- The Learning Environment
## The Learning Environment

Mathematics courses are usually delivered in a lecture format. Many courses include recitations that allow students to seek additional explanation of course material and practice their problem solving skills. Enrollment in lower level mathematics courses is limited to 90 students. Most advanced courses have no more than 30-40 students. The Mathematics Department also offers individual study courses that let students research specialized topics under guidance of a faculty member. For the last few years the department has been increasing the number of courses that are offered remotely.

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, 131, or 141 must complete the ALEKS assessment tool and achieve a required score of 76 as a prerequisite.

### About Our Facilities

All faculty and administrative offices of the Math Department are located in the Mathematics Building. The Mathematics Building also has a few classrooms that are used for seminars and more advanced courses. Large mathematics courses meet in other buildings on UB North Campus. Rooms 107 and 110 in the Mathematics Building house the Math Help Center that offers free, walk-in help on lower level mathematics courses.

### About Our Faculty

The Department of Mathematics is proud of its excellent teaching and active research programs. Several of our faculty members have received awards for excellence in teaching. Faculty members are currently involved in research in areas such as mathematical analysis, abstract algebra, number theory, geometry, topology, and applied mathematics. Each year approximately one third of the faculty hold external research grants, from agencies such as the National Science Foundation, National Security Agency and the Department of Defense. The Department offers research opportunities for undergraduates including writing honors theses. The undergraduate Teaching Assistants program allows undergraduate students to gain college level teaching experience.

The Director of Undergraduate Studies, Professor Joseph Hundley, is available to talk one-on-one with any student interested in finding out about our mathematics program. For appointments call 716-645-8785.

#### Faculty List Directory

Please visit the Mathematics department website for additional information about our faculty.

## Department of Mathematics

244 Mathematics BuildingNorth Campus

Buffalo, NY 14260

**Gino Biondini**

Chair

**Joseph Hundley**

Director of Undergraduate Studies

math-undergrad@buffalo.edu

### Associated Programs

- Statistics BA
- Physics BS
- Mathematics BA - Computing and Applied Mathematics Concentration
- Mathematics BA - General Study in Mathematics Concentration
- Statistics Minor
- Mathematics BS - General Study in Applied Mathematics Concentration
- Mathematics BS - Actuarial Science Concentration
- Mathematics BA/Mathematics Adolescence Education EdM
- Mathematics BA/MA
- Physics Minor
- Mathematics BS - General Study in Mathematics Concentration
- Mathematics BA - General Study in Mathematics and Education Concentration
- Mathematics BA - General Study in Applied Mathematics Concentration
- Education Minor
- Mathematics BA - General Curriculum
- Economics Minor
- Economics BA
- Physics BA
- Mechanical Engineering BS
- Economics BA/Econometrics and Quantitative Economics (STEM) MS
- Mathematics Minor
- Economics BA/MA

### Associated Subjects

## MTH Courses

- MTH 101LR Introduction to Contemporary Mathematics
**Lecture**

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.

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Spring

- MTH 101LR Introduction to Contemporary Mathematics
**Lecture**

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.

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Spring

- MTH 115LR Survey of Algebra and Trigonometry
**Lecture**

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.

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Spring

- MTH 115LR Survey of Algebra and Trigonometry
**Lecture**

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.

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Spring

- MTH 120LEC Selected Topics in Calculus
**Lecture**

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

**Credits:**1 - 3

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Winter, Spring, Summer

- MTH 120LEC Selected Topics in Calculus
**Lecture**

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

**Credits:**1 - 3

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Winter, Spring, Summer

- MTH 121LR Survey of Calculus and Its Applications I
**Lecture**

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.

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Winter, Spring, Summer

**Requisites:**Pre-Requisite: Score of 70/100 or better on both the Fundamentals and the Algebra portions of the Math Readiness Assessment, or a C or better in ULC 148, MTH 115, MTH 131, or MTH 141 or a score of 4 on AP Calculus. - MTH 121LR Survey of Calculus and Its Applications I
**Lecture**

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.

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Winter, Spring, Summer

**Requisites:**Pre-Requisite: Score of 70/100 or better on both the Fundamentals and the Algebra portions of the Math Readiness Assessment, or a C or better in ULC 148, MTH 115, MTH 131, or MTH 141 or a score of 4 on AP Calculus. - MTH 122LR Survey of Calculus and Its Applications II
**Lecture**

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.

- MTH 122LR Survey of Calculus and Its Applications II
**Lecture**

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.

- MTH 131LR Mathematical Analysis for Management
**Lecture**

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.

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Winter, Spring, Summer

**Requisites:**Pre-Requisite: Score of 70/100 or better on both the Fundamentals and the Algebra portions of the Math Readiness Assessment, or a C or better in ULC 148, MTH 115, MTH 131, or MTH 141 or a score of 4 on AP Calculus. - MTH 131LR Mathematical Analysis for Management
**Lecture**

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.

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Winter, Spring, Summer

**Requisites:**Pre-Requisite: Score of 70/100 or better on both the Fundamentals and the Algebra portions of the Math Readiness Assessment, or a C or better in ULC 148, MTH 115, MTH 131, or MTH 141 or a score of 4 on AP Calculus. - MTH 141LR College Calculus I
**Lecture**

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.

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Winter, Spring, Summer

**Requisites:**Pre-Requisite: Score of 70/100 or better on all three components of the Math Readiness Assessment (exam results are valid for one year), or a C or better in ULC 148, MTH 115, MTH 121, MTH 131, or MTH 141, or a score of 4 on AP Calculus. - MTH 141LR College Calculus I
**Lecture**

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.

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Winter, Spring, Summer

**Requisites:**Pre-Requisite: Score of 70/100 or better on all three components of the Math Readiness Assessment (exam results are valid for one year), or a C or better in ULC 148, MTH 115, MTH 121, MTH 131, or MTH 141, or a score of 4 on AP Calculus. - MTH 142LR College Calculus 2
**Lecture**

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.

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Winter, Spring, Summer

**Requisites:**Pre-Requisite: MTH 141. - MTH 142LR College Calculus 2
**Lecture**

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.

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Winter, Spring, Summer

**Requisites:**Pre-Requisite: MTH 141. - MTH 153LR Honors Calculus I
**Lecture**

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.

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Fall

- MTH 153LR Honors Calculus I
**Lecture**

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.

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Fall

- MTH 154LR Honors Calculus 2
**Lecture**

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.

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Spring

- MTH 154LR Honors Calculus 2
**Lecture**

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.

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Spring

- MTH 191LR Introduction to Discrete Structures
**Lecture**

Foundational material for further studies in computer science. Topics include logic, proofs, sets, functions, relations, recursion, recurrence relations, mathematical induction, graphs, trees, basic counting theory, regular languages, and context free grammars. This course is the same as CSE 191 and course repeat rules will apply. Students should consult with their major department regarding any restrictions on their degree requirements.

- MTH 191LR Introduction to Discrete Structures
**Lecture**

Foundational material for further studies in computer science. Topics include logic, proofs, sets, functions, relations, recursion, recurrence relations, mathematical induction, graphs, trees, basic counting theory, regular languages, and context free grammars. This course is the same as CSE 191 and course repeat rules will apply. Students should consult with their major department regarding any restrictions on their degree requirements.

- MTH 198SEM UB Seminar
**Seminar**

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.

**Credits:**1

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Spring

**Requisites:**Students who have already successfully completed the UB seminar course may not repeat this course. If you have any questions regarding enrollment for this course, please contact your academic advisor. - MTH 198SEM UB Seminar
**Seminar**

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.

**Credits:**1

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Spring

**Requisites:**Students who have already successfully completed the UB seminar course may not repeat this course. If you have any questions regarding enrollment for this course, please contact your academic advisor. - MTH 199SEM UB Seminar
**Seminar**

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.

**Credits:**3

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Spring

**Requisites:**Students who have already successfully completed the first year seminar course may not repeat this course. If you have any questions regarding enrollment for this course, please contact your academic advisor. - MTH 199SEM UB Seminar
**Seminar**

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.

**Credits:**3

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Spring

**Requisites:**Students who have already successfully completed the first year seminar course may not repeat this course. If you have any questions regarding enrollment for this course, please contact your academic advisor. - MTH 241LR College Calculus 3
**Lecture**

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.

- MTH 241LR College Calculus 3
**Lecture**

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.

- MTH 251LR Honors Calculus 3
**Lecture**

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

- MTH 251LR Honors Calculus 3
**Lecture**

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

- MTH 306LR Introduction to Differential Equations
**Lecture**

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.

- MTH 306LR Introduction to Differential Equations
**Lecture**

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.

- MTH 309LR Introductory Linear Algebra
- MTH 309LR Introductory Linear Algebra
- MTH 311LR Introduction to Higher Mathematics
**Lecture**

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.

- MTH 311LR Introduction to Higher Mathematics
**Lecture**

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.

- MTH 323LR Honors Linear Algebra
- MTH 323LR Honors Linear Algebra
- MTH 326LR Honors Diff Equations
- MTH 326LR Honors Diff Equations
- MTH 335LR Elements of Geometry
**Lecture**

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

- MTH 335LR Elements of Geometry
**Lecture**

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

- MTH 337LEC Introduction to Scientific and Mathematical Computing
**Lecture**

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.

- MTH 337LEC Introduction to Scientific and Mathematical Computing
**Lecture**

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.

- MTH 353LR Introduction to Combinatorics I
**Lecture**

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

- MTH 353LR Introduction to Combinatorics I
**Lecture**

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

- MTH 399SEM Junior Seminar
**Seminar**

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.

**Credits:**3

**Grading:**Graded (GRD)

**Typically Offered:**Varies

**Requisites:**Pre-Requisite: MTH 241 and Permission of Instructor Required for Registration. - MTH 399SEM Junior Seminar
**Seminar**

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.

**Credits:**3

**Grading:**Graded (GRD)

**Typically Offered:**Varies

**Requisites:**Pre-Requisite: MTH 241 and Permission of Instructor Required for Registration. - MTH 411LR Probability Theory
**Lecture**

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.

- MTH 411LR Probability Theory
**Lecture**

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.

- MTH 412LR Introduction to Statistical Inference
**Lecture**

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.

- MTH 412LR Introduction to Statistical Inference
**Lecture**

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.

- MTH 417LR Survey of Multivariable Calculus
**Lecture**

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.

- MTH 417LR Survey of Multivariable Calculus
**Lecture**

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.

- MTH 418LR Survey of Partial Differential Equations
**Lecture**

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.

- MTH 418LR Survey of Partial Differential Equations
**Lecture**

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.

- MTH 419LR Introduction to Abstract Algebra
**Lecture**

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.

- MTH 419LR Introduction to Abstract Algebra
**Lecture**

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.

- MTH 420LR Abstract Linear Algebra
- MTH 420LR Abstract Linear Algebra
- MTH 424LEC Survey of Fourier Series Methods
- MTH 424LEC Survey of Fourier Series Methods
- MTH 425LR Introduction to Complex Variables I
**Lecture**

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.

- MTH 425LR Introduction to Complex Variables I
**Lecture**

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.

- MTH 427LR Introduction to Topology I
**Lecture**

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

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Fall

**Requisites:**Pre-Requisite: Permission of the instructor - MTH 427LR Introduction to Topology I
**Lecture**

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

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Fall

**Requisites:**Pre-Requisite: Permission of the instructor - MTH 428LEC Introduction to Topology II
**Lecture**

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

- MTH 428LEC Introduction to Topology II
**Lecture**

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

- MTH 429LR Introduction to the Theory of Numbers I
**Recitation**

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

**Credits:**3

**Grading:**Graded (GRD)

**Typically Offered:**Fall

**Requisites:**Pre-Requisite: MTH 311 Recommended - MTH 429LR Introduction to the Theory of Numbers I
**Recitation**

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

**Credits:**3

**Grading:**Graded (GRD)

**Typically Offered:**Fall

**Requisites:**Pre-Requisite: MTH 311 Recommended - MTH 431LR Introduction to Real Variables I
**Lecture**

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.

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Spring

**Requisites:**Pre-Requisite: MTH 311. - MTH 431LR Introduction to Real Variables I
**Lecture**

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.

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Spring

**Requisites:**Pre-Requisite: MTH 311. - MTH 432LR Introduction to Real Variables II
**Lecture**

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.

- MTH 432LR Introduction to Real Variables II
**Lecture**

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.

- MTH 435LEC Introduction to Cryptography
**Lecture**

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.

- MTH 435LEC Introduction to Cryptography
**Lecture**

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.

- MTH 437LR Introduction to Numerical Analysis I
**Lecture**

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.

- MTH 437LR Introduction to Numerical Analysis I
**Lecture**

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.

- MTH 438LR Introduction to Numerical Analysis II
- MTH 438LR Introduction to Numerical Analysis II
- MTH 443LEC Fundamentals of Applied Mathematics I
**Lecture**

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.

- MTH 443LEC Fundamentals of Applied Mathematics I
**Lecture**

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.

- MTH 444LEC Fundamentals of Applied Mathematics II
- MTH 444LEC Fundamentals of Applied Mathematics II
- MTH 448LEC Data-Oriented Computing for Mathematics
**Lecture**

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.

- MTH 448LEC Data-Oriented Computing for Mathematics
**Lecture**

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.

- MTH 450LEC Network Theory
**Lecture**

Network Theory - the science of mapping physical systems to mathematical graphs - provides an attractive methodology to describe and quantify real-world systems. In this course, we will explore the mathematical foundations of network theory and the network statistics used to quantify network structure. The course will especially emphasize connections to real-world data and the importance of interpreting network statistics in the context of the system being studied. Students will identify a physical system of interest and ultimately prepare a research paper that uses tools from network theory to quantify the structure of their system and provide a meaningful interpretation of their findings. Dual listed with MTH 550.

- MTH 450LEC Network Theory
**Lecture**

Network Theory - the science of mapping physical systems to mathematical graphs - provides an attractive methodology to describe and quantify real-world systems. In this course, we will explore the mathematical foundations of network theory and the network statistics used to quantify network structure. The course will especially emphasize connections to real-world data and the importance of interpreting network statistics in the context of the system being studied. Students will identify a physical system of interest and ultimately prepare a research paper that uses tools from network theory to quantify the structure of their system and provide a meaningful interpretation of their findings. Dual listed with MTH 550.

- MTH 456LEC The Mathematics of Voting
**Lecture**

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.

**Credits:**3

**Grading:**Graded (GRD)

**Typically Offered:**Spring

**Requisites:**Junior or Senior Standing Required - MTH 456LEC The Mathematics of Voting
**Lecture**

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.

**Credits:**3

**Grading:**Graded (GRD)

**Typically Offered:**Spring

**Requisites:**Junior or Senior Standing Required - MTH 458LEC Mathematical Finance
**Lecture**

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.

- MTH 458LEC Mathematical Finance
**Lecture**

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.

- MTH 459LEC Mathematical Finance 2
**Lecture**

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.

- MTH 459LEC Mathematical Finance 2
**Lecture**

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.

- MTH 460LR Theory of Games
**Lecture**

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.

- MTH 460LR Theory of Games
**Lecture**

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.

- MTH 461LEC Topics in Algebra
- MTH 461LEC Topics in Algebra
- MTH 463LEC Topics in Applied Mathematics
**Lecture**

**Credits:**3

**Grading:**Graded (GRD)

**Typically Offered:**Varies

- MTH 463LEC Topics in Applied Mathematics
**Lecture**

**Credits:**3

**Grading:**Graded (GRD)

**Typically Offered:**Varies

- MTH 464LEC Topics in Combinatorial Analysis
- MTH 464LEC Topics in Combinatorial Analysis
- MTH 465LEC Lectures in Geometry
**Lecture**

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.

**Credits:**3

**Grading:**Graded (GRD)

**Typically Offered:**Varies

**Requisites:**Permission of Instructor Required to Register - MTH 465LEC Lectures in Geometry
**Lecture**

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.

**Credits:**3

**Grading:**Graded (GRD)

**Typically Offered:**Varies

**Requisites:**Permission of Instructor Required to Register - MTH 469LEC Topics in Topology
**Lecture**

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

**Credits:**3

**Grading:**Graded (GRD)

**Typically Offered:**Varies

**Requisites:**Permission of Instructor Required to Register - MTH 469LEC Topics in Topology
**Lecture**

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

**Credits:**3

**Grading:**Graded (GRD)

**Typically Offered:**Varies

**Requisites:**Permission of Instructor Required to Register - MTH 495TUT Undergraduate Supervised Teaching
**Tutorial**

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.

**Credits:**3

**Grading:**Pass/Not Pass (PNP)

**Typically Offered:**Fall, Spring

**Requisites:**Permission of Department - MTH 495TUT Undergraduate Supervised Teaching
**Tutorial**

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.

**Credits:**3

**Grading:**Pass/Not Pass (PNP)

**Typically Offered:**Fall, Spring

**Requisites:**Permission of Department - MTH 496TUT Internship in Mathematics
**Tutorial**

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.

**Credits:**1 - 4

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Spring

- MTH 496TUT Internship in Mathematics
**Tutorial**

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.

**Credits:**1 - 4

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Spring

- MTH 497TUT Honors Thesis in Mathematics
**Tutorial**

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.

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Spring

- MTH 497TUT Honors Thesis in Mathematics
**Tutorial**

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.

**Credits:**4

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Spring

- MTH 499TUT Independent Study
**Tutorial**

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

**Credits:**1 - 4

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Spring

- MTH 499TUT Independent Study
**Tutorial**

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

**Credits:**1 - 4

**Grading:**Graded (GRD)

**Typically Offered:**Fall, Spring