# Mathematics Degree Requirements

## 2021 - 2022 Catalog

We have the following degrees:

## Mathematics major leading to BA degree

A **major in mathematics** leading to a Bachelor of Arts degree requires the completion of at least 33 credits as follows:

- MATH 201, 221, 222, 311, 321, 343
- One course chosen from MATH 391, 392, and 393
- One course selected from BIOL 282; CHEM 260, 261; CSCI 211, 313; ECON 302, 320; ENGN 203; GEOL 250; MATH 270, 310, 332, 333; PHYS 112
- Nine additional credits selected from mathematics courses numbered above 300.

Additional courses required as prerequisites for completion of the above include MATH 101 and 102, or their equivalents. Furthermore, the course selected to fulfill requirement 2 above may have prerequisites.

- Required courses:
- MATH 201 - Bridges to Advanced Mathematics
FDR SC Credits 3 Prerequisite 6 credits of MATH courses or MATH 221 or 222 Faculty Staff The course explores various important mathematical constructions and ideas, with a particular emphasis on mathematical inquiry and reasoning. Topics include: sets, functions, equivalence relations, modular arithmetic, and basic properties of the integers, real numbers, and complex numbers.

- MATH 221 - Multivariable Calculus
FDR SC Credits 3 Prerequisite The equivalent of MATH 102 with a C grade or better or MATH 201 or 222 Motion in three dimensions, differential calculus of multivariable functions, multiple integrals, line integrals, and Green's Theorem.

- MATH 222 - Linear Algebra
FDR SC Credits 3 Prerequisite The equivalent of MATH 102 with a C grade or better or MATH 201 or 221 Linear algebra is the backbone of much of mathematics. Students in this course learn to identify and explain the basic principles, terminology, and theories used in linear algebra, and apply quantitative and/or qualitative reasoning skills to solve problems posed in linear algebra, primarily through applications of to both mathematics and the sciences, and also by writing proofs In mathematics.

- MATH 311 - Real Analysis
Credits 3 Prerequisite MATH 201 (or 301) and 221 Faculty Staff A systematic study of concepts basic to calculus, such as topology of the real numbers, limits, differentiation, integration, sequences and series. Additional topics vary by instructor.

- MATH 321 - Abstract Algebra
Credits 3 Prerequisite MATH 201 (or 301) and 222 Faculty Staff An introduction to basic algebraic structures common throughout mathematics. These include rings, fields, groups, homomorphisms and quotient structures. Additional topics vary by instructor.

- MATH 343 - Geometry
Credits 3 Prerequisite MATH 201 (or 301) , 221, and 222 Faculty Staff This course is an introduction to geometric techniques through study of Euclidean and non-Euclidean geometries and their transformations. Additional topics vary by instructor.

- One course chosen from:
- MATH 391 - Topics in Analysis
Credits 3 Prerequisite MATH 311 Topics vary but can include complex analysis, topology, differential equations, differential topology, numerical analysis, functional analysis, measure theory, fractal geometry, Lebesgue integration and Fourier analysis, harmonic analysis, and analytic number theory. May be repeated for degree credit if the topic is different.

**Fall 2021, MATH 391A-01: Topics in Analysis: Numerical Mathematics for Data Science (3).***Prerequisite: MATH 311.*This course is designed to introduce knowledge of numerical computation and analysis, in order to equip students with necessary numerical techniques to address practical questions arising from data science and other fields. We will discuss useful methods to construct mathematical models from given data and powerful algorithms to solve large scale systems of linear equations which are formulated during the creation of mathematical models. Students will also learn computational complexity, accuracy, stability, conditioning, and other mathematical concepts of numerical analysis which are fundamental in developing an efficient numerical algorithm. MATLAB will be the programming language used for this course.*Wang.* - MATH 392 - Topics in Abstract Algebra
Credits 3 Prerequisite MATH 321 Topics vary but can include field and Galois theory, geometric and combinatorial group theory, representation theory, number theory, algebraic number theory, commutative algebra, algebraic geometry, arithmetic geometry, advanced linear algebra, algebraic coding theory and cryptography, algebraic topology, homological algebra, and graph theory, May be repeated for degree credit if the topic is different.

**Winter 2022, Math 392A-01: Topics in Abstract Algebra: Rings, Fields, and Galois Theory (3).***Prerequisite: MATH 321.*Rings, ring homomorphisms, modules, module homomorphisms, Euclidean domains, PIDs, UFDs, field extensions, degree, algebraic numbers, automorphisms, irreducible polynomials, Galois groups, Galois correspondence.*Colbert.* - MATH 393 - Topics in Geometry and Topology
Credits 3 Prerequisite MATH 342 or 343 Topics vary but can include knot theory, topology and geometry of surfaces, differential geometry, Riemann surfaces, 3-manifolds, tilings, geometric probability, geometry of spacetime, finite geometry, computational geometry, differential topology, and projective geometry. May be repeated for degree credit if the topic is different.

**Winter 2022, Math 393A-01: Topics in Geometry and Topology: Differential Topology (3).***Prerequisite: Math 342 or 343.*This course builds on material from multivariable calculus, linear algebra and geometry. We cover a range of topics: an introduction to manifolds with boundary, derivatives as linear transformations, tangent spaces, inverse and implicit function theorems, transversality, and intersections of manifolds. Then integration on manifolds, differential forms, and the generalized Stokes's Theorem.*Denne.***Winter 2021, MATH 393A-01: Topics in Geometry and Topology: Experimenting with Geometry (3).***Prerequisite: MATH 342 or 343.*This course will be run in an experimental format modeled on the notion of a "Geometry Lab." Students will study unsolved problems in geometry, learning whatever background material is relevant for understanding and approaching the selected problems. Topics are likely to include algebraic geometry, hyperbolic geometry, and projective geometry.*Abrams.* - One course selected from:
- BIOL 282 - Modeling and Simulations in Public Health
FDR SL Credits 4 Prerequisite MATH 101 Faculty Toporikova Where are infections spreading? How many people will be affected? What are some different ways to stop the spread of an epidemic? These are questions that all of us ask during an outbreak or emergency. In a process known as modeling, scientists analyze data using complex mathematical methods to provide answers to these and other questions during an emergency response. Models provide the foresight that can help decision-makers better prepare for the future. In this course you will learn how to develop a simple mathematical models using data. You will learn basic epidemiological concepts, computational data analysis tools and relevant mathematical techniques to integrate existing data into the model and generate relevant predictions. In an open-ended project, you and several of your classmates will develop a model and recommendation about potential public health threat. No prior programming experience required - you will learn to use Python, a popular open-source programming language and Jupyter Notebook data analysis environment, to interactively explore data. Laboratory course.

- CHEM 261 - Physical Chemistry: Quantum & Computational Chemistry
Credits 3 Prerequisite CHEM 110 and MATH 102 and junior standing Faculty Tuchler An introduction to quantum mechanics as it applies to atomic and molecular systems. The emphasis is placed on spectroscopic methods and the modern picture of chemical bonding and molecular structure and computational methods.

- CSCI 211 - Algorithm Design and Analysis
Credits 3 Prerequisite CSCI 112 and MATH 121 or MATH 201 Faculty Staff Methods for designing efficient algorithms, including divide-and-conquer, dynamic programming, and greedy algorithms. Analysis of algorithms for correctness and estimating running time and space requirements. Topics include advanced data structures, graph theory, network flow, and computational intractability.

- CSCI 313 - Theory of Computation
Credits 3 Prerequisite MATH 121 or MATH 201 or instructor consent Faculty Levy A study of the principles of computer science embodied in formal languages, automata, computability, and computational complexity. Topics include context-free grammars, Turing machines, and the halting problem.

- ECON 302 - Game Theory
Credits 3 Prerequisite MATH 101 or equivalent and ECON 210 Faculty Guse This course abandons the assumptions of perfect competition. Buyers and sellers may be few; information may be privately held; property rights may poorly enforced; externalities abound and uncertainty is the rule. Game theory is a general framework for analyzing the messy world of strategic interactions. Standard solution concepts such as Nash Equilibrium, subgame perfection, and Bayesian equilibrium are introduced in the context of a broad array of microeconomic topics. These include auctions, bargaining, oligopoly, labor market signaling, public finance and insurance. Class time combines lectures, problem-solving workshops, and classroom experiments.

- ECON 320 - Mathematical Methods in Economics
Credits 3 Prerequisite Either ECON 210 or MATH 221. Preference to ECON majors during the first round of registration. Other majors are encouraged to add to the waiting list after registration re-opens for all class years Faculty Grajzl An introduction to fundamental mathematical methods of economic analysis with a variety of applications from both microeconomics and macroeconomics. Topics covered include theory and applications of linear algebra, multivariable calculus, static optimization, and comparative statics. The course is highly recommended for anyone planning to undertake graduate studies in economics or a closely related field.

*Should not be taken if completed ECON 220: Mathematical Economics.* - ENGN 203 - Mechanics I: Statics
Credits 3 Prerequisite Grade of C or better in MATH-101 and PHYS-111 (PHYS 111 as corequisite with instructor consent) Faculty Staff The science of mechanics is used to study bodies in equilibrium under the action of external forces. Emphasis is on problem solving: trusses, frames and machines, centroids, area moments of inertia, beams, cables, and friction.

- GEOL 250 - Structural Geology and Tectonics
Credits 4 Prerequisite MATH 101 and GEOL 100, 101, or 102 Faculty Connors Description and methods of analysis of large- and small-scale structural features of the Earth's crust. Topics also include the analysis of geometry, strain and stress as they relate to deformation in the earth. Rock mechanics, application of structural geology in environmental engineering and resource exploration, geometric and computational techniques used in structural analysis, interpretation of geologic maps, and the structural development of mountain systems are also covered. Laboratory course.

- MATH 270 - Financial and Actuarial Mathematics
Credits 3 Prerequisite MATH 102 Faculty Staff An introduction to some of the fundamental topics in financial and actuarial mathematics. Possible topics include calculating present and accumulated values for various streams of cash and the theoretical basis of corporate finance and financial models and the application of those models to insurance and other financial risks.

- MATH 310 - Mathematical Statistics
Credits 3 Prerequisite MATH 309 Sampling distributions, point and interval estimation, testing hypotheses, regression and correlation, and analysis of variance.

- MATH 332 - Ordinary Differential Equations
Credits 3 Prerequisite MATH 221 with C grade or better First and second order differential equations, systems of differential equations, and applications. Techniques employed are analytic, qualitative, and numerical.

- MATH 333 - Partial Differential Equations
Credits 3 Prerequisite MATH 332 An introduction to the study of boundary value problems and partial differential equations. Topics include modeling heat and wave phenomena, Fourier series, separation of variables, and Bessel functions. Techniques employed are analytic, qualitative, and numerical.

- PHYS 112 - General Physics II
FDR SL Credits 4 Prerequisite PHYS 111 Faculty Staff A continuation of PHYS 111. Topics include thermodynamics, electricity, magnetism, and optics. Laboratory course with fee.

- Nine additional credits selected from mathematics courses numbered above 300.

## Mathematics major leading to BS degree

A **major in mathematics** leading to a Bachelor of Science degree requires the completion of at least 51 credits as follows:

- MATH 201, 221, 222, 311, 321, 343
- One course chosen from MATH 391, 392, and 393
- PHYS 111 and 112
- CSCI 111 or 121
- 12 additional credits selected from mathematics courses numbered above 300
- Six additional credits selected from courses in biology, chemistry, computer science, engineering, geology, mathematics (numbered 200 and above), and physics, except courses excluded from degree programs in those subjects.

Additional courses required as prerequisites for completion of the above include MATH 101 and 102 or their equivalents.

- Required courses:
- MATH 201 - Bridges to Advanced Mathematics
FDR SC Credits 3 Prerequisite 6 credits of MATH courses or MATH 221 or 222 Faculty Staff The course explores various important mathematical constructions and ideas, with a particular emphasis on mathematical inquiry and reasoning. Topics include: sets, functions, equivalence relations, modular arithmetic, and basic properties of the integers, real numbers, and complex numbers.

- MATH 221 - Multivariable Calculus
FDR SC Credits 3 Prerequisite The equivalent of MATH 102 with a C grade or better or MATH 201 or 222 Motion in three dimensions, differential calculus of multivariable functions, multiple integrals, line integrals, and Green's Theorem.

- MATH 222 - Linear Algebra
FDR SC Credits 3 Prerequisite The equivalent of MATH 102 with a C grade or better or MATH 201 or 221 Linear algebra is the backbone of much of mathematics. Students in this course learn to identify and explain the basic principles, terminology, and theories used in linear algebra, and apply quantitative and/or qualitative reasoning skills to solve problems posed in linear algebra, primarily through applications of to both mathematics and the sciences, and also by writing proofs In mathematics.

- MATH 311 - Real Analysis
Credits 3 Prerequisite MATH 201 (or 301) and 221 Faculty Staff A systematic study of concepts basic to calculus, such as topology of the real numbers, limits, differentiation, integration, sequences and series. Additional topics vary by instructor.

- MATH 321 - Abstract Algebra
Credits 3 Prerequisite MATH 201 (or 301) and 222 Faculty Staff An introduction to basic algebraic structures common throughout mathematics. These include rings, fields, groups, homomorphisms and quotient structures. Additional topics vary by instructor.

- MATH 343 - Geometry
Credits 3 Prerequisite MATH 201 (or 301) , 221, and 222 Faculty Staff This course is an introduction to geometric techniques through study of Euclidean and non-Euclidean geometries and their transformations. Additional topics vary by instructor.

- One course chosen from:
- MATH 391 - Topics in Analysis
Credits 3 Prerequisite MATH 311 Topics vary but can include complex analysis, topology, differential equations, differential topology, numerical analysis, functional analysis, measure theory, fractal geometry, Lebesgue integration and Fourier analysis, harmonic analysis, and analytic number theory. May be repeated for degree credit if the topic is different.

**Fall 2021, MATH 391A-01: Topics in Analysis: Numerical Mathematics for Data Science (3).***Prerequisite: MATH 311.*This course is designed to introduce knowledge of numerical computation and analysis, in order to equip students with necessary numerical techniques to address practical questions arising from data science and other fields. We will discuss useful methods to construct mathematical models from given data and powerful algorithms to solve large scale systems of linear equations which are formulated during the creation of mathematical models. Students will also learn computational complexity, accuracy, stability, conditioning, and other mathematical concepts of numerical analysis which are fundamental in developing an efficient numerical algorithm. MATLAB will be the programming language used for this course.*Wang.* - MATH 392 - Topics in Abstract Algebra
Credits 3 Prerequisite MATH 321 Topics vary but can include field and Galois theory, geometric and combinatorial group theory, representation theory, number theory, algebraic number theory, commutative algebra, algebraic geometry, arithmetic geometry, advanced linear algebra, algebraic coding theory and cryptography, algebraic topology, homological algebra, and graph theory, May be repeated for degree credit if the topic is different.

**Winter 2022, Math 392A-01: Topics in Abstract Algebra: Rings, Fields, and Galois Theory (3).***Prerequisite: MATH 321.*Rings, ring homomorphisms, modules, module homomorphisms, Euclidean domains, PIDs, UFDs, field extensions, degree, algebraic numbers, automorphisms, irreducible polynomials, Galois groups, Galois correspondence.*Colbert.* - MATH 393 - Topics in Geometry and Topology
Credits 3 Prerequisite MATH 342 or 343 Topics vary but can include knot theory, topology and geometry of surfaces, differential geometry, Riemann surfaces, 3-manifolds, tilings, geometric probability, geometry of spacetime, finite geometry, computational geometry, differential topology, and projective geometry. May be repeated for degree credit if the topic is different.

**Winter 2022, Math 393A-01: Topics in Geometry and Topology: Differential Topology (3).***Prerequisite: Math 342 or 343.*This course builds on material from multivariable calculus, linear algebra and geometry. We cover a range of topics: an introduction to manifolds with boundary, derivatives as linear transformations, tangent spaces, inverse and implicit function theorems, transversality, and intersections of manifolds. Then integration on manifolds, differential forms, and the generalized Stokes's Theorem.*Denne.***Winter 2021, MATH 393A-01: Topics in Geometry and Topology: Experimenting with Geometry (3).***Prerequisite: MATH 342 or 343.*This course will be run in an experimental format modeled on the notion of a "Geometry Lab." Students will study unsolved problems in geometry, learning whatever background material is relevant for understanding and approaching the selected problems. Topics are likely to include algebraic geometry, hyperbolic geometry, and projective geometry.*Abrams.* - Take:
- PHYS 111 - General Physics I
FDR SL Credits 4 Prerequisite or co-requisite: MATH 101 or equivalent Faculty Staff An introduction to classical mechanics. Topics include kinematics, Newton's laws, solids, fluids, and wave motion. Laboratory course with fee.

- PHYS 112 - General Physics II
FDR SL Credits 4 Prerequisite PHYS 111 Faculty Staff A continuation of PHYS 111. Topics include thermodynamics, electricity, magnetism, and optics. Laboratory course with fee.

- Take one of the following courses:
- CSCI 111 - Fundamentals of Programming I
FDR FM Credits 4 Faculty Staff A disciplined approach to programming with Python. Emphasis is on problem-solving methods, algorithm development, and object-oriented concepts. Lectures and formal laboratories.

- CSCI 121 - Scientific Computing
FDR FM Credits 4 Faculty Levy *Not open to students who have taken CSCI 211 or higher.*An introduction to computer programming for scientific applications and a survey of the main methodological areas of scientific computation. The course provides the tools needed for students to use computers effectively in scientific work, whether in physics, chemistry, mathematics, economics, biology, psychology, or any field involving quantitative work. Programming in Matlab, a scientific-computing software package, with a focus on topics relevant to students' major fields of study. Lectures and formal labs. - 12 additional credits selected from mathematics courses numbered above 300
- Six additional credits selected from courses in

biology, chemistry, computer science, engineering, geology, mathematics (numbered 200 and above), and physics, except courses excluded from degree programs in those subjects.