legendre polynomials). D'Alambert solution of the wave equation.

*Prerequisites:* C or higher in the following: MAT 203 or 205 or AMS 261; MAT 303 or 305 or AMS 361

*Advisory Prerequisite:* MAT 200

3 credits

### MAT 342: Applied Complex Analysis

Functions of a complex variable, calculus of residues including evaluation of real integrals, power and Laurent series, conformal mappings and applications, Laplace and Cauchy-Riemann equations, the Dirichlet and Neumann problems, and the Laplace and Hilbert transforms and their applications to ordinary and partial differential equations.

*Prerequisites:* C or higher in the following: MAT 203 or 205 or AMS 261; MAT 303 or 305 or AMS 361

*Advisory Prerequisite:* MAT 200

3 credits

### MAT 351: Differential Equations: Dynamics and Chaos

A study of the long-term behavior of solutions to ordinary differential equations or of iterated mappings, emphasizing the distinction between stability on the one hand and sensitive dependence and chaotic behavior on the other. The course describes examples of chaotic behavior and of fractal attractors, and develops some mathematical tools for understanding them.

*Prerequisites:* C or higher in the following: MAT 203 or 205 or AMS 261; MAT 303 or 305 or AMS 361; MAT 200 or permission of instructor

3 credits

### MAT 360: Geometric Structures

Formal geometries and models. Topics selected from projective, affine, Euclidean, and non-Euclidean geometries.

*Pre- or Corequisites:* C or higher in the following: MAT 203 or 205 or AMS 261; MAT 211 or AMS 210; MAT 200 or permission of instructor

3 credits

### MAT 362: Differential Geometry of Surfaces

The local and global geometry of surfaces: geodesics, parallel transport, curvature, isometries, the Gauss map, the Gauss-Bonnet theorem.

*Prerequisite:* C or higher in MAT 319 or 320 or 364

3 credits

### MAT 364: Topology and Geometry

A broadly based introduction to topology and geometry, the mathematical theories of shape, form, and rigid structure. Topics include intuitive knot theory, lattices and tilings, non-Euclidean geometry, smooth curves and surfaces in Euclidean 3-space, open sets and continuity, combinatorial and algebraic invariants of spaces, higher dimensional spaces.

*Prerequisites:* MAT 203 or 205 or AMS 261

*Advisory Prerequisite:* MAT 319 or 320

3 credits

### MAT 371: Logic

A survey of the logical foundations of mathematics: development of propositional calculus and quantification theory, the notions of a proof and of a model, the completeness theorem, Goedel's incompleteness theorem. This course is offered as both CSE 371 and MAT 371.

*Prerequisite:* CSE 150 or CSE 215 or MAT 200

3 credits

### MAT 373: Analysis of Algorithms

Mathematical analysis of a variety of computer algorithms including searching, sorting, matrix multiplication, fast Fourier transform, and graph algorithms. Time and space complexity. Upper-bound, lower- bound, and average-case analysis. Introduction to NP completeness. Some machine computation is required for the implementation and comparison of algorithms. This course is offered as CSE 373 and MAT 373.

*Prerequisites:* MAT 211 or AMS 210; CSE 214 or CSE 260

3 credits

### MAT 401: Seminar in Mathematics

Discussions of a specific area of interest in mathematics. The work of each semester covers a different area of mathematics. May be repeated as topic changes. Prerequisites will be announced with the topic each time the course is offered.

*Prerequisites:* U3 or U4 standing; additional prerequisites announced with topic

3 credits

### MAT 402: Seminar in Mathematics

Discussions of a specific area of interest in mathematics. The work of each semester covers a different area of mathematics. May be repeated as topic changes. Prerequisites will be announced with the topic each time the course is offered.

*Prerequisites:* U3 or U4 standing; additional prerequisites announced with topic

3 credits

### MAT 475: Undergraduate Teaching Practicum

Each student assists in teaching a lower-division mathematics course or works in the Mathematics Learning Center. The student's work is regularly supervised by a faculty member. In addition, a weekly seminar is conducted. Responsibilities may include preparation of materials for student use and discussions, helping students with problems, and involvement in "alternative" teaching projects. Intended for upper-division students who have excelled in the calculus sequence. May not be used for major credit.

*Prerequisite:* Permission of the director of undergraduate studies

3 credits, S/U grading

### MAT 487: Independent Study in Special Topics

A reading course for juniors and seniors. The topics may be chosen by the student with the approval of a supervising member of the faculty, who also takes responsibility for evaluation. A topic that is covered in a course regularly offered by the department is not appropriate for independent study. May be repeated.

*Prerequisite:* Permission of the director of undergraduate studies

0-6 credits

### MAT 495: Honors Thesis

The student and a supervising faculty member together choose a topic in mathematics, and the student writes a substantial paper expounding the topic in a new way.

*Prerequisite:* Permission of the director of undergraduate studies

3 credits