# Mathematics

## Learning Goals

**Understanding Higher Mathematics**

- Understanding the need to
**transition**from a procedural/computational understanding of mathematics to a broader**conceptual understanding**encompassing logical reasoning, generalization, abstraction, and formal proof.

- Understanding the
**core of mathematical culture**: the value and validity of careful reasoning, of precise definition, and close argument. Understanding mathematics as a growing body of knowledge, driven by creativity, a search for fundamental structure and interrelationships, and a methodology that is both powerful and intellectually compelling.

- Understanding the fundamentals of
**mathematical reasoning and logical argument**, including the role of hypotheses, conclusions, counterexamples, and other forms of mathematical evidence in the development and formulation of mathematical ideas.

- Understanding the methods and fundamental role of mathematics in
**modeling and solution**of critical real-world challenges in science and social science.

- Understanding the basic insights and methods of a broad variety of mathematical areas. All students of mathematics must achieve such understanding in
**calculus, naïve set and function theory, and linear algebra**, and ideally will further achieve such understanding in**probability and statistics, differential equations, analysis, and algebra.**

- Understanding in greater depth of at least one important subfield of mathematics such as abstract algebra, real analysis, geometry, topology, statistics, optimization, modeling, numerical methods, and dynamical systems.

**Skills Required for Effective Use of Mathematical Knowledge**

**Problem Solving**—to develop confidence in one’s ability to tackle difficult problems in both theoretical and applied mathematics, to translate between intuitive understandings and formal definitions and proofs, to formulate precise and relevant conjectures based on examples and counterexamples, to prove or disprove conjectures, to learn from failure, and to realize solutions are often multi-staged and require creativity, time, and patience.

**Modeling**—to interactively construct, modify, and analyze mathematical models of systems encountered in the natural and social sciences, to assess a model's accuracy and usefulness, and to draw contextual conclusions from them.

**Technology**—to recognize and appreciate the important role of technology in mathematical work, and to achieve proficiency with the technological tools of most value in one’s chosen area of concentration.

**Data and Observation**—to be cognizant of the uses of data and empirical observation in forming mathematical and statistical models, providing context for their use, and establishing their limits.

**Presentation**—to produce clear, precise, motivated, and well-organized expositions, in both written and oral form, using precise reasoning and genuine analysis.

**Oral Argumentation**—to think on one’s feet, to field questions, and to defend mathematical ideas and arguments before a range of audiences.

**Mathematical Literature**—to know how to effectively search the mathematical literature and how to appropriately combine and organize information from a variety of sources.

Mary Lou Zeeman, *Department Chair*

Suzanne M. Theberge, *Senior Department Coordinator*

*Professors:* William Barker**, Adam B. Levy, Jennifer Taback, Mary Lou Zeeman

*Associate Professors:* John D. O’Brien, Thomas Pietraho‡, Manuel L. Reyes‡

*Assistant Professors:* Christopher Chong‡, Naomi Tanabe

*Senior Lecturer:* Eric C. Gaze

*Lecturers:* James M. Broda II, Michael King

*Visiting Faculty:* Subhadip Chowdhury, Behrang Forghani, Patricia Garmirian

## Mathematics Major

A major consists of at least nine courses numbered 1800 or higher.

Code | Title | Credits |
---|---|---|

Required Courses | ||

MATH 2000 | Linear Algebra | 1 |

MATH 2020 | Introduction to Mathematical Reasoning | 1 |

at least one advanced course (3000–3969) | 1 | |

Select at least six additional courses numbered 1800 or higher. | 6 |

The requirement of an advanced course (3000–3969) is meant to ensure that all majors have sufficient experience in at least one specific area of mathematics as listed below:

Code | Title | Credits |
---|---|---|

Algebra | ||

Intermediate Linear Algebra | ||

Number Theory and Cryptography | ||

Group Theory | ||

Rings and Fields | ||

Advanced Topics in Group Theory | ||

Advanced Topics in Rings and Number Theory | ||

Analysis | ||

Functions of a Complex Variable | ||

MATH 2501 Vector Calculus | ||

Introduction to Analysis | ||

Advanced Complex Analysis | ||

Advanced Analysis | ||

Modeling and Dynamics | ||

Biomathematics | ||

Ordinary Differential Equations | ||

Advanced Topics in Modeling | ||

Advanced Topics in Dynamical Systems | ||

Optimization and Numerical Methods | ||

Optimization | ||

Numerical Methods | ||

Optimal Control | ||

Partial Differential Equations | ||

Probability and Statistics | ||

Probability | ||

Statistics | ||

Advanced Topics in Probability and Statistics | ||

Geometry and Topology | ||

Geometry | ||

Topology | ||

Projective and Non-Euclidean Geometries |

Students who have already mastered the material in Mathematics 2000 and 2020 may substitute a more advanced course after receiving approval from the department chair. Students must submit a planned program of courses to the department when they declare a major.

## Mathematics Minor

A minor in mathematics consists of a minimum of five courses numbered 1800 or higher.

**Additional Information and Department Policies**

- Each of the courses required for the major or minor must be taken for a regular letter grade, not Credit/D/Fail, with a minimum earned grade of C-.
- At most, two of the nine courses required for the major, or one of the five courses required for the minor, can be transfer credits from other institutions.
- Independent studies and honors projects can count toward major and minor requirements with prior departmental approval.
- Advanced Placement and International Baccalaureate scores, in addition to the mathematics placement questionnaire, are only used for placement.

### Recommended Courses

Listed below are some of the courses recommended to students with the indicated interests. For a full and rich mathematical experience, it is recommended that students also take courses from across the mathematical spectrum of pure mathematics, applied mathematics, and statistics.

Code | Title | Credits |
---|---|---|

Pure Mathematics | ||

MATH 2303 | Functions of a Complex Variable | 1 |

MATH 2501 Vector Calculus | ||

MATH 2602 | Group Theory | 1 |

MATH 2603 | Introduction to Analysis | 1 |

MATH 2702 | Rings and Fields | 1 |

Select at least two advanced courses from the following: | 2 | |

Topology | ||

Advanced Complex Analysis | ||

Projective and Non-Euclidean Geometries | ||

Advanced Topics in Group Theory | ||

Advanced Analysis | ||

Advanced Topics in Rings and Number Theory | ||

Engineering and Applied Mathematics | ||

MATH 1808 | Biomathematics | 1 |

MATH 2208 | Ordinary Differential Equations | 1 |

MATH 2209 | Numerical Methods | 1 |

MATH 2301 | Intermediate Linear Algebra | 1 |

MATH 2303 | Functions of a Complex Variable | 1 |

MATH 2501 Vector Calculus | ||

MATH 2601 Combinatorics and Graph Theory | ||

MATH 2603 | Introduction to Analysis | 1 |

MATH 2606 | Statistics | 1 |

MATH 3108 | Advanced Topics in Modeling | 1 |

MATH 3109 | Optimal Control | 1 |

MATH 3208 | Advanced Topics in Dynamical Systems | 1 |

MATH 3209 | Partial Differential Equations | 1 |

Mathematical Economics and Econometrics | ||

MATH 2000 | Linear Algebra | 1 |

MATH 2109 | Optimization | 1 |

MATH 2206 | Probability | 1 |

MATH 2209 | Numerical Methods | 1 |

MATH 2301 | Intermediate Linear Algebra | 1 |

MATH 2603 | Introduction to Analysis | 1 |

MATH 2606 | Statistics | 1 |

MATH 3108 | Advanced Topics in Modeling | 1 |

MATH 3109 | Optimal Control | 1 |

MATH 3208 | Advanced Topics in Dynamical Systems | 1 |

MATH 3209 | Partial Differential Equations | 1 |

MATH 3603 | Advanced Analysis | 1 |

MATH 3606 | Advanced Topics in Probability and Statistics | 1 |

Statistics | ||

MATH 2000 | Linear Algebra | 1 |

MATH 2206 | Probability | 1 |

MATH 2208 | Ordinary Differential Equations | 1 |

MATH 2209 | Numerical Methods | 1 |

MATH 2603 | Introduction to Analysis | 1 |

MATH 2606 | Statistics | 1 |

MATH 3606 | Advanced Topics in Probability and Statistics | 1 |

Operations Research and Management Science | ||

MATH 2000 | Linear Algebra | 1 |

MATH 2020 | Introduction to Mathematical Reasoning | 1 |

MATH 2109 | Optimization | 1 |

MATH 2206 | Probability | 1 |

MATH 2301 | Intermediate Linear Algebra | 1 |

MATH 2601 Combinatorics and Graph Theory | ||

MATH 2606 | Statistics | 1 |

MATH 3109 | Optimal Control | 1 |

MATH 3606 | Advanced Topics in Probability and Statistics | 1 |

**MATH 1040** **(a, FYS) ** **Educated Guessing**

Every Fall. Fall 2019. Enrollment limit: 16.

A writing-intensive course that explores the many ways in which randomness affects everyday life. Introduces historical and computational aspects of mathematical logic, probability, and statistics. Addresses decision-making strategies as well as sources of flawed reasoning, including cognitive biases and logical fallacies. Topics include: games of chance, weather phenomena, financial markets, legal proceedings, and medical diagnostics. Students engage in all facets of the writing process: from invention, library research, drafting, and revision to final editing.

Previous terms offered: Fall 2018.

**MATH 1050** **(a, MCSR) ** **Quantitative Reasoning**

Every Semester. Fall 2019. Enrollment limit: 30.

Explores the ways and means by which we communicate with numbers; the everyday math we encounter on a regular basis. The fundamental quantitative skill set is covered in depth providing a firm foundation for further coursework in mathematics and the sciences. Topics include ratios, rates, percentages, units, descriptive statistics, linear and exponential modeling, correlation, logic, and probability. A project-based course using Microsoft Excel, emphasizing conceptual understanding and application. Reading of current newspaper articles and exercises involving personal finance are incorporated to place the mathematics in real-world context.

Prerequisites: Placement in MATH 1050 or 1051 (S/M) or Placement in MATH 1050 (S/M).

Previous terms offered: Spring 2019, Fall 2018, Spring 2018, Fall 2017, Spring 2017, Fall 2016, Spring 2016, Fall 2015.

**MATH 1051** **(a, MCSR) ** **Quantitative Reasoning: Mathematics for Sustainability**

Every Semester. Fall 2019. Enrollment limit: 30.

Provides students with a comprehensive overview of the quantitative skills required to cope with the practical demands of daily life. Explores the connection between mathematics and real-world problems through the study of topics related to sustainability in a variety of social, economic, and ecological systems. Develops the ability to recognize, define, and solve problems within a quantitative framework and provides a foundation for further coursework in mathematics and the sciences. Topics include: rates, units, system diagrams, networks, logistic and exponential models, descriptive statistics, random processes, and elementary game theory.

Prerequisites: Placement in MATH 1050 or 1051 (S/M) or Placement in MATH 1050 (S/M).

**MATH 1300** **(a, MCSR) ** **Biostatistics**

Every Semester. Fall 2019. Enrollment limit: 30.

An introduction to the statistical methods used in the life sciences. Emphasizes conceptual understanding and includes topics from exploratory data analysis, the planning and design of experiments, probability, and statistical inference. One and two sample t-procedures and their non-parametric analogs, one-way ANOVA, simple linear regression, goodness of fit tests, and the chi-square test for independence are discussed. An average of four to five hours of class meetings and computer laboratory sessions per week. Not open to students who have credit for Economics 2557, Psychology 2520, or Mathematics 1200 or have credit or are concurrently enrolled in Mathematics 1400.

Prerequisites: MATH 1050 or Placement in MATH 1300 (S) or Placement in MATH 1300 or 1400 (S) or Placement in MATH 1300 or 2206(S) or MATH 1051.

Previous terms offered: Spring 2019, Fall 2018, Spring 2018, Fall 2017, Spring 2017, Fall 2016, Spring 2016, Fall 2015.

**MATH 1400** **(a, MCSR) ** **Statistics in the Sciences**

Formatted in consideration of the use of statistics as a means for principled argumentation in the natural and social sciences, and examines historical, computational, mathematical, and practical examples. Readings from the scientific literature are paired with techniques to interpret data in a variety of contexts. Explorations of the interconnections between statistics, mathematics, scientific practice, and computation underlie all aspects. Topics include: probability, Bayesian reasoning, random variables, standard statistical tests, such as t-tests, regression, and ANOVA, p-values, hypothesis testing, computation, data visualization, and scientific writing. Not open to students who have credit for Economics 2557, Psychology 2520, Mathematics 1200, or have credit or are concurrently enrolled in Mathematics 1300.

Prerequisites: MATH 1600 or MATH 1700 or MATH 1750 or MATH 1800 or Placement in MATH 1300 or 1400 (S) or Placement in MATH 1400 or 2206 (S).

Previous terms offered: Spring 2019, Spring 2018.

**MATH 1600** **(a, MCSR) ** **Differential Calculus**

Every Semester. Fall 2019. Enrollment limit: 30.

Functions, including the trigonometric, exponential, and logarithmic functions; the derivative and the rules for differentiation; the anti-derivative; applications of the derivative and the anti-derivative. Four to five hours of class meetings and computer laboratory sessions per week, on average. Open to students who have taken at least three years of mathematics in secondary school.

Prerequisites: MATH 1050 or Placement in MATH 1600 (M) or PHYS 1093 (same as CHEM 1093) or MATH 1051.

Previous terms offered: Spring 2019, Fall 2018, Spring 2018, Fall 2017, Spring 2017, Fall 2016, Spring 2016, Fall 2015.

**MATH 1700** **(a, MCSR) ** **Integral Calculus**

Every Semester. Fall 2019. Enrollment limit: 30.

The definite integral; the Fundamental theorems; improper integrals; applications of the definite integral; differential equations; and approximations including Taylor polynomials and Fourier series. An average of four to five hours of class meetings and computer laboratory sessions per week.

Prerequisites: MATH 1600 or Placement in MATH 1700 (M).

**MATH 1750** **(a, MCSR) ** **Integral Calculus, Advanced Section**

Every Semester. Fall 2019. Enrollment limit: 30.

A review of the exponential and logarithmic functions, techniques of integration, and numerical integration. Improper integrals. Approximations using Taylor polynomials and infinite series. Emphasis on differential equation models and their solutions. An average of four to five hours of class meetings and computer laboratory sessions per week. Open to students whose backgrounds include the equivalent of Mathematics 1600 and the first half of Mathematics 1700. Designed for first-year students who have completed an AB Advanced Placement calculus course in their secondary schools.

Prerequisites: Placement in MATH 1750 (M).

**MATH 1800** **(a, MCSR) ** **Multivariate Calculus**

Every Semester. Fall 2019. Enrollment limit: 30.

Multivariate calculus in two and three dimensions. Vectors and curves in two and three dimensions; partial and directional derivatives; the gradient; the chain rule in higher dimensions; double and triple integration; polar, cylindrical, and spherical coordinates; line integration; conservative vector fields; and Green’s theorem. An average of four to five hours of class meetings and computer laboratory sessions per week.

Prerequisites: MATH 1700 or MATH 1750 or Placement in MATH 1800 (M).

**MATH 1808** **(a, MCSR) ** **Biomathematics**

Every Fall. Fall 2019. Enrollment limit: 30.

A study of mathematical modeling in biology, with a focus on translating back and forth between biological questions and their mathematical representation. Biological questions are drawn from a broad range of topics, including disease, ecology, genetics, population dynamics, and neurobiology. Mathematical methods include discrete and continuous (ODE) models and simulation, box models, linearization, stability analysis, attractors, oscillations, limiting behavior, feedback, and multiple time-scales. Within the biology major, this course may count as the mathematics credit or as biology credit, but not both. Students are expected to have taken a year of high school or college biology prior to this course. (Same as: BIOL 1174)

Prerequisites: MATH 1600 or higher or Placement in MATH 1700 (M) or Placement in MATH 1750 (M) or Placement in MATH 1800 (M) or Placement in 2000, 2020, 2206 (M).

Previous terms offered: Fall 2017, Spring 2017, Fall 2015.

**MATH 2000** **(a, MCSR) ** **Linear Algebra**

Every Semester. Fall 2019. Enrollment limit: 30.

A study of linear algebra in the context of Euclidean spaces and their subspaces, with selected examples drawn from more general vector spaces. Topics will include: vectors, linear independence and span, linear transformations, matrices and their inverses, bases, dimension and rank, determinants, eigenvalues and eigenvectors, diagonalization and change of basis, and orthogonality. Applications drawn from linear systems of equations, discrete dynamical systems, Markov chains, computer graphics, and least-squares approximation.

Prerequisites: MATH 1800 or Placement in 2000, 2020, 2206 (M).

**MATH 2020** **(a, MCSR) ** **Introduction to Mathematical Reasoning**

Every Semester. Fall 2019. Enrollment limit: 20.

An introduction to logical deductive reasoning and mathematical proof through diverse topics in higher mathematics. Specific topics include set and function theory, modular arithmetic, proof by induction, and the cardinality of infinite sets. May also consider additional topics such as graph theory, number theory, and finite state automata.

Prerequisites: MATH 1800 or Placement in 2000, 2020, 2206 (M).

**MATH 2109** **(a, MCSR) ** **Optimization**

Every Other Spring. Fall 2019. Enrollment limit: 30.

A study of optimization problems arising in a variety of situations in the social and natural sciences. Analytic and numerical methods are used to study problems in mathematical programming, including linear models, but with an emphasis on modern nonlinear models. Issues of duality and sensitivity to data perturbations are covered, and there are extensive applications to real-world problems.

Prerequisites: MATH 2000.

Previous terms offered: Spring 2019, Spring 2017.

**MATH 2206** **(a, MCSR) ** **Probability**

Every Semester. Fall 2019. Enrollment limit: 35.

A study of the mathematical models used to formalize nondeterministic or “chance” phenomena. General topics include combinatorial models, probability spaces, conditional probability, discrete and continuous random variables, independence and expected values. Specific probability densities, such as the binomial, Poisson, exponential, and normal, are discussed in depth.

Prerequisites: MATH 1800 or Placement in 2000, 2020, 2206 (M) or Placement in MATH 1400 or 2206 (S).

**MATH 2208** **(a, MCSR) ** **Ordinary Differential Equations**

Every Semester. Fall 2019. Enrollment limit: 30.

A study of some of the ordinary differential equations that model a variety of systems in the physical, natural and social sciences. Classical methods for solving differential equations with an emphasis on modern, qualitative techniques for studying the behavior of solutions to differential equations. Applications to the analysis of a broad set of topics, including population dynamics, oscillators and economic markets. Computer software is used as an important tool, but no prior programming background is assumed.

Prerequisites: MATH 2000.

**MATH 2209** **(a, MCSR) ** **Numerical Methods**

An introduction to the theory and application of numerical analysis. Topics include approximation theory, numerical integration and differentiation, iterative methods for solving equations, and numerical analysis of differential equations.

Prerequisites: MATH 2000.

Previous terms offered: Fall 2018, Spring 2016.

**MATH 2301** **(a, MCSR) ** **Intermediate Linear Algebra**

A continuation of Linear Algebra focused on the interplay of algebra and geometry as well as mathematical theory and its applications. Topics include matrix decompositions, eigenvalues and spectral theory, vector and Hilbert spaces, norms and low-rank approximations. Applications to biology, computer science, economics, and statistics, including artificial learning and pattern recognition, principal component analysis, and stochastic systems. Course and laboratory work balanced between theory and application.

Prerequisites: Two of: MATH 2000 and MATH 2020.

Previous terms offered: Spring 2018, Fall 2016.

**MATH 2303** **(a, MCSR) ** **Functions of a Complex Variable**

Every Other Fall. Fall 2019. Enrollment limit: 35.

The differential and integral calculus of functions of a complex variable. Cauchy’s theorem and Cauchy’s integral formula, power series, singularities, Taylor’s theorem, Laurent’s theorem, the residue calculus, harmonic functions, and conformal mapping.

Prerequisites: MATH 1800 or Placement in 2000, 2020, 2206 (M).

Previous terms offered: Fall 2018, Spring 2016.

**MATH 2351** **(a) ** **Lie Theory**

A Lie group is a smooth n-dimensional surface with a multiplication that is differentiable. Allowing for a theory of "continuous symmetry" of objects, Lie groups and their associated algebras are central tools of modern mathematics and theoretical physics. Although highly sophisticated in general, the most common Lie groups are groups of matrices under matrix multiplication. Matrix groups can be studied with only a background in multivariable calculus and linear algebra. Basic course topics include, among others, real and complex matrix Lie groups and Lie algebras, one-parameter subgroups, the exponential map, the adjoint representations, and applications in geometry and physics.

Prerequisites: Two of: MATH 2000 and MATH 2020.

Previous terms offered: Spring 2017.

**MATH 2404** **(a, MCSR) ** **Geometry**

Every Other Spring. Fall 2019. Enrollment limit: 30.

A survey of modern approaches to Euclidean geometry in two dimensions. Axiomatic foundations of metric geometry. Transformational geometry: isometries and similarities. Klein’s Erlanger Programm. Symmetric figures. Other topics may be chosen from three-dimensional geometry, ornamental groups, area, volume, fractional dimension, and fractals.

Prerequisites: MATH 2020.

Previous terms offered: Spring 2018, Fall 2015.

**MATH 2502** **(a, MCSR) ** **Number Theory and Cryptography**

A survey of number theory from Euclid’s proof that there are infinitely many primes through Wiles’s proof of Fermat’s Last Theorem in 1994. Prime numbers, unique prime factorization, and results on counting primes. The structure of modular number systems. Continued fractions and “best” approximations to irrational numbers. Investigation of the Gaussian integers and other generalizations. Squares, sums of squares, and the law of quadratic reciprocity. Applications to modern methods of cryptography, including public key cryptography and RSA encryption.

Prerequisites: MATH 2020.

Previous terms offered: Spring 2019, Spring 2017.

**MATH 2602** **(a, MCSR) ** **Group Theory**

Every Other Fall. Fall 2019. Enrollment limit: 35.

An introduction to the theory of finite and infinite groups, with examples ranging from symmetry groups to groups of polynomials and matrices. Properties of mappings that preserve algebraic structures are studied. Topics include cyclic groups, homomorphisms and isomorphisms, normal subgroups, factor groups, the structure of finite abelian groups, and Sylow theorems.

Prerequisites: Two of: MATH 2000 and MATH 2020.

Previous terms offered: Fall 2018, Fall 2016.

**MATH 2603** **(a, MCSR) ** **Introduction to Analysis**

Every Fall. Fall 2019. Enrollment limit: 35.

Building on the theoretical underpinnings of calculus, develops the rudiments of mathematical analysis. Concepts such as limits and convergence from calculus are made rigorous and extended to other contexts, such as spaces of functions. Specific topics include metric spaces, point-set topology, sequences and series, continuity, differentiability, the theory of Riemann integration, and functional approximation and convergence.

Prerequisites: MATH 2020.

Previous terms offered: Fall 2018, Fall 2017, Fall 2016, Fall 2015.

**MATH 2606** **(a, MCSR) ** **Statistics**

An introduction to the fundamentals of mathematical statistics. General topics include likelihood methods, point and interval estimation, and tests of significance. Applications include inference about binomial, Poisson, and exponential models, frequency data, and analysis of normal measurements.

Prerequisites: MATH 2206.

Previous terms offered: Spring 2019, Spring 2018, Spring 2017, Spring 2016.

**MATH 2702** **(a, MCSR) ** **Rings and Fields**

An introduction to algebraic structures based on the study of rings and fields. Structure of groups, rings, and fields, with an emphasis on examples. Fundamental topics include: homomorphisms, ideals, quotient rings, integral domains, polynomial rings, field extensions. Further topics may include unique factorization domains, rings of fractions, finite fields, vector spaces over arbitrary fields, and modules. Mathematics 2502 is helpful but not required.

Prerequisites: Two of: MATH 2000 and MATH 2020.

Previous terms offered: Fall 2017, Fall 2015.

**MATH 3108** **(a) ** **Advanced Topics in Modeling**

Every Other Spring. Fall 2019. Enrollment limit: 16.

A study of mathematical modeling, with emphasis on how to identify scientific questions appropriate for modeling, how to develop a model appropriate for a given scientific question, and how to interpret model predictions. Applications drawn from the natural, physical, environmental, and sustainability sciences. Model analysis uses a combination of computer simulation and theoretical methods and focuses on predictive capacity of a model.

Prerequisites: Three of: MATH 2000 and MATH 2020 and MATH 2208.

Previous terms offered: Spring 2018.

**MATH 3109** **(a) ** **Optimal Control**

A study of infinite-dimensional optimization, including calculus of variations and optimal control. Classical, analytic techniques are covered, as well as numerical methods for solving optimal control problems. Applications in many topic areas, including economics, biology, and robotics.

Prerequisites: Three of: MATH 2000 and MATH 2020 and MATH 2208.

Previous terms offered: Spring 2019, Spring 2016, Fall 2015.

**MATH 3204** **(a) ** **Topology**

A mathematical study of shape. Examination of surfaces, knots, and manifolds with or without boundary. Topics drawn from point-set topology, algebraic topology, knot theory, and computational topology, with possible applications to differential equations, graph theory, topological data analysis, and the sciences.

Prerequisites: Three of: either MATH 2602 or MATH 2603 or MATH 2702 and MATH 2000 and MATH 2020.

Previous terms offered: Fall 2017, Spring 2016.

**MATH 3208** **(a) ** **Advanced Topics in Dynamical Systems**

A study of nonlinear dynamical systems arising in applications, with emphasis on modern geometric, topological, and analytical techniques to determine global system behavior, from which predictions can be made. Topics chosen from local stability theory and invariant manifolds, limit cycles and oscillation, global phase portraits, bifurcation and resilience, multiple time scales, and chaos. Theoretical methods supported by simulations. Applications drawn from across the sciences.

Prerequisites: Four of: MATH 2000 and MATH 2020 and MATH 2208 and MATH 2603.

Previous terms offered: Spring 2017.

**MATH 3209** **(a) ** **Partial Differential Equations**

A study of some of the partial differential equations that model a variety of systems in the natural and social sciences. Classical methods for solving partial differential equations are covered, as well as modern, numerical techniques for approximating solutions. Applications to the analysis of a broad set of topics, including air quality, traffic flow, and imaging. Computer software is used as an important tool.

Prerequisites: Three of: MATH 2000 and MATH 2020 and MATH 2208.

Previous terms offered: Fall 2018, Fall 2016.

**MATH 3303** **(a, MCSR) ** **Advanced Complex Analysis**

A second course in complex analysis. Topics may include conformal mappings, harmonic functions, and analytic functions. Applications drawn from boundary value problems, elliptic functions, two-dimensional potential theory, Fourier analysis, and topics in analytic number theory.

Prerequisites: Two of: MATH 2020 and MATH 2303.

Previous terms offered: Spring 2019.

**MATH 3404** **(a) ** **Projective and Non-Euclidean Geometries**

A survey of affine, projective, and non-Euclidean geometries in two-dimensions, unified by the transformational viewpoint of Klein’s Erlanger Programm. Special focus placed on conic sections and projective embeddings. Additional topics as time permits: complex numbers in plane geometry, quaternions in three-dimensional geometry, and the geometry of four-dimensional space-time in special relativity. Mathematics 2404 is helpful but not required.

Prerequisites: Two of: MATH 2000 and MATH 2020.

Previous terms offered: Fall 2018, Fall 2016.

**MATH 3602** **(a) ** **Advanced Topics in Group Theory**

The study of group actions on geometric objects; understanding finite and discrete groups via generators and presentations. Applications to geometry, topology, and linear algebra, focusing on certain families of groups. Topics may include Cayley graphs, the word problem, growth of groups, and group representations.

Prerequisites: MATH 2602.

Previous terms offered: Spring 2019, Spring 2017.

**MATH 3603** **(a) ** **Advanced Analysis**

Measure theory and integration with applications to probability and mathematical finance. Topics include Lebesgue measure and integral, measurable functions and random variables, convergence theorems, analysis of random processes including random walks and Brownian motion, and the Ito integral.

Prerequisites: Two of: MATH 2000 and MATH 2603.

Previous terms offered: Spring 2018, Spring 2017.

**MATH 3606** **(a) ** **Advanced Topics in Probability and Statistics**

Every Other Fall. Fall 2019. Enrollment limit: 15.

One or more specialized topics in probability and statistics. Possible topics include regression analysis, nonparametric statistics, logistic regression, and other linear and nonlinear approaches to modeling data. Emphasis is on the mathematical derivation of the statistical procedures and on the application of the statistical theory to real-life problems.

Prerequisites: Two of: MATH 2000 and MATH 2606.

Previous terms offered: Fall 2017.

**MATH 3702** **(a) ** **Advanced Topics in Rings and Number Theory**

Advanced topics in modern algebra based on rings and fields. Possible topics include: Galois theory with applications to geometric constructions and (in)solvability of polynomial equations; algebraic number theory and number fields such as the p-adic number system; commutative algebra; algebraic geometry and solutions to systems of polynomial equations.

Prerequisites: MATH 2702.

Previous terms offered: Spring 2018, Spring 2016.