Mathematics

This course is intended for students who do not plan to major in programs that require calculus courses. The aim of this course is to expose students to the utility and beauty of mathematics, and strengthen their quantitative and analytical skills. The material is organized as a series of independent modules exploring various topics in modern mathematics, its real-world applications, and directions of current research. Topics of the modules are selected at the discretion of the course instructor. This course fulfills the Math and Quantitative Reasoning requirement of the UB Curriculum.

A pre-calculus 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. Emphasizes applications to problems. This fast-paced course reviews Course III and prepares students for further courses in mathematics. Students with three years of high school math but with weak algebra skills should take ULC 147 before MTH 115. Students who have had only two years of high school mathematics may take MTH 115, or may prefer to take a two-semester sequence covering this material more thoroughly and at a more moderate pace: ULC 147 and ULC 148.

Allows transfer students to efficiently learn specific topics from UB calculus courses that were not covered in calculus courses they took at other institutions.

For students in social, biological, and management sciences. Limits, continuity, differentiation of algebraic and exponential functions; applications; introduces integration. This course is a controlled enrollment (impacted) course. Students who have previously attempted the course and received a grade other than W may repeat the course in the summer or winter; or only in the fall or spring semester with a petition to the College of Arts and Sciences Deans' Office. To help students achieve greater success, all new UB students are required to complete the ALEKS math assessment tool. New and continuing UB students seeking to enroll in MTH 121 must complete the ALEKS assessment tool within the last 12 months and achieve a required score of 61 as a prerequisite. (ALEKS is not required in Summer/Winter).

Continuation of MTH 121. Maximization of functions of several variables using both calculus and elementary linear programming techniques. Elementary integration, simple differential equations, matrix algebra.

For students in Management. Limits, continuity, differentiation of algebraic and exponential functions. Applications, partial derivatives and applications. Introduces integration. This course is a controlled enrollment (impacted) course. Students who have previously attempted the course and received a grade other than W may repeat the course in the summer or winter; or only in the fall or spring semester with a petition to the College of Arts and Sciences Deans' Office. To help students achieve greater success, all new UB students are required to complete the ALEKS math assessment tool. New and continuing UB students seeking to enroll in MTH 131 must complete the ALEKS assessment tool within the last 12 months and achieve a required score of 61 as a prerequisite.(ALEKS is not required in Summer/Winter).

Beginning of a three-semester sequence in calculus for students of mathematics, natural sciences, and engineering. Covers differentiation and integration with applications. This course is a controlled enrollment (impacted) course. Students who have previously attempted the course and received a grade other than W may repeat the course in the summer or winter; or only in the fall or spring semester with a petition to the College of Arts and Sciences Deans' Office. To help students achieve greater success, all new UB students are required to complete the ALEKS math assessment tool. New and continuing UB students seeking to enroll in MTH 141 must complete the ALEKS assessment tool within the last 12 months and achieve a required score of 76 as a prerequisite. (ALEKS is not required in Summer/Winter).

Differentiation and integration of transcendental functions; infinite sequences; series and power series; integration methods; additional topics in analytic geometry. This course is a controlled enrollment (impacted) course. Students who have previously attempted the course and received a grade other than W may repeat the course in the summer or winter; or only in the fall or spring semester with a petition to the College of Arts and Sciences Deans' Office.

First course in the honors sequence for intended math majors or for others with suitable preparation. Emphasizes proofs and concepts of calculus. Note: Credit will not be given for both MTH 153 and MTH 121/131/141. Credit for MTH 153 may be given in addition to AP Calculus credit for MTH 141.

A more theoretically oriented version of College Calculus II. Differentiation and integration of transcendental functions; infinite sequences; series and power series; integration methods. Topics enhance those of MTH 142 and concepts are often developed with proofs. This is designed to be a challenging course for bright students, including students who might be interested in graduate work in the sciences, engineering, and mathematics. Note: Credit for MTH 154 may be given in addition to AP Calculus credit for MTH 142. Credit will not be given for both MTH 154 and MTH 122/142. Pre-Requisite: A 4 or 5 on an AP Calculus Exam, or MTH 153, or an outstanding performance in MTH 141 College Calculus I. Contact the Math Undergraduate Office or the Honors College for registration in this course.

First part of a two-semester sequence. Provides the mathematical foundations for the study of computer science. Also approved for mathematics majors in Concentration GS/ED. Topics include sets, relations, functions, mathematical induction, fundamental counting methods, difference equations, and sequences and series.

Second part of a two-semester sequence. Provides the mathematical foundations for the study of computer science. Topics include discrete probability, mathematical logic, linear algebra, and graph theory. Same as CSE 192.

The one credit UB Seminar is focused on a big idea or challenging issue to engage students with questions of significance in a field of study and, ultimately, to connect their studies with issues of consequence in the wider world. Essential to the UB Curriculum, the Seminar helps transition to UB through an early connection to UB faculty and the undergraduate experience at a comprehensive, research university. This course is equivalent to any 198 offered in any subject. This course is a controlled enrollment (impacted) course. Students who have previously attempted the course and received a grade of F or R may not be able to repeat the course during the fall or spring semester.

The three credit UB Seminar is focused on a big idea or challenging issue to engage students with questions of significance in a field of study and, ultimately, to connect their studies with issues of consequence in the wider world. Essential to the UB Curriculum, the Seminar helps students with common learning outcomes focused on fundamental expectations for critical thinking, ethical reasoning, and oral communication, and learning at a university, all within topic focused subject matter. The Seminars provide students with an early connection to UB faculty and the undergraduate experience at a comprehensive, research university. This course is equivalent to any 199 offered in any subject. This course is a controlled enrollment (impacted) course. Students who have previously attempted the course and received a grade of F or R may not be able to repeat the course during the fall or spring semester.

Geometry and vectors of n-dimensional space; Green's theorem, Gauss theorem, Stokes theorem; multidimensional differentiation and integration; application to 2- and 3-D space. This course is a controlled enrollment (impacted) course. Students who have previously attempted the course and received a grade other than W may repeat the course in the summer or winter; or only in the fall or spring semester with a petition to the College of Arts and Sciences Deans' Office.

Third-semester calculus course for honors students and students with an excellent record in previous calculus courses. Emphasizes proofs and concepts of calculus.

Analytic solutions, qualitative behavior of solutions to differential equations. First-order and higher-order ordinary differential equations, including nonlinear equations. Covers analytic, geometric, and numerical perspectives as well as an interplay between methods and model problems. Discusses necessary matrix theory and explores differential equation models of phenomena from various disciplines. Uses a mathematical software system designed to aid in the numerical and qualitative study of solutions, and in the geometric interpretation of solutions. This course is a controlled enrollment (impacted) course. Students who have previously attempted the course and received a grade other than W may repeat the course in the summer or winter; or only in the fall or spring semester with a petition to the College of Arts and Sciences Deans' Office.

Linear equations, matrices, determinants, vector spaces, linear mappings, inner products, eigenvalues, eigenvectors.

Develops the student's ability to read, comprehend and construct rigorous proofs. Topics may include the following: the number systems N, Z, Q, R and the existence of irrational numbers; sets and functions; size of sets(finite/infinite, countable/uncountable); the countability of the rationals and the uncountability of the real numbers; boundedness; upper and lower bounds; lub's and glb's; lub and glb property; density of the rationals in the reals; Archimedean property of the reals; mathematical induction, including strong induction and the well-ordering of the natural numbers; sequences of real numbers, including the Monotone Convergence Theorem, Cauchy sequences, and the Bolzano-Weierstrass Theorem.

Cardinals, ordinals, order-types, and operations on them. Axiom of choice. Sets.

Euclidean and non-Euclidean geometries. Studies the Hilbert postulates and various models, emphasizing Euclidean and Lobachevskian geometries.

Computing now plays an essential and ever-expanding role in science and mathematics. This course provides a broad introduction to computing in the sciences and in both abstract and applied mathematics. It is accessible to students early in their undergraduate program, thereby opening the door to the profitable use of computation throughout the junior and senior years.

Permutations, combinations, and other problems of selecting and arranging objects subject to various restrictions; generating functions; recurrence relations; inclusion-exclusion theorem.

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.

Seminar based around a specific topic or area of mathematics appropriate to juniors in mathematics and the mathematical sciences. the format is determined by the instructor or team of instructors. Sessions include lectures by UB faculty in Mathematics and other departments around the university, talks by outside experts presentations by the students registered in the seminar on readings and/or research work they have done in relation to the subject matter of the seminar, and occasional field trips. Open discussion during the sessions is a key feature.

A first course in probability. 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. This course is dual-listed with MTH 511.

Rigorous derivation of statistical results, clarification of limitations of statistical analysis, extensive use of computational software, application of statistical methods to case studies. Topics include: Graphical and numerical techniques for exploring data. Use and accuracy of population samples using parametric and nonparametric methods. Determination of probability distributions from statistical data. Use of computational methods based on resampling of data to determine reliability of statistical information. Classical statistical inference methods: probability distribution estimation, confidence intervals for statistical results, hypothesis testing for statistical significance. Fitting of data using linear regression and determining the accuracy of fit. Bayesian methods for estimating probability distributions using prior information. Advanced topics such as importance sampling for understanding the probability of rare events.

Informal and formal development of propositional calculus; predicate calculus and predicate calculus with equality; completeness theorem and some consequences.

For math majors in Concentration C, and majors of science and engineering. Surveys 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. Introduces functions of a complex variable; curves and regions in the complex plane; analytic functions, Cauchy-Riemann equations, Cauchy integral formula. Applications.

Surveys elementary differential equations of physics; separation of variables and superposition of solutions; orthogonal functions and Fourier series. Introduces boundary value problems, Fourier and Laplace transforms.

A theoretical introduction to the basic ideas of modern abstract algebra. Topics include groups, rings, fields, quotient groups and rings, and the fundamental homomorphism theorems. Also may include applications to number theory.

Topics in advanced linear algebra.

For mathematics, science, and engineering majors with strong mathematics backgrounds. Theory of Fourier series and transforms, orthogonal sets, special functions, applications.

For students of physics, electrical and other areas of engineering, and mathematics. Analyticity; calculus over the complex numbers. Cauchy theorems, residues, singularities, conformal mapping. Weierstrass convergence theorem; analytic continuation.

Continuation of MTH 425. 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.

Metric spaces and abstract topological spaces. Continuous functions and homeomorphisms. Subspace, product, and quotient topologies. Separations axioms. Connectedness and path connectedness. Compactness.

Homotopy and homotopy equivalence. Fundamental group. Covering spaces and their classification. Seifert-van Kampen theorem. CW complexes and their fundamental groups. Topological manifolds.

The Euclidean algorithm and unique factorization; arithmetical functions; congruences, reduced residue systems; primitive roots; certain diophantine equations.

Continuation of MTH 429. 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.

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. A prerequisite for most advanced courses in analysis.

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. A prerequisite for most advanced courses in analysis.

Rigorous course in analyzing dimensions greater than one. Includes details of three basic theorems: the inverse function theorem, the implicit function theorem, and the change of variables theorem in multiple integrals. Topics include continuously differentiable functions, the chain rule, inverse and implicit function theorems, Riemann integration, partitions of unity, change of variables theorem.

Explains the basics of cryptography, which is the systematic study of methods of concealing messages from people who are not authorized to read them. Topics include the following: cryptosystem definitions and basic types of attack; substitution ciphers. Hill ciphers; congruences and modular exponentiation; digital encryption standard; public key and RSA cryptosystems; pseudoprimes and primality testing; Pollard rho method; basic finite field theory; discrete log; and digital signatures.

A first course on the design and implementation of numerical methods to solve the most common types of problem arising in science and engineering. Most such problems cannot be solved in terms of a closed analytical formula, but many can be handled with numerical methods learned in this course. Topics for the two semesters include: how a computer does arithmetic, solving systems of simultaneous linear or nonlinear equations, finding eigenvalues and eigenvectors of (large) matrices, minimizing a function of many variables, fitting smooth functions to data points (interpolation and regression), computing integrals, solving ordinary differential equations (initial and boundary value problems), and solving partial differential equations of elliptic, parabolic, and hyperbolic types. We study how and why numerical methods work, and also their errors and limitations. Students gain practical experience through course projects that entail writing computer programs. This course is the same as CSE 437.

Second part of the 2-semester sequence described under MTH 437. This course is the same as CSE 438.

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.

Explores other topics described in MTH 443.

A practical hands-on introduction for mathematics majors to tools and methods for acquiring, storing, manipulating, and exploring data - both big and small. It also introduces students to the use of data in a variety of fields where mathematical talent can be productively applied: examples are bioinformatics (e.g. genomics), health care informatics, urban and regional planning, and data journalism. Extensive writing of reports is a central feature of the course.

Introduces the use of mathematical modeling in applied mathematics using a case study approach. Population ecology; chemical kinetics; traffic dynamics.

Introduces the mathematical theory of voting - the systemic analysis of the ways to determine a choice of a group from the choices of individuals within a group - with applications to economics and politics. Examines voting procedures including the standard plurality vote, the antiplurality vote, the Borda count, Condorcets's method, and run-off elections. Provides an understanding of how different procedures effect group decisions. Uses convex geometry in the plane and in three spaces. Also discusses political power.

Introduces the mathematical theory and computation of modern financial products used in the banking and corporate world. Derives and analyzes mathematical models for the valuation of derivative products.

Describes the mathematical development of both the theoretical and the computational techniques used to analyze financial instruments. Specific topics include utility functions; forwards, futures, and swaps; and modeling of derivatives and rigorous mathematical analysis of the models, both theoretically and computationally. Develops, as needed, the required ideas from partial differential equations and numerical analysis.

Introduces the mathematical theory of games--a systematic approach to modeling conflict, competition, cooperation, and negotiation--with applications to mathematics, economics, politics and evolutionary biology. A game, in mathematical terms, consists of a starting point and various choices made by 'players.' Each choice might lead to new choices or to an outcome that ends the game. Some choices might be random; some might be made without full information about what has transpired. The players are each trying to maximize their own payoff, but the play of each might influence the results of the others. The approaches Game Theory uses to analyze conflict between two or more people lead to results that can seem paradoxical as well as illuminating. The most important thing a student can take from this course is a useful way of approaching decisions, from the trivial-- how does a couple decide which movie to see--to the critical--how should countries pursue their goals in cooperation or conflict with their allies and enemies. Partial list of topics: Prisoner's Dilemma, game trees, pure and mixed strategies, backward induction, normal form, Nash equilibrium, chance moves, utility functions, domination, convexity, payoff regions, strictly competitive games, separating hyperplanes, repeating games, and cooperative bargaining theory.

Treats problems, methods, and recent developments pertaining to a specific area of algebra. Topics courses can be taken more than once for credit.

Treats problems, methods, and recent developments pertaining to analysis. Topics courses can be taken more than once for credit.

Treats problems, methods, and recent developments pertaining combinatorial analysis. Topics courses can be taken more than once for credit.

Provides a broader understanding of differential geometry. Comprehensively introduces the theory of curves and surfaces in space. Moves toward the goal of viewing surfaces as special concrete examples of differentiable manifolds, reached by studying surfaces using tools that are basic to studying manifolds. Topics include curves in 3-D space, differential forms, Frenet formulae, patch computations, curvature, isometries, intrinsic geometry of surfaces. Serves as an introduction to more advanced courses involving differentiable manifolds.

Treats problems, methods, and recent developments pertaining logic and set theory. Topics courses can be taken more than once for credit.

Treats problems, methods, and recent developments pertaining number theory. Topics courses can be taken more than once for credit.

Treats problems, methods, and recent developments pertaining numerical analysis. Topics courses can be taken more than once for credit.

Treats problems, methods, and recent developments pertaining topology. Topics courses can be taken more than once for credit.

Treats problems, methods, and recent developments in any area of mathematics that does not fit nearly or fully under the title of any other "Topics in..." course.

Students who have at least junior status and satisfy the department's pre-requisites may apply to serve as undergraduate teaching assistants in one of the calculus courses (MTH 121/MTH 122, MTH 131, MTH 141/MTH 142, MTH 241). Under the supervision of the professor, undergraduate teaching assistants will lead two recitation sections each week of approximately 30 students each. Some grading of homework will be expected.

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.

Open only to math majors intending to seek an honors degree in mathematics. For information, consult the director of undergraduate studies in the Department of Mathematics.

Individual study arranged between student and faculty member in an area of mathematics of particular interest to the student.