Reviews material presented in MATH 1106 lectures and provides further instruction for students who need reinforcement, including problem-solving techniques and tips as well as prelim review. Not a substitute for attending MATH 1106 lectures or discussions. Students should contact their college for the most up-to-date information regarding if and how credits for this course will count toward graduation, and/or be considered regarding academic standing.

Full details for MATH 1006 - Academic Support for MATH 1106

Reviews material presented in MATH 1110 lectures and provides further instruction for students who need reinforcement, including problem-solving techniques and tips as well as prelim review. Not a substitute for attending MATH 1110 lectures or discussions. Students should contact their college for the most up-to-date information regarding if and how credits for this course will count toward graduation, and/or be considered regarding academic standing.

Full details for MATH 1011 - Academic Support for MATH 1110

Introduces topics in calculus: limits, rates of change, definition of and techniques for finding derivatives, relative and absolute extrema, and applications. The calculus content of the course is similar to 1/3 of the content covered in MATH 1106 and MATH 1110. In addition, the course includes a variety of topics of algebra, with emphasis on the development of linear, power, exponential, logarithmic, and trigonometric functions. Because of the strong emphasis on graphing, students will have a better understanding of asymptotic behavior of these functions.

Full details for MATH 1101 - Calculus Preparation

The goal of this course is to give students a strong basis in quantitative skills needed in the life and social sciences. We will focus on modeling using fundamental concepts from calculus developed in the course, including derivatives, integrals, and introductory differential equations. Examples from the life sciences are used throughout the course, including predator-prey populations. We will discuss mathematical models describing the evolution of these populations, analyze quantitative and qualitative properties to make predictions about these populations, and discuss assumptions and limitations of these models. Derivatives and integrals will be covered with a more applied focus than in MATH 1110 or a typical high school calculus course. Students who plan to take more than one semester of calculus should take MATH 1110 rather than MATH 1106.

Full details for MATH 1106 - Modeling with Calculus for the Life Sciences

MATH 1110 can serve as a one-semester introduction to calculus or as part of a two-semester sequence in which it is followed by MATH 1120. Topics include functions and graphs, limits and continuity, differentiation and integration of algebraic, trigonometric, inverse trig, logarithmic, and exponential functions; applications of differentiation, including graphing, max-min problems, tangent line approximation, implicit differentiation, and applications to the sciences; the mean value theorem; and antiderivatives, definite and indefinite integrals, the fundamental theorem of calculus, and the area under a curve.

Full details for MATH 1110 - Calculus I

Focuses on integration: applications, including volumes and arc length; techniques of integration, approximate integration with error estimates, improper integrals, differential equations and their applications. Also covers infinite sequences and series: definition and tests for convergence, power series, Taylor series with remainder, and parametric equations.

Full details for MATH 1120 - Calculus II

For students who wish to experience how mathematical ideas naturally evolve. The course emphasizes ideas and imagination rather than techniques and calculations. Homework involves students in actively investigating mathematical ideas. Topics vary depending on the instructor. Some assessment through writing assignments.

Full details for MATH 1300 - Mathematical Explorations

Essentially a second course in calculus and the first in a sequence designed for engineers that assumes familiarity with differential calculus at the level of MATH 1110. Topics include techniques of integration, finding areas and volumes by integration, exponential growth, partial fractions, infinite sequences and series, tests of convergence, and power series.

Full details for MATH 1910 - Calculus for Engineers

Introduction to multivariable calculus. Topics include partial derivatives, double and triple integrals, line and surface integrals, vector fields, Green's theorem, Stokes' theorem, and the divergence theorem.

Full details for MATH 1920 - Multivariable Calculus for Engineers

An introduction to linear algebra for students who plan to major or minor in mathematics or a related field. Topics include vector algebra, linear transformations, matrices, determinants, orthogonality, eigenvalues, and eigenvectors. Applications are made to linear differential or difference equations. Lectures will introduce students to formal proofs, and students will be required to produce some proofs in their homework and on exams. For a more applied version of this course, see MATH 2310.

Full details for MATH 2210 - Linear Algebra

An introduction to multivariable calculus for students who plan to major or minor in mathematics or a related field. Topics include differential and integral calculus of functions in several variables, line and surface integrals as well as the theorems of Green, Stokes and Gauss.

Full details for MATH 2220 - Multivariable Calculus

Designed for students who have been extremely successful in their previous calculus courses and for whom the notion of solving very hard problems and writing careful proofs is highly appealing, MATH 2230-MATH 2240 provides an integrated treatment of linear algebra and multivariable calculus at a higher theoretical level than in MATH 2210-MATH 2220. Topics covered in MATH 2230 include vectors, matrices, and linear transformations; differential calculus of functions of several variables; inverse and implicit function theorems; quadratic forms, extrema, and manifolds; multiple and iterated integrals.

Full details for MATH 2230 - Theoretical Linear Algebra and Vector Calculus I

An introduction to linear algebra for students interested in applications to data science. The course diverges from traditional linear algebra courses by emphasizing data science applications while teaching similar concepts. Key topics include matrices as data tables, high-dimensional datasets, singular value decomposition for data compression, and linear transformations in computer graphics. Students who take MATH 2310 may need more foundational coursework before pursuing further study in mathematics.

Full details for MATH 2310 - Linear Algebra for Data Science

An introduction to ordinary and partial differential equations. Topics include first-order equations (separable, linear, homogeneous, exact); mathematical modeling (e.g., population growth, terminal velocity); qualitative methods (slope fields, phase plots, equilibria and stability); numerical methods; second-order equations (method of undetermined coefficients, application to oscillations and resonance, boundary-value problems and eigenvalues); and Fourier series. A substantial part of this course involves partial differential equations, such as the heat equation, the wave equation, and Laplace's equation. MATH 2930 and MATH 2940 are independent and can be taken in either order; they should not be taken in the same semester.

Full details for MATH 2930 - Differential Equations for Engineers

Linear algebra and its applications. Topics include matrices, determinants, vector spaces, eigenvalues and eigenvectors, orthogonality and inner product spaces. Applications include brief introductions to difference equations, Markov chains, and systems of linear ordinary differential equations. May include computer use in solving problems. MATH 2930 and MATH 2940 are independent and can be taken in either order; they should not be taken in the same semester.

Full details for MATH 2940 - Linear Algebra for Engineers

A useful course for students who wish to improve their skills in mathematical proof and exposition, or who intend to study more advanced topics in mathematics. The methodology of proof provides a central tool for confirming the validity of mathematical assertions, functioning much as the experimental method does in the physical sciences. We will study various methods of mathematical proof, starting with basic techniques in propositional and predicate calculus and in set theory and combinatorics, then moving to applications and illustrations of these via topics in one or more of the three main pillars of mathematics: algebra, analysis, and geometry. Because cogent communication of mathematical ideas is important in the presentation of proofs, the course emphasizes clear, concise exposition.

Full details for MATH 3040 - Prove It!

Provides a transition from calculus to real analysis. Topics include rigorous treatment of fundamental concepts in calculus: including limits and convergence of sequences and series, compact sets; continuity, uniform continuity and differentiability of functions. Emphasis is placed upon understanding and constructing mathematical proofs.

Full details for MATH 3110 - Introduction to Analysis

A manifold is a type of subset of Euclidean space that has a well-defined tangent space at every point. Such a set is amenable to the methods of multivariable calculus. After reviewing some relevant calculus, this course investigates manifolds and the structures they are endowed with, such as tangent vectors, boundaries, orientations, and differential forms. The notion of a differential form encompasses such ideas as area forms and volume forms, the work exerted by a force, the flow of a fluid, and the curvature of a surface, space or hyperspace. We re-examine the integral theorems of vector calculus (Green, Gauss, and Stokes) in the light of differential forms and apply them to problems in partial differential equations, topology, fluid mechanics, and electromagnetism.

Full details for MATH 3210 - Manifolds and Differential Forms

A one-semester introduction to the theory and techniques of ordinary differential equations. Topics may include first-order and second-order differential equations, systems of linear differential equations, initial-value and two-point boundary-value problems, Sturm-Liouville theory, Sturm oscillation and comparison theory, the basic existence and uniqueness theorems, series solutions, special functions, and Laplace transforms. Applications from science and engineering may be included at the instructor’s discretion.

Full details for MATH 3270 - Introduction to Ordinary Differential Equations

An introductory course on number theory, the branch of algebra that studies the deeper properties of integers and their generalizations. Usually includes most of the following topics: the Euclidean algorithm, continued fractions, Pythagorean triples, Diophantine equations such as Pell's equation, congruences, quadratic reciprocity, binary quadratic forms, Gaussian integers, and factorization in quadratic number fields. May include a brief introduction to Fermat's Last Theorem.

Full details for MATH 3320 - Introduction to Number Theory

Introduction to the theory and practice of mathematical modeling. We compare and contrast different types of mathematical models (discrete vs. continuous, deterministic vs. stochastic), focusing on advantages, disadvantages, and limits of applicability for each approach. Case-study format covers a variety of application areas including economics, physics, sociology, traffic engineering, urban planning, robotics, and resource management. Students learn how to implement mathematical models on the computer and how to interpret/describe the results of their computational experiments.

Full details for MATH 3610 - Mathematical Modeling

Basic problem-solving techniques and strategies for contest problems. Contest problems differ from standard mathematical instruction in that they are "contextless": the problem does not tell you which field it is in and sometimes goes to great lengths to disguise it. Understanding how to solve contest problems has applications to research, where knowing how to work with unknown concepts, how to attack a problem fluidly from the perspective of multiple fields, and different tricks for simplifying problems is extremely useful. This course will cover topics that appear in standard competitions (e.g., the Putnam exam), including combinatorics, number theory, geometry, and calculus. It will also teach strategies for understanding and attacking problems, more advanced proof techniques, and effective proof writing.

Full details for MATH 4040 - Patterns, Proofs, and Problems

Introduction to the rigorous theory underlying calculus, covering the real number system and functions of one variable. Topics typically include construction of the real number system, properties of the real number system, continuous functions, differential and integral calculus of functions of one variable, sequences and series of functions. Based entirely on proofs. The student is expected to know how to read and, to some extent, construct proofs before taking this course. More experience with proofs may be gained by first taking a 3000-level MATH course.

Full details for MATH 4130 - Honors Introduction to Analysis I

An introduction to nonlinear dynamics with applications to physics, engineering, biology, and chemistry. Emphasizes analytical methods, concrete examples, and geometric thinking. Topics include one-dimensional systems; bifurcations; phase plane; nonlinear oscillators; and Lorenz equations, chaos, strange attractors, fractals, iterated mappings, period doubling, renormalization. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.

Full details for MATH 4210 - Nonlinear Dynamics and Chaos

Covers complex variables, Fourier transforms, Laplace transforms and applications to partial differential equations. Additional topics may include an introduction to generalized functions. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course. Undergraduates who plan to attend graduate school in mathematics should take MATH 4180 rather than MATH 4220.

Full details for MATH 4220 - Applied Complex Analysis

Introduction to the fundamentals of numerical analysis: error analysis, approximation, interpolation, and numerical integration. In the second half of the course, we use these to build approximate solvers for ordinary and partial differential equations. Strong emphasis is placed on understanding the advantages, disadvantages, and limits of applicability for all the covered techniques. Computer programming is used to test the theoretical concepts throughout the course. Students will be expected to be comfortable writing proofs and have knowledge of programming. MATH 4250/CS 4210 and MATH 4260/CS 4220 can be taken independently from each other and in either order. Together they provide a comprehensive introduction to numerical analysis.

Full details for MATH 4250 - Numerical Analysis and Differential Equations

Introduction to linear algebra, including the study of vector spaces, linear transformations, matrices, and systems of linear equations. Additional topics include quadratic forms and inner product spaces, canonical forms for various classes of matrices and linear transformations. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course. Undergraduates who plan to attend graduate school in mathematics should take MATH 4330 instead of MATH 4310.

Full details for MATH 4310 - Linear Algebra

Honors version of a course in advanced linear algebra, which treats the subject from an abstract and axiomatic viewpoint. Topics include vector spaces, linear transformations, polynomials, determinants, tensor and wedge products, canonical forms, inner product spaces, and bilinear forms. Emphasis is on understanding the theory of linear algebra; homework and exams include at least as many proofs as computational problems. Strong proficiency in writing proofs is expected. More experience with proofs may be gained by first taking a 3000-level MATH course. MATH 4330-MATH 4340 is recommended for undergraduates who plan to attend graduate school in mathematics. For a less theoretical course that covers approximately the same subject matter as MATH 4330, see MATH 4310.

Full details for MATH 4330 - Honors Linear Algebra

Introduction to algebraic geometry and computational algebra. Students will learn how to compute a Gröbner basis for polynomials in many variables. Covers the following applications: solving systems of polynomial equations in many variables, solving diophantine equations in many variables, 3-colorable graphs, and integer programming. Such applications arise, for example, in computer science, engineering, economics, and physics. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.

Full details for MATH 4370 - Computational Algebra

Combinatorics studies discrete structures arising in mathematics, computer science, and many areas of application. Key topics include counting objects with specific properties (e.g., trees) and proving the existence of structures (e.g., matchings of all vertices in a graph). We cover basic questions in graph theory, including extremal graph theory (how large a graph must be to have a certain subgraph) and Ramsey theory (large objects are forced to have structure). An introduction to network flow theory and variations on matching theory, including theorems of Dilworth, Hall, König, and Birkhoff, are discussed. Methods of enumeration (inclusion/exclusion, Möbius inversion, and generating functions) are applied to problems of counting permutations, partitions, and triangulations. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.

Full details for MATH 4410 - Introduction to Combinatorics I

Topology may be described briefly as qualitative geometry. This course begins with basic point-set topology, including connectedness, compactness, and metric spaces. Later topics may include the classification of surfaces (such as the Klein bottle and Möbius band), elementary knot theory, or the fundamental group and covering spaces. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.

Full details for MATH 4530 - Introduction to Topology

An introduction to probability theory that prepares the student to take MATH 4720. The course begins with basics: combinatorial probability, mean and variance, independence, conditional probability, and Bayes formula. Density and distribution functions and their properties are introduced. The law of large numbers and central limit theorem are stated and their implications for statistics are discussed.

Full details for MATH 4710 - Basic Probability

Introduction to classical theory of parametric statistical inference that builds on the material covered in STSCI 3080. Topics include: sampling distributions, principles of data reduction, likelihood, parameter estimation, hypothesis testing, interval estimation, and basic asymptotic theory.

Full details for MATH 4720 - Theory of Statistics

First course in mathematical logic providing precise definitions of the language of mathematics and the notion of proof (propositional and predicate logic). The completeness theorem says that we have all the rules of proof we could ever have. The Gödel incompleteness theorem says that they are not enough to decide all statements even about arithmetic. The compactness theorem exploits the finiteness of proofs to show that theories have unintended (nonstandard) models. Possible additional topics: the mathematical definition of an algorithm and the existence of noncomputable functions; the basics of set theory to cardinality and the uncountability of the real numbers. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking CS 2800 or a 3000-level MATH course.

Full details for MATH 4810 - Mathematical Logic

An independent research course by arrangement with an individual professor. The goal is for the student to perform an independent investigation into a specific mathematical question. The student and professor will set expectations and grading policies at the beginning of the term.

Full details for MATH 4900 - Supervised Research

An independent reading course by arrangement with an individual professor. The goal is for the student to master a body of mathematics outside the normal curriculum. The student and professor will set expectations and grading policies at the beginning of the term.

Full details for MATH 4901 - Supervised Reading

This independent study course offers math majors (i.e., undergraduates whose applications to affiliate with the math major have been approved) an opportunity to reflect on concepts from mathematics as they were encountered and applied in a recent internship. Students write a short paper describing their work experience and how it connects to the educational objectives of the mathematics major.

Full details for MATH 4997 - Practical Training in Mathematics

Covers complex variables, Fourier transforms, Laplace transforms and applications to partial differential equations. Additional topics may include an introduction to generalized functions. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.

Full details for MATH 5220 - Applied Complex Analysis

Introduction to the fundamentals of numerical analysis: error analysis, approximation, interpolation, and numerical integration. In the second half of the course, we use these to build approximate solvers for ordinary and partial differential equations. Strong emphasis is placed on understanding the advantages, disadvantages, and limits of applicability for all the covered techniques. Computer programming is required to test the theoretical concepts throughout the course. Students will be expected to be comfortable writing proofs and have knowledge of programming.

Full details for MATH 5250 - Numerical Analysis and Differential Equations

Combinatorics studies discrete structures arising in mathematics, computer science, and many areas of application. Key topics include counting objects with specific properties (e.g., trees) and proving the existence of structures (e.g., matchings of all vertices in a graph). We cover basic questions in graph theory, including extremal graph theory (how large a graph must be to have a certain subgraph) and Ramsey theory (large objects are forced to have structure). An introduction to network flow theory and variations on matching theory, including theorems of Dilworth, Hall, König, and Birkhoff, are discussed. Methods of enumeration (inclusion/exclusion, Möbius inversion, and generating functions) are applied to problems of counting permutations, partitions, and triangulations. Students will be expected to be comfortable writing proofs. More experience with proofs may be gained by first taking a 3000-level MATH course.

Full details for MATH 5410 - Introduction to Combinatorics I

MATH 6110-MATH 6120 are the core analysis courses in the mathematics graduate program. MATH 6110 covers abstract measure and integration theory, and related topics such as the Lebesgue differentiation theorem, the Radon-Nikodym theorem, the Hardy-Littlewood maximal function, the Brunn-Minkowski inequality, rectifiable curves and the isoperimetric inequality, Hausdorff dimension and Cantor sets, and an introduction to ergodic theory.

Full details for MATH 6110 - Real Analysis

Covers measure theory, integration, and Lp spaces.

Full details for MATH 6210 - Measure Theory and Lebesgue Integration

A mathematically rigorous course on lattices. Lattices are periodic sets of vectors in high-dimensional space. They play a central role in modern cryptography, and they arise naturally in the study of high-dimensional geometry (e.g., sphere packings). We will study lattices as both geometric and computational objects. Topics include Minkowski's celebrated theorem, the famous LLL algorithm for finding relatively short lattice vectors, Fourier-analytic methods, basic cryptographic constructions, and modern algorithms for finding shortest lattice vectors. We may also see connections to algebraic number theory.

Full details for MATH 6302 - Lattices: Geometry, Cryptography, and Algorithms

MATH 6310-MATH 6320 are the core algebra courses in the mathematics graduate program. MATH 6310 covers group theory, especially finite groups; rings and modules; ideal theory in commutative rings; arithmetic and factorization in principal ideal domains and unique factorization domains; introduction to field theory; tensor products and multilinear algebra. (Optional topic: introduction to affine algebraic geometry.)

Full details for MATH 6310 - Algebra

An introduction to the theory of noncommutative rings and modules. Topics vary by semester and include semisimple modules and rings, the Jacobson radical and Artinian rings, group representations and group algebras, characters of finite groups, representations of the symmetric group, central simple algebras and the Brauer group, representation theory of finite-dimensional algebras, and Morita theory.

Full details for MATH 6330 - Noncommutative Algebra

An introduction to number theory focusing on the algebraic theory. Topics include, but are not limited to, number fields, Dedekind domains, class groups, Dirichlet's unit theorem, local fields, ramification, decomposition and inertia groups, and the distribution of primes.

Full details for MATH 6370 - Algebraic Number Theory

Lie groups, Lie algebras, and their representations play an important role in much of mathematics, particularly in number theory, mathematical physics, and topology. This is an introductory course, meant to be useful for more advanced topics and applications. The relationship between Lie groups and Lie algebras will be highlighted throughout the course. A different viewpoint is that of algebraic groups. We will endeavor to discuss this along with the smooth viewpoint. Some knowledge of differential and algebraic geometry are helpful.

Full details for MATH 6390 - Lie Groups and Lie Algebras

An introduction to enumerative combinatorics from an algebraic, geometric and topological point of view. Topics include, but are not limited to, permutation statistics, partitions, generating functions and combinatorial species, various types of posets and lattices (distributive, geometric, and Eulerian), Mobius inversion, face numbers, shellability, and relations to the Stanley-Reisner ring.

Full details for MATH 6410 - Enumerative Combinatorics

MATH 6510-MATH 6520 are the core topology courses in the mathematics graduate program. This course is an introduction to geometry and topology from a differentiable viewpoint, suitable for beginning graduate students. The objects of study are manifolds and differentiable maps. The collection of all tangent vectors to a manifold forms the tangent bundle, and a section of the tangent bundle is a vector field. Alternatively, vector fields can be viewed as first-order differential operators. We will study flows of vector fields and prove the Frobenius integrability theorem. We will examine the tensor calculus and the exterior differential calculus and prove Stokes' theorem. If time permits, de Rham cohomology, Morse theory, or other optional topics will be covered.

Full details for MATH 6520 - Differentiable Manifolds

Measure theory, independence, distribution of sums of iid random variables, laws of large numbers, and central limit theorem. Other topics as time permits.

Full details for MATH 6710 - Probability Theory I

First course in axiomatic set theory at the level of the book by Kunen.

Full details for MATH 6870 - Set Theory

Selection of advanced topics from analysis. Course content varies.

Full details for MATH 7110 - Topics in Analysis

Selection of advanced topics from partial differential equations. Content varies.

Full details for MATH 7160 - Topics in Partial Differential Equations

Selection of advanced topics from dynamical systems. Content varies.

Full details for MATH 7280 - Topics in Dynamical Systems

Talks on various methods in scientific computing, the analysis of their convergence properties and computational efficiency, and their adaptation to specific applications.

Full details for MATH 7290 - Seminar on Scientific Computing and Numerics

A seminar on an advanced topic in topology or a related subject. Content varies. The format is usually that the participants take turns to present.

Full details for MATH 7510 - Berstein Seminar in Topology

Selection of advanced topics from modern geometry. Content varies.

Full details for MATH 7610 - Topics in Geometry

Supervised research for the doctoral dissertation.

Full details for MATH 7900 - Supervised Reading and Research