Department of Mathematics
College of Arts and Sciences

Mathematics Building
North Campus
Buffalo, NY 14260
(716) 645-MATH (645-6284)
E-mail: UndergraduateStudies@math.buffalo.edu

Jonathan Bell, Chair of Department
James Faran, Associate Chair
Richard E. Vesley, Director of Undergraduate Studies

Mathematics is a broad discipline with many diverse applications in social, managerial, and life sciences, as well as in the physical sciences and engineering. The Department of Mathematics provides a variety of programs leading to a baccalaureate degree.

Programs Leading to B.A. Degree

Program C—Mathematics major with a concentration in computing and applied mathematics; this program is designed for students interested in careers in applying mathematics or doing scientific programming.

Program TC—Mathematics major for prospective high school teachers.

Program GS—Mathematics major for intended graduate study in mathematics; a basic theoretical course of study for students planning careers in mathematics or planning graduate study.

Program GS/AM—Mathematics major for intended graduate study in applied mathematics; a basic theoretical course of study for students planning a career in applied mathematics or planning graduate study.

Program M/E—Joint major with economics, for students intending graduate study in economics.

Programs Leading to B.S. Degree

Program BSc—Same as Program GS with four additional electives.

Program BSc/AM—Same as Program GS/AM with four additional electives.

Program M/P—Joint major with physics, for students intending graduate study in mathematical physics.

Program Leading to Combined B.A./M.A. Degree
Five-year, 138-credit program. Admission requires a 3.0 GPA in courses through differential equations and linear algebra, as well as letters of recommendation. For program details contact the director of undergraduate studies.

Academic Counseling

Admission Advisement

Note: If you are majoring in mathematics, see the director of undergraduate studies at least once a year. If you are intending to major in mathematics, contact the director of undergraduate studies as soon as possible regarding your program. Even if your grades are too low for acceptance to the department, you should see the director of undergraduate studies regarding conditions for acceptance and suggestions for your schedule while waiting for admittance.

The Department of Mathematics provides academic advice to majors and nonmajors. The Office of Undergraduate Studies is open 9:00 a.m.–5:00 p.m. Monday–Friday. Students in need of advisement on acceptance to the department, choice of courses, program or degree decisions, or career advisement, should see the director of undergraduate studies.

Upon acceptance to the department, it is recommended that students meet with the director of undergraduate studies to map out their program. Students in all programs except computing and applied math are required to discuss with the undergraduate director the appropriateness of their choice of program; this is an opportunity for students to clarify their understanding of the more theoretically oriented programs.

Course Advisement

1. MTH115 Survey of Algebra and Trigonometry or three years of high school math is a prerequisite for the freshman calculus course. A placement exam is given to entering freshmen to assist them in determining their level of preparedness. Students whose performance on this exam is low are encouraged to enroll in MTH115. Students whose mathematical preparation is so weak that they are not prepared for MTH115 are encouraged to take remedial courses in the university’s Learning Center located in 103 Talbert Hall.
2. The Department of Mathematics offers five calculus sequences:
• MTH141-142 College Calculus I-II: the standard basic calculus course for engineering and science (including computer science) students. It is a prerequisite for more advanced work in mathematics.
• MTH121-122 Survey of Calculus and Applications I-II: a basic calculus course for general health sciences,  and arts and letters. It is somewhat easier than MTH141-142 and does not go into the subject as deeply. MTH121-122 does not meet the requirements of most science majors.
• MTH131 Mathematical Analysis for Management: a one semester calculus course for management students. It meets the calculus requirement for students in the School of Management.
• MTH151-152 Enriched Calculus I-II: a more challenging version of MTH141-142, taught in small classes.
• MTH153-154 Honors Calculus I-II: emphasizes proofs and concepts of calculus.

A student who is unsure whether to take MTH141-142 or MTH121-122, should take MTH141-142. Taking MTH121-122 might delay graduation by as much as a year.

Since each of the calculus sequences has its own specific requirements and purpose, a change in sequence (MTH121-142 or MTH141-122) is not recommended. Students who take MTH121 and later decide that they need to switch into the MTH141-142 track will usually need to take MTH141 first, since even a good grasp of MTH121 can fail to provide adequate preparation for MTH142.

3. There are two sequences of upper-division analysis courses:
• MTH417 Survey of Multivariable Calculus, and MTH418 Survey of Partial Differential Equations. These courses are advanced versions of MTH241 and MTH306, respectively. These are terminal courses, designed for students requiring no further advanced courses in mathematical analysis, such as MTH445 or MTH449.
• MTH301 Basic Analysis and Proofs, and MTH431-432 Introduction to Real Variables I-II. This is a theoretically oriented sequence, stressing proofs, which is designed for students who want preparation for more advanced analysis courses, such as MTH445 or MTH449, including students intending to go to graduate school in mathematics.
4. Students taking freshman mathematics courses can get tutoring help on a drop-in basis from The Math Place. This facility is located in the Learning Center in Talbert Hall.

Majoring in Mathematics

Students, including freshmen, considering majoring in mathematics are encouraged to consult as early as possible with the departmental director of undergraduate studies. Admission to the department as far in advance of graduation as possible ensures that the department and the student can plan together a program best suited to the student’s needs.

Transfer Policy

Transfer students who intend to major in mathematics are urged to consult the director of undergraduate studies before or very early in their first semester at the University at Buffalo to determine equivalency of transferred mathematics courses. Transfer students are required to complete at least four upper-division courses in the Department of Mathematics at the University at Buffalo.

Recommendations Concerning Related Courses

The Department of Mathematics strongly recommends that its majors in the theoretically oriented programs (Programs GS and GS/AM or Programs BSc and BSc/AM) take at least two semesters of French, German, or Russian, since these, along with English, are the basic languages of the mathematical literature.

All mathematicians need some facility with computers. The department, therefore, recommends that its students take an introductory course in scientific programming. Specifically, a one year sequence, CSE113-114 or CSE113-115 or CSE115-116, is strongly recommended for students majoring in mathematics. Some major programs have specific computer science requirements.

Acceptance Criteria

Applicants to the mathematics program should bring a copy of their current UB DARS reports directly to the Department of Mathematics. Students may apply at any time after they have completed the three-course calculus sequence MTH141-142-241 or the equivalent at a transfer institution, but should apply, if possible, while enrolled in MTH241 to receive appropriate advisement.

The department’s requirement for admission is a 2.5 GPA in the three calculus courses. Students who have taken more than these three courses usually need a minimum grade of "C" in any additional required courses they have completed.

Note: Admittance to the department guarantees admittance to Program C only—all other programs require specific approval from the director of undergraduate studies. Students who expect difficulty in being admitted to the department should follow the requirements for Program C or see the director of undergraduate studies for advice on which program to follow, pending admittance.

Departmental Probation

Students whose math GPA drops below 2.0 will be placed on departmental probation for one semester. If their GPA remains below 2.0 at the end of the probationary semester, they will be dropped from the department. They may reapply after one semester.

Departmental Requirements for Graduation

After a student has been accepted as a major, progress toward fulfillment of the degree requirements is monitored by the department. A student will be recommended for a baccalaureate degree in mathematics only if (1) a program has been approved by the director of undergraduate studies; (2) all required/approved courses have been completed; (3) A minimum departmental GPA of 2.0 has been attained in the specific program; and (4) completion of 16 credit hours from natural sciences outside of mathematics to be distributed among two or more departments.

Note: Some out-of-department courses will count toward both the departmental requirements and the natural sciences requirement.

Mathematics (MTH)

Note: Mathematics courses are not numbered according to difficulty. For example, MTH301 is a theoretically oriented course, stressing proofs. Some students find it to be much more difficult than MTH417 and MTH418.

115 Survey of Algebra and Trigonometry (4) (F; Sp)
A precalculus course; covers topics from the NYS Regents Course III: order, absolute value, inequalities, exponents, radicals, polynomials, rational expressions, solving systems of linear equations, quadratic equations and inequalities, functions (rational, logarithmic, exponential, trigonometric), graphing, trigonometric identities.
    Applications to problems are emphasized throughout the course. This fast-paced course is designed to review Course III and prepare students for further courses in mathematics.
    Students with three years of high school math but with weak algebra skills should take ULC147 before MTH115. Students who have had only two years of high school mathematics may take MTH115, or may prefer to take a two-semester sequence covering this material more thoroughly and at a more moderate pace: ULC147 and ULC148. LEC/LAB

117 Applications of Mathematical Concepts (3)
Prerequisite: three years of NYS Regents-level high school mathematics
Intended to be nontechnical, with an emphasis on conceptually sophisticated and aesthetically appealing mathematical discoveries of the twentieth century, such as mathematical aspects of social choice, management science, and growth and symmetry. LEC/REC

119 Survey of Finite Mathematics (4)
Prerequisite: three years of NYS Regents-level high school mathematics
Elementary topics in topology, probability, numerical sequences, and groups. LEC/LAB

121 Survey of Calculus and Its Applications I (4) (F; Sp)
Prerequisite: NYS Regents Course III or MTH115
For students in social, biological, and management sciences. Limits, continuity, differentiation of algebraic and exponential functions; applications; introduction to integration. LEC/LAB

122 Survey of Calculus and Its Applications II (4) (F; Sp)
Prerequisite: MTH121 or MTH131 with recommended grade of "C" or higher
Continuation of MTH121. Maximization of functions of several variables using both calculus and elementary linear programming techniques. Elementary integration, simple differential equations, matrix algebra. LEC/LAB

131 Mathematical Analysis for Management (4) (F; Sp)
Prerequisite: NYS Regents Course III or MTH 115
For students in Management. Limits, continuity, differentiation of algebraic and exponential functions. Applications, partial derivatives and applications. Introduction to integration. LEC/LAB/REC

141 College Calculus I (4) (F; Sp)
Prerequisite: trigonometry or NYS Regents Course III or MTH115
Beginning of a three-semester sequence in calculus for students of mathematics, natural sciences, and engineering. Covers differentiation and integration with applications. LEC/LAB

142 College Calculus II (4) (F; Sp)
Prerequisite: MTH141 with recommended grade of "C" or higher; MTH121 is usually not adequate preparation for MTH142
Differentiation and integration of transcendental functions; infinite sequences; series and power series; methods of integration; additional topics in analytic geometry. LEC/LAB

145 Introduction to Mathematical Computing and Programming (4) (F)
Basic introduction to computers and computer organization; algorithms and their development in a structured manner; procedures and functions; programming; aspects and topics from computer science. LEC/LAB

151 Enriched Calculus I (4) (F)
Prerequisites: consent of instructor and a math SAT score of at least 650
Similar to MTH141 but more challenging. May include additional material and more proofs. LEC/LAB

152 Enriched Calculus II (4) (Sp)
Prerequisite: MTH151 and/or consent of instructor
Continuation of MTH151. Covers all the material in MTH142 and additional material as described in MTH151. LEC/LAB

153 Honors Calculus I (4) (F)
Prerequisite: consent of instructor or a score of 4 or 5 on the advanced placement calculus exam or equivalent. Credit may be given in addition to advanced placement credit
First course in the honors sequence for intended math majors or for others with suitable preparation. Emphasizes proofs and concepts of calculus. LEC/LAB

154 Honors Calculus II (4) (Sp)
Prerequisite: MTH153 and/or consent of instructor
Continuation of MTH153. May be taken in addition to advanced placement credit. LEC/LAB

172 Principles of Mathematical Computing (4) (Sp)
Prerequisite: MTH145
Programming through data structures (sorting algorithms) and applications in scientific computing. LEC/LAB
    Note: Credit is not given for both MTH172 and CSE114.

181 Conceptual Mathematics I (3)
Prerequisite: high school algebra
Intended primarily for students not contemplating a major in the natural sciences, this course concentrates on the conceptual nature of mathematics, and should aid in understanding science as part of the general culture, as well as in learning to analyze and reason more clearly. It differs substantially from other beginning courses, which must emphasize the computational facility needed for advanced work. Topics include categories; adding and multiplying numbers, objects, and propositions; transformations of space and quantity. LEC/LAB

191 Introduction to Discrete Mathematics I (4) (F)
Prerequisites: working knowledge of a programming language
First part of a two-semester sequence intended to provide the mathematical foundations for the study of computer science. This course is also approved for mathematics majors in Program TC. Topics include sets, relations, functions, mathematical induction, fundamental counting methods, difference equations, and sequences and series. Same as CSE191. LEC/LAB

192 Introduction to Discrete Mathematics II (4) (S)
Prerequisite: MTH191 or CSE191
Second part of a two-semester sequence in discrete mathematics intended to provide the mathematical foundations for the study of computer science. Topics include discrete probability, mathematical logic, linear algebra, and graph theory. Same as CSE192. LEC/LAB

241 College Calculus III (4) (F; Sp)
Prerequisite: MTH142 with recommended grade of "C" or higher
Geometry and vectors of n-dimensional space; Green's theorem, Gauss' theorem, Stokes' theorem; multidimensional differentiation and integration; application to two- and three-dimensional space. LEC/LAB

242 Survey of Differential Equations (4) (F; Sp)
Prerequisite: MTH142 with recommended grade of "C" or higher; MTH242 may be taken before MTH241
First-order linear and nonlinear equations; second-order homogeneous and nonhomogeneous equations, variation of parameters, undetermined coefficients; applications of first- and second-order equations; series solutions; linear systems; topics from among Laplace transforms, numerical methods, qualitative theory of differential equations, partial differential equations. LEC/LAB

251 Honors Calculus III (4) (F)
Prerequisite: permission of instructor
Third-semester calculus course for honors students and students with an excellent record in previous calculus courses. Emphasizes proofs and concepts of calculus. LEC/LAB/REC

272 Mathematics of Data Manipulation (4) (Sp)
Prerequisite: MTH172 or CSE114
Aspects of data representation and manipulation needed for applications programming together with an introduction to their mathematical theory. LEC/LAB

301 Basic Analysis and Proofs (4) (F; Sp)
Prerequisite: MTH241
Prerequisite for MTH431 and MTH335, and is recommended for MTH419. An introduction to mathematical proofs. Problem solving and proof construction in the context of geometry of Rn, convergence, basic topological definitions in Rn, completeness, compactness (with emphasis on examples in R, R2, and R3). LEC/LAB

306 Introduction to Differential Equations (4)
Prerequisite: MTH142
Analytic solutions, qualitative behavior of solutions to differential equations. First-order and higher order ordinary differential equations, including nonlinear equations. Analytic, geometric, and numerical perspectives will be covered, as well as an interplay between methods and model problems. Necessary matrix theory will be discussed, and differential equation models of phenomena from various disciplines will be explored. Integrated into the course is the use of a mathematical software system designed to aid in the numerical and qualitative study of solutions, and to aid in the geometric interpretation of solutions. LEC/LAB

309 Introductory Linear Algebra (4) (F; Sp)
Prerequisite: MTH142, MTH192, or CSE192
Prerequisite for MTH310, MTH419, and MTH335. Linear equations, matrices, determinants, vector spaces, linear mappings, inner products, eigenvalues, eigenvectors. LEC/LAB

310 Applications of Linear Algebra (4) (F; Sp)
Prerequisites: MTH242, MTH309
Topics selected by the instructor. LEC/LAB

313 Elements of Set Theory (4)
Prerequisite: MTH241 or permission of instructor
Cardinals, ordinals, order-types, and operations on them. Axiom of choice. Sets. LEC/REC

314 Logic for Computer Science (3) (F)
Prerequisites: at least one year of previous training in college mathematics and one year of programming
An introduction to the predicate calculus emphasizing proof theory rather than model theory. Some programming exercises will be given. LEC

335 Elements of Geometry (4) (Sp)
Prerequisites: MTH301, MTH309
Euclidean and non-Euclidean geometries. Study of the Hilbert postulates and various models, with emphasis on Euclidean and Lobachevskian geometries. LEC/LAB

336 Projective Geometry (4)
Prerequisite: MTH241 or permission of instructor
Projective geometry is both a generalization and a simplification of familiar Euclidean geometry, omitting concepts of distance and angle. The small set of axioms has an important duality property. There are several famous theorems relating figures in the plane. Three-dimensional geometry and specialization to Euclidean and other geometries may be considered, as time permits. No previous geometry is required. LEC

337 Introduction to Scientific Computing (4)
Prerequisite: MTH141
For students in all scientific disciplines. Offers an introduction to the fundamental numerical techniques used in high performance scientific computing. These methods include sorting, numerical integration, root finding for nonlinear systems, elementary linear algebra, and basic ordinary differential equations. The emphasis of this course is on an experimental approach to scientific computing. The students will learn the basic of UNIX, will write and debug programs in FORTRAN, and will use computational and visualization tools in Mathlab. Cross-listed as CSE237. LEC/LAB

343 Codes (3)
Prerequisite: MTH309
Study of the characteristics and mathematical foundations of several codes widely used in information handling for data compression, error detection, error correction and other special purposes. The codes studied include binary, comma-free, cryptographic, cyclic, Hamming, and Huffman. LEC

353 Introduction to Combinatorics I (3) (F)
Prerequisite: MTH241
Permutations, combinations, and other problems of selecting and arranging objects subject to various restrictions; generating functions; recurrence relations; inclusion-exclusion theorem. LEC

354 Introduction to Combinatorics II (3) (Sp)
Prerequisite: MTH241
Theory of graphs: Eulerian and Hamiltonian circuits; trees; planarity; colorability; directed graphs and tournaments; isomorphism; adjacency matrix; applications to problems in communication, scheduling, and traffic flow. LEC

411 Probability Theory (4) (F)
Prerequisite: MTH141-142 or equivalent
Introduces the basic concepts of probability theory and addresses many concrete problems. A list of basic concepts includes axioms of probability, conditional probability, independence, random variables (continuous and discrete), distribution functions, expectation, variance, joint distribution functions, limit theorems. LEC/REC

412 Introduction to Statistical Inference (4) (Sp)
Prerequisite: MTH141 (or equivalent), and MTH411 (or STA401) or permission of instructor
Topics include: review of probability, conditional probability, Bayes' Theorem; random variables and distributions; expectation and properties; covariance, correlation and conditional expectation; special distributions; Central Limit Theorem and applications; estimations, including Bayes estimators, maximum likelihood estimators and their properties. The course will also include use of sufficient statistics to 'improve' estimators, unbiasedness, hypothesis testing, linear statistical models, and statistical inference from the Bayesian point of view. REC

413 Introduction to Mathematical Logic I (3) (F)
Prerequisite: MTH313 or permission of instructor
Informal and formal development of propositional calculus; predicate calculus and predicate calculus with equality; completeness theorem and some consequences. LEC

414 Introduction to Mathematical Logic II (3) (Sp)
Prerequisite: MTH314 or MTH413
Godel’s incompleteness theorem; decidability and recursiveness; consistency problems. LEC

417 Survey of Multivariable Calculus (4) (F; Sp)
Prerequisite: MTH241
Survey of functions of several variables; differentiation, composite, and implicit functions; critical points; line integrals; Green’s theorem. Vector field theory; gradient, divergence, and curl; integral theorems. Introduction to functions of a complex variable; curves and regions in the complex plane; analytic functions, Cauchy-Riemann equations, Cauchy integral formula. Applications. LEC/LAB
    Note: For math majors in Program C, and majors of science and engineering.

418 Survey of Partial Differential Equations (4) (F; Sp)
Prerequisite: MTH242 or MTH306
Survey of elementary differential equations of physics; separation of variables and superposition of solutions; orthogonal functions and Fourier series. Introduction to boundary value problems, Fourier and Laplace transforms. LEC/LAB
    Note: For math majors in Program C, and majors of science and engineering.

419 Introduction to Algebra I (4) (F; Sp)
Prerequisite: MTH309; MTH301 recommended
Emphasizes theorems and proofs. Groups, cosets, group-homomorphisms. Rings, modules, and morphisms between them. Ideals in a commutative ring (with 1). The residue-class ring R/a. Minimal and maximal ideals. LEC/REC
    Note: The MTH309 prerequisite for this course is strictly enforced. Students who have not completed MTH309, but who have had an equivalent course, need to obtain a waiver from the director of undergraduate studies.

420 Introduction to Algebra II (4) (Sp)
Prerequisite: MTH419
Continuation of MTH419. Quotient field of an integral domain; fields; vector spaces; field extensions; finite fields; modules; irreducible modules; Wedderburn-
Artin structure theorems. LEC/REC

425 Introduction to Complex Variables I (3)
Prerequisite: MTH417 or MTH431 or permission of instructor
Analyticity; calculus over the complex numbers. Cauchy theorems, residues, singularities, conformal mapping. Weierstrass convergence theorem; analytic continuation. LEC
    Note: For students of physics, electrical and other areas of engineering, and mathematics.

426 Introduction to Complex Variables II (3)
Prerequisite: MTH425
Continuation of MTH425. Weierstrass and Mittag-Leffler theorems, harmonic functions, conformal mapping and Green’s function, analytic equivalence, and Riemann’s mapping theorem. Montel’s theorem, external mappings. LEC

427 Introduction to Topology I (3) (F)
Prerequisite: MTH431
Abstract topological spaces, bases, convergence, filters, and nets; separation axioms, continuity, and homeomorphisms; connectedness, separability, compactness. LEC

428 Introduction to Topology II (3) (Sp)
Prerequisite: MTH427
Continuation of MTH427. Product and quotient topologies; compac-tification; complete semi-metric spaces; metrization; topological algebra. Application of results to such fields as differential equations, numerical analysis, probability theory. LEC

429 Introduction to the Theory of Numbers I (3) (F)
Prerequisite: MTH419
The Euclidean algorithm and unique factorization; arithmetical functions; congruences, reduced residue systems; primitive roots; certain diophantine equations. LEC

430 Introduction to the Theory of Numbers II (3) (Sp)
Prerequisite: MTH429
Continuation of MTH429. Irrational numbers; continued fractions from a geometric viewpoint; best rational approximations to real numbers; the Fermat-Pell equation; quadratic fields and integers. Applications to diophantine equations. LEC

431 Introduction to Real Variables I (4) (F)
Prerequisite: MTH301
Comprehensive and rigorous course in the study of real valued functions of one real variable. Topics include sequences of numbers, limits and the Cauchy criterion, continuous functions, differentiation, inverse function theorem, Riemann integration, sequences and series, uniform convergence. This course is a prerequisite for most advanced courses in analysis. LEC/LAB
    Note: The MTH301 prerequisite for this course is strictly enforced. Students who have not completed MTH301, but who have had an equivalent course, need to obtain a waiver from the director of undergraduate studies.

432 Introduction to Real Variables II (4) (Sp)
Prerequisite: MTH431
Rigorous course in the study of analysis in dimensions greater than one. Three basic theorems: the inverse function theorem, the implicit function theorem, and the change of variables theorem in multiple integrals are among the subjects studied in detail. Topics in this course include continuously differentiable functions, the chain rule, inverse and implicit function theorems, Riemann integration, partitions of unity, change of variables theorem. LEC/REC

437 Introduction to Numerical Analysis I (4) (F; Sp)
Prerequisite: CSE113, MTH241, MTH242 or MTH306, MTH309, or permission of instructor
Lagrangian interpolation; quadrature formulae; difference equations; numerical solution of ordinary differential equations; predictor-corrector and Runge-Kutta methods. LEC

438 Introduction to Numerical Analysis II (4) (F; Sp)
Prerequisite: MTH437
Solution of linear and nonlinear systems of equations; calculation of eigenvalues and eigenvectors; introduction to the numerical solution of boundary-value problems for both ordinary and partial differential equations. LEC/LAB

443 Fundamentals of Applied Mathematics I (4)
Prerequisite: MTH242 or MTH306
Mathematical formulation and analysis of models for phenomena in the natural sciences. Includes derivation of relevant differential equations from conservation laws and constitutive relations. Potential topics include diffusion, stationary solutions, traveling waves, linear stability analysis, scaling and dimensional analysis, perturbation methods, variational and phase-space methods, kinematics, and laws of motion for continuous media. Examples from areas might include, but are not confined to, biology, fluid dynamics, elasticity, chemistry, astrophysics, geophysics. LEC

444 Fundamentals of Applied Mathematics II (4)
Prerequisite: MTH443
Continuation of topics described in MTH443. LEC

445 Ordinary Differential Equations (3)
Prerequisite: MTH431 or permission of instructor
Existence and uniqueness of solutions; continuation of solutions; dependence on initial conditions and parameters; linear systems of equations with constant and variable coefficients; autonomous systems, phase space, and stability. LEC
    Note: Course designed for students with strong mathematics backgrounds.

449 Introduction to Partial Differential Equations (3)
Prerequisite: MTH431 or permission of instructor
Classical linear equations of mathematical physics, well-posed problems, qualitative properties of solutions. Techniques for solving problems. LEC
    Note: Course designed for students with strong mathematics backgrounds. Students must know the material of both MTH431 and MTH306 or 242.

455 Mathematical Modeling (3)
Prerequisites: MTH242 or MTH306, MTH309
An introduction to the use of mathematical modeling in applied mathematics using a case study approach. Population ecology; chemical kinetics; traffic dynamics. LEC

457 Problem Solving Seminar (3)
Prerequisites: MTH242 or MTH306, MTH411 or STA401 and some experience with scientific programming; minimum grade of B in these courses.
Based on open discussion and solving of a series of quantitatively based problems involving both individual and student team efforts. Central to the active and cooperative learning experience is the communication of results in the form of carefully written technical reports and oral presentations. Analytic, geometric, and numerical problem solving techniques may all be used. LEC

458 Mathematical Finance (3)
Prerequisite: MTH241, MTH309, MTH306 or MTH242
Introduce the mathematical theory and computation of modern financial products used in the banking and corporate world. Mathematical models for the valuation of derivative products will be derived and analyzed. LEC

460 Theory of Games (3)
Prerequisite: MTH419, MTH431, or permission of instructor
Finite, N-person, non-zero-sum, and infinite games. Two-person games with complete information. Incomplete information; mixed strategies; minimax theorem. LEC

465 Lectures on Geometry (3)
Prerequisites: MTH309, permission of instructor
Provides a broader understanding of differential geometry. It includes a comprehensive introduction to the theory of curves and surfaces in space. Moves toward the goal of viewing surfaces as special concrete examples of differentiable manifolds. This goal is reached by studying surfaces using tools that are basic to the study of manifolds. Topics include curves in three-dimensional space, differential forms, Frenet formulae, patch computations, curvature, isometries, intrinsic geometry of surfaces. Serves as an introduction to more advanced courses involving differentiable manifolds. LEC

461-464 and 466-469 Lecture Courses
Each of these courses is a topics course that treats problems, methods, and recent developments pertaining to a specific area of mathematics indicated below. Each course can be taken more than once for credit.

461 Lectures on Algebra (3)
Prerequisite: MTH420
LEC

462 Lectures on Analysis (3)
Prerequisite: MTH432
LEC

463 Lectures on Applied Mathematics (3)
Prerequisite: permission of instructor
LEC

464 Lectures on Combinatorial Analysis (3)
Prerequisite: permission of instructor
LEC

466 Lectures on Logic and Set Theory (3)
Prerequisite: permission of instructor
LEC

467 Lectures on Number Theory (3)
Prerequisite: permission of instructor
LEC

468 Lectures on Computational Mathematics (3)
Prerequisite: permission of instructor
LEC

469 Lectures on Topology (3)
Prerequisite: permission of instructor
LEC

497 Honors Thesis in Mathematics (4) (F; Sp)
This course is only open to math majors intending to seek an honors degree in mathematics. For information, consult the director of undergraduate studies in the Department of Mathematics. TUT

498 Internship in Mathematics (1-4) (F; Sp)
Open only to accepted majors in mathematics, in the third or fourth year of their programs (with minimum 24 credits in mathematics), and then only after consultation with the director of undergraduate studies. Students get field experience in mathematical employment, in business, industry, or education, working under the joint supervision of an off campus supervisor and a university faculty member, usually the director of undergraduate studies. May be taken once only. TUT

499 Independent Study (1-4) (F; Sp)
Individual study arranged between student and faculty member in an area of mathematics of particular interest to the student. TUT

Checklist for Math Majors

Complete MTH141, 142, 241 with a minimum GPA of 2.5.
    Apply to the department while enrolled in MTH241, or any time after that. Applicants should bring a copy of their current UB DARS report directly to the Department of Mathematics.
    Upon application to the department, you will receive a department decision form informing you of (1) your department status, and (2) your current status pertaining to your intended program (with appropriate advice in some cases).

If you choose to be in Program C – Concentration in Computing and Applied Math, no further approval is necessary.

If you intend to major in mathematics with a concentration in:

• Program TC – Preparation for Teacher Certification in Mathematics - Also apply to Teacher Education Program by the middle of third year
• Program GS – Preparation for Graduate Study in Mathematics
• Program GS/AM – Preparation for Graduate Study in Applied Mathematics
• Program M/E – Joint Major in Mathematics and Economics - Also apply to Economics Department
• Program BSc – (B.S. degree) Preparation for Graduate Study in Mathematics
• Program BSc/AM – (B.S. degree) Preparation for Graduate Study in Applied Mathematics
• Program M/P – Joint Major in Mathematics and Physics - Also apply to Physics Department
• Program Leading to Combined B.A./M.A. Degree

You must obtain approval from the director of undergraduate studies no later than the end of the semester in which you are admitted.

If you do not receive approval and do not meet with the director of undergraduate studies to review your program, you will be placed in Program C and your progress will be monitored according to the requirements for Program C.

Schedule an appointment with the director of undergraduate studies at least once a year to review your progress toward a degree.

If you plan to attend graduate school, you should discuss your plans with the director of undergraduate studies in spring of your junior year, and take the GREs by December of your senior year.

Complete the Natural Sciences requirement: 16 credit hours from natural sciences outside of mathematics to be distributed among two or more departments.

In fall of your final year, talk to Career Development Office regarding job interviews, resumes, etc.

File for degree with the Office of Records and Registration one semester before your intended graduation date (check the deadline dates in the class schedule).

Mathematics — B.S.

Program BSc: Preparation for Graduate Study in Mathematics

Requirements for this program are the same as listed in Program GS for the freshman–junior years. The change in requirements occurs in the senior year regarding mathematics electives (see below). Important: Admission to this program is by departmental approval only. See the director of undergraduate studies regarding approval.

Total Required Credit Hours in Program BSc - 64-68
   (17 courses in math and related areas)

Fourth Year
1. Select three 300-400 level mathematics courses (plus MTH301)
2. Select four 300-400 level courses in mathematics or computer science (not MTH417, 418)
    (Note: all eight electives need not necessarily be taken in the senior year.)

Natural Sciences Requirement - 16 cr
(from natural sciences outside of mathematics to be distributed among two or more departments)

Electives - 13-27 cr

See "Baccalaureate Degree Requirements" for remaining university requirements.

 Mathematics — B.S.

Program BSc/AM: Preparation for Graduate Study in Applied Mathematics

Requirements for this program are the same as listed in Program GS/AM for the freshman—junior years. The change in requirements occurs in the senior year regarding mathematics electives (see below). Important: Admission to this program is by departmental approval only. See the director of undergraduate studies regarding approval.

Total Required Credit Hours in Program BSc/AM - 77-83
   (23 courses in math and related areas)

Fourth Year
1. Select two 300-400 level mathematics courses (plus MTH301)
2. Select four 300-400 level courses in mathematics or computer science (not MTH417, 418)
    (Note: all seven electives need not necessarily be taken in the senior year.)

Natural Sciences Requirement - 16 cr
(from natural sciences outside of mathematics to be distributed among two or more departments)

Electives - 2-14 cr

See "Baccalaureate Degree Requirements" for remaining university requirements.

Mathematics — B.A.

Program C: Concentration in Computing and Applied Mathematics

This program is designed to serve those students interested in careers as applied mathematicians or scientific applications programmers.

Total Required Credit Hours in Program C - 64-68
   (17 courses in math and related areas)

Upon admittance to the department, no further approval is needed to enter Program C. The courses in this program are like the calculus and computing courses, though more difficult. A student who can handle MTH141, 142, 241, and CSE115–116 or MTH145–172 should be able to handle Program C.

First Year

Second Year

Third Year

Fourth Year

*MTH301 is a prerequisite for MTH431. Students intending to go to graduate school in applied mathematics should take MTH301-431-432 instead of MTH417-418. MTH301 would then count as a technical elective.

Approved technical electives recommended:

Any other 300-400-level MTH course would also be acceptable. MTH301 may be taken as an elective; it is the first abstract theoretical course with proofs. Note that MTH431-432 and MTH419-420 are designed for students who want intensive preparation in analysis and modern algebra, and are not good choices to just "fill out your schedule." MTH431-432 may be taken as electives (in addition to MTH417 or MTH418) or as requirements (in place of MTH417 and MTH418).

Approved technical electives outside the mathematics department:

Note: Both CSE480 Computer Graphics and MTH463 Computer Graphics may be taken as technical electives, one in and one outside of the department.

Natural Sciences Requirement - 16 cr
(from natural sciences outside of mathematics to be distributed among two or more departments)

Electives - 12-27 cr

See "Baccalaureate Degree Requirements" for remaining university requirements.

 Mathematics — B.A.

Program GS: Preparation for Graduate Study in Mathematics

This program describes a basic theoretical course of study for mathematics majors intending to pursue graduate study in mathematics.
Important: Admission to this program is by departmental approval only. See the director of undergraduate studies regarding approval.

Total Required Credit Hours in Program GS - 49-52
   (13 courses in math and related areas)

Advisement Note: Program GS has the fewest courses of any of the math programs, but many courses are quite difficult; MTH301, 431-432 and MTH419-420 are abstract theoretical courses emphasizing proofs. A student must do well in MTH301 to enter this program.

Students planning to go to graduate school should meet with the director of undergraduate studies in mathematics in the fall semester of their junior year. They should plan to take the GREs in either spring of the junior year or fall of the senior year. It is strongly advised that students in Program GS take MTH425, MTH427-428, MTH465, and MTH531 as part of this program.

First Year

Second Year

Third Year

Fourth Year

1. Select one 300-400 level mathematics course (plus MTH301)
2. Select two 300-400 level courses in mathematics or computer science (Not MTH417, 418)

Natural Sciences Requirement - 16 cr
(from natural sciences outside of mathematics to be distributed among two or more departments)

Electives - 29-40 cr

See "Baccalaureate Degree Requirements" for remaining university requirements.

 Mathematics — B.A.

Program GS/AM: Preparation for Graduate Study in Applied Mathematics

This program describes a course of study for mathematics majors intending to pursue graduate study in applied mathematics.
Important: Admission to this program is by departmental approval only. See the director of undergraduate studies regarding approval.

Total Required Credit Hours in Program GS/AM - 65-67 cr
   (17 courses in math and related areas)

Advisement Note: MTH301-431-432 and MTH419 are abstract theoretical courses emphasizing proofs. A student must do well in MTH301 to enter this program.

Students planning to go to graduate school should meet with the director of undergraduate studies in mathematics in the fall semester of their junior year. They should plan to take the GREs in either spring of the junior year, or fall of the senior year.

First Year

Second Year

Third Year

Fourth Year
1. Two 300-400 level courses in mathematics or computer science (plus MTH301) (not MTH417, 418)

Natural Sciences Requirement - 16 cr
(from natural sciences outside of mathematics to be distributed among two or more departments)

Electives - 14-26 cr

See "Baccalaureate Degree Requirements" for remaining university requirements.

 Mathematics-Economics — B.A.

Program M/E: Joint Major in Mathematics/Economics

This is a program for students intending graduate study in economics. Acceptance into this program requires at least a 2.5 GPA in both the mathematics and economics courses during the first year. Separate applications must be submitted to both the Departments of Mathematics and Economics.
Important: Admission to this program is by departmental approval only. See the director of undergraduate studies regarding approval.

Total Required Credit Hours in Program M/E - 58-61
   (15 courses in math and economics)

Advisement Note: A student must be able to handle MTH301 to enter this program. Students should consult the advisor in each department concerning senior-level courses. Exceptions in the curriculum may be allowed through consultation with the appropriate department. Interested students should contact the departments upon application for up-to-date information on possible changes.

First Year

Second Year

Third Year

Fourth Year
1. Select one 300-400 level mathematics course (plus MTH301) (not MTH417, 418)
2. Select two 300-400 level courses in economics (not ECO480 or 481)

Natural Sciences Requirement - 16 cr
(from natural sciences outside of mathematics to be distributed among two or more departments)

Electives - 23-26 cr

See "Baccalaureate Degree Requirements" for remaining university requirements.

 Mathematical Physics — B.S.

Program M/P: Joint Major in Mathematics/Physics

Acceptance into this program requires at least a 2.5 GPA in MTH141, 142, 241 and PHY107, 108, 158. Separate applications must be submitted to both the Departments of Mathematics and Physics.
Important: Admission to this program is by departmental approval only. See the director of undergraduate studies regarding approval.

Total Required Credit Hours in Program M/P - 77-78
   (21 courses in math and physics)

Advisement Note: Students should consult with the advisor in each department regarding approved electives. Caution: This is a joint program. A student who follows this program but does not complete it, will have difficulty completing either a math major or a physics major without substantial additional coursework.

First Year

Second Year

Third Year

Fourth Year

1. Mathematics Elective: One 300-400 level mathematics course (recommended: MTH301)
2. Physics Elective: One course from PHY302, 402, 404, 406
3. Physics lab: PHY407 or 408

Natural Sciences Requirement: no additional coursework required

Electives - 3-14 cr

See "Baccalaureate Degree Requirements" for remaining university requirements.

Mathematics — B.A.

Program TC: Preparation for Teacher Certification in Mathematics

This program describes a basic course of study for students seeking accreditation to teach in secondary schools. Students in this program should file an application for a minor in teacher education with their academic advisor (this should include transcripts from all universities and colleges attended, including UB, through the most recent fall semester). In addition, an application for admission to the teacher education program must also be filed with TEI* (379 Baldy Hall) by February 1.

Important: Admission to this program is by departmental approval only. See the director of undergraduate studies regarding approval.

Total Required Credit Hours in Program TC - 55-56
   (14 courses in math and related areas)

Teacher Education Minor - 24 cr

Advisement Note: The courses in this program are more theoretical than those in Program C; MTH301, MTH419, and MTH431 are difficult. A student must be able to handle MTH301 in order to enter this program. An overall GPA of 2.7 is required for admittance to the minor in teacher education. WARNING: In order to graduate with a degree in mathematics in Program TC, a student must be accepted into and complete the teacher education program through TEI prior to graduation. The teacher education program has a limited number of openings; admission is based in part on interviews and can be very competitive in some years. Students who complete Program TC math requirements but do not gain admission to the teacher education program will have to change programs, usually to Program C or Program GS, in order to graduate. It is recommended that students interested in this program contact the department upon application for up-to-date information on possible changes in requirements.

First Year

Second Year

Third Year

Fourth Year

Electives - 2-12 cr

*Teacher Education Institute (TEI) is located in the Graduate School of Education and administers and coordinates the NYS Education Department-approved teacher education programs.

See "Baccalaureate Degree Requirements" for remaining university requirements.

Mathematics — Minor

Acceptance Criteria
GPA of 2.5 in MTH141, 142, and 241 (or approved transfer equivalent)

Total Required Credit Hours - 26-28
   (7 courses from within the mathematics department)

Application Procedure
Students must complete an application for a minor (available in the Math Undergraduate Office) and submit it together with a copy of the most recent UB DARS report. Applicants will be contacted by the Department of Mathematics regarding acceptance.

Required Courses
MTH141 College Calculus I
MTH142 College Calculus II
MTH241 College Calculus III
MTH309 Introductory Linear Algebra
One of the following:
    MTH242 Survey of Differential Equations
    MTH306 Introduction to Differential Equations

Mathematics Electives:
Two additional 300- or 400-level courses from within the Department of Mathematics
(Note: MTH306 is not an approved elective.)

Note: A GPA of 2.0 is required in these courses for departmental recommendation for a minor in mathematics.

© 2000 University at Buffalo Undergraduate Catalog 2000-2001