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

Reviews material presented in MATH 1120 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 1120 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 1012 - Academic Support for MATH 1120

Reviews material presented in MATH 2210 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 2210 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 1021 - Academic Support for MATH 2210

Introduction to linear algebra, probability, and Markov chains that develops the parts of the theory most relevant for applications. Topics include equations of lines, the method of least squares, solutions of linear systems, matrices; basic concepts of probability, permutations, combinations, binomial distribution, mean and variance, and the normal approximation to the binomial distribution. Examples from biology and the social sciences are used.

Full details for MATH 1105 - Finite Mathematics for the Life and Social Sciences

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

We apply mathematical reasoning to problems arising in the social sciences. We discuss game theory and its applications to questions of governing and the analysis of political conflicts. The problem of finding fair election procedures to choose among three or more alternatives is analyzed.

Full details for MATH 1340 - Strategy, Cooperation, and Conflict

How can you use writing to solve mathematical problems? Or even to discover problems that are worth solving? This course revisits high school mathematics with the goal of learning how mathematicians think about math. We will use writing to probe familiar math problems as well as discover new ones, and we will refine our ability to communicate their solutions. Readings will include excerpts from Lang’s Basic Mathematics and works by Gelfand. (These are high school math textbooks written by mathematicians with research careers.) The class will culminate in a research project: you will use writing to develop, investigate, and present your progress on a problem.

Full details for MATH 1890 - FWS: Writing in Mathematics

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 2240 include vector fields; line integrals; differential forms and exterior derivative; work, flux, and density forms; integration of forms over parametrized domains; and Green's, Stokes', and divergence theorems.

Full details for MATH 2240 - Theoretical Linear Algebra and Vector Calculus II

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

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

An introduction to structures of abstract algebra, including groups, rings, fields, factorization of polynomials and integers, congruences, and the structure of finite abelian groups. Additional topics include modules over Euclidean domain and Sylow theorems. Students considering graduate school in mathematics might consider taking MATH 4330 after MATH 3340.

Full details for MATH 3340 - Abstract Algebra

Introduction to the concepts and methods of abstract algebra that are of interest in applications. Covers the basic theory of groups, rings, and fields and their applications to such areas as public-key cryptography and error-correcting codes. Applications include the RSA cryptosystem and use of finite fields to construct error-correcting codes. Topics include elementary number theory, Euclidean algorithm, prime factorization, congruences, theorems of Fermat and Euler, elementary group theory, Chinese remainder theorem, factorization in the ring of polynomials, and classification of finite fields.

Full details for MATH 3360 - Applicable Algebra

A mathematical study of the formal languages of standard first-order propositional and predicate logic, including their syntax, semantics, and deductive systems. The basic apparatus of model theory will be presented. Various formal results will be established, most importantly soundness and completeness.

Full details for MATH 3810 - Deductive Logic

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

A proof-based introduction to further topics in analysis. Topics may include the Lebesgue measure and integration, functions of several variables, differential calculus, implicit function theorem, infinite dimensional normed and metric spaces, Fourier series, and ordinary differential equations.

Full details for MATH 4140 - Honors Introduction to Analysis II

A theoretical and rigorous introduction to complex variable theory recommended for students who plan to attend graduate school in mathematics. Topics include complex numbers, differential and integral calculus for functions of a complex variable, including Cauchy's theorem and the calculus of residues, elements of conformal mapping. Students will be expected to be comfortable writing proofs. For applications of complex analysis, consider MATH 4220 rather than MATH 4180.

Full details for MATH 4180 - Complex Analysis

Covers ordinary differential equations in one and higher dimensions: qualitative, analytic, and numerical methods. Emphasis is on differential equations as models and the implications of the theory for the behavior of the system being modeled and includes an introduction to bifurcations. 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 4200 - Differential Equations and Dynamical Systems

Introduction to the fundamentals of numerical linear algebra: direct and iterative methods for linear systems, eigenvalue problems, singular value decomposition. In the second half of the course, the above are used to build iterative methods for nonlinear systems and for multivariate optimization. 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.

Full details for MATH 4260 - Numerical Analysis: Linear and Nonlinear Problems

Topics are selected from first-order quasilinear equations, classification of second-order equations, with emphasis on maximum principles, existence, uniqueness, stability, and Fourier series methods. Additional topics as time permits. 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 4280 - Introduction to Partial 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 abstract algebra, which treats the subject from an abstract and axiomatic viewpoint, including universal mapping properties. Topics include groups, groups acting on sets, Sylow theorems; rings, Euclidean domains, factorization, structure theorem of finitely generated modules over a principal ideal domain; fields, root adjunction, finite fields, introduction to Galois theory. The course emphasizes understanding the theory with proofs in both homework and exams. MATH 4330-MATH 4340 is recommended for undergraduates who plan to attend graduate school in mathematics. For a less theoretical course that covers subject matter similar to MATH 4340, see MATH 3340.

Full details for MATH 4340 - Honors Introduction to Algebra

Algebraic combinatorics studies combinatorial problems with algebraic tools and algebraic problems with combinatorial tools. Topics covered include the study of the ring of symmetric functions, the symmetric group, and tableaux combinatorics.

Full details for MATH 4430 - Introduction to Algebraic Combinatorics

Matrix groups are central to mathematics and important in physics and engineering. The objects of study are classes of matrices (e.g., orthogonal, unitary, or symplectic) with both algebraic (groups) and geometric/topological (manifolds) structure. Thus the course is a mixture of algebra, geometry/topology, and a little analysis. Topics include Lie algebras (an extension of the notion of vector multiplication in three-dimensional space), the exponential mapping (a generalization of the exponential function of calculus), and representation theory (which studies different ways groups can be represented by matrices). Concrete examples will be emphasized. Background not included in the prerequisites will be developed as needed. 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 4500 - Matrix Groups

Differential geometry involves using calculus to study geometric concepts such as curvature and geodesics. This introductory course focuses on the differential geometry of curves and surfaces. It may also touch upon the higher-dimensional generalizations, Riemannian manifolds, which underlie the study of general relativity. 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 4540 - Introduction to Differential Geometry

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

A one-semester introduction to stochastic processes which develops the theory together with applications. Covers Markov chains in discrete and continuous time and Poisson processes. Other topics may include queuing theory, martingales, Brownian motion, and option pricing. This course may be useful to graduate students in the biological sciences or other disciplines who encounter stochastic models in their work but who do not have the background for more advanced courses such as ORIE 6500. 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 4740 - Stochastic Processes

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

Covers ordinary differential equations in one and higher dimensions: qualitative, analytic, and numerical methods. Emphasis is on differential equations as models and the implications of the theory for the behavior of the system being modeled and includes an introduction to bifurcations. 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 5200 - Differential Equations and Dynamical Systems

Algebraic combinatorics studies combinatorial problems with algebraic tools and algebraic problems with combinatorial tools. Topics covered include the study of the ring of symmetric functions, the symmetric group, and tableaux combinatorics.

Full details for MATH 5430 - Introduction to Algebraic Combinatorics

Provides graduate students with an introduction to core concepts in teaching and learning mathematics. Participants read and discuss articles and videos, reflect on their own teaching and learning experiences, and engage in collaborative activities that help them become more effective teachers, learners, and communicators. Students will observe peers and faculty and reflect on their observations. Especially valuable for those considering teaching mathematics at some point in their careers, or who want to improve their own mathematics learning skills. Students are expected to have taken the Math TA training prior to enrolling in the course.

Full details for MATH 6080 - Teaching and Learning Mathematics

MATH 6110-6120 are the core analysis courses in the mathematics graduate program. MATH 6120 covers complex analysis, Fourier analysis, and distribution theory.

Full details for MATH 6120 - Complex Analysis

Functional analysis is a branch of mathematical analysis that mainly focuses on the study of infinite-dimensional vector spaces and the operators acting upon them. It builds upon results and ideas from linear algebra and real and complex analysis to develop general frameworks that can be used to study analytical problems. Functional analysis plays a pivotal role in several areas of mathematics, physics, engineering, and even in some areas of computer science and economics. This course will cover the basic theory of Banach, Hilbert, and Sobolev spaces, as well as explore several notable applications, from analyzing partial differential equations (PDEs), numerical analysis, inverse problems, control theory, optimal transportation, and machine learning.

Full details for MATH 6220 - Applied Functional Analysis

This course is a self-contained introduction to the modern theory of optimal control and differential games. Dynamic programming uses Hamilton-Jacobi partial differential equations (PDEs) to encode the optimal behavior in cooperative and adversarial sequential decision-making problems. The same PDEs have an alternative interpretation in the context of front propagation problems. We show how both interpretations are useful in constructing efficient numerical methods. We also consider a wide range of applications, including robotics, computational geometry, path-planning, computer vision, photolithography, economics, seismic imaging, ecology, financial engineering, crowd dynamics, and aircraft collision avoidance. Assumes no prior knowledge of non-linear PDEs or numerical analysis.

Full details for MATH 6230 - Differential Games and Optimal Control

A high-level survey course on ten influential algorithms that shaped modern computational science and optimization. Each week focuses on a single algorithm - such as the Metropolis algorithm, simplex method, conjugate gradient, QR factorization, fast Fourier transform, fast multipole method - covering the historical context, key mathematical ideas, and modern applications. Students must have a background in linear algebra, calculus, and basic analysis.

Full details for MATH 6250 - Top Ten Algorithms of the 20th Century

This course studies the iteration of holomorphic maps in one complex variable, with emphasis on the dynamics of polynomials and rational maps on the Riemann sphere. We will cover Julia and Fatou sets, normal families, local and global behavior of periodic points, polynomial dynamics, puzzle techniques and local connectivity problems, quasiconformal methods, and renormalization. Central results include Sullivan’s no wandering domains theorem, the Douady– Hubbard straightening theorem, Yoccoz’s proof of the Siegel Brjuno theorem, and Thurston’s topological characterization of rational functions. The course will also give a glimpse of the connection between complex dynamics with Teichmuller theory, hyperbolic geometry, and geometric group theory.

Full details for MATH 6280 - Complex Dynamics

MATH 6310-MATH 6320 are the core algebra courses in the mathematics graduate program. MATH 6320 covers Galois theory, representation theory of finite groups, and introduction to homological algebra.

Full details for MATH 6320 - Algebra

MATH 6510-MATH 6520 are the core topology courses in the mathematics graduate program. MATH 6510 is an introductory study of certain geometric processes for associating algebraic objects such as groups to topological spaces. The most important of these are homology groups and homotopy groups, especially the first homotopy group or fundamental group, with the related notions of covering spaces and group actions. The development of homology theory focuses on verification of the Eilenberg-Steenrod axioms and on effective methods of calculation such as simplicial and cellular homology and Mayer-Vietoris sequences. If time permits, the cohomology ring of a space may be introduced.

Full details for MATH 6510 - Algebraic Topology

This course is an introduction to some of the fundamentals of homotopy theory. Homotopy theory studies spaces up to homotopy equivalence, not just up to homeomorphism. This allows for a variety of algebraic techniques which are not available when working up to homeomorphism. This class studies the fundamentals and tools of homotopy theory past homology and cohomology. Topics may include computations of higher homotopy groups, simplicial sets, model categories, spectral sequences, and rational homotopy theory.

Full details for MATH 6540 - Homotopy Theory

Topics include linear connections, Riemannian metrics and parallel translation; covariant differentiation and curvature tensors; the exponential map, the Gauss Lemma and completeness of the metric; isometries and space forms, Jacobi fields and the theorem of Cartan-Hadamard; the first and second variation formulas; the index form of Morse and the theorem of Bonnet-Myers; the Rauch, Hessian, and Laplacian comparison theorems; the Morse index theorem; the conjugate and cut loci; and submanifolds and the Second Fundamental form.

Full details for MATH 6620 - Riemannian Geometry

The second course in a graduate probability series. Topics include conditional expectation, martingales, Markov chains, Brownian motion, and (time permitting) elements of stochastic integration.

Full details for MATH 6720 - Probability Theory II

Introduction to model theory at the level of David Marker's text.

Full details for MATH 6830 - Model Theory

An introduction to (mostly Euclidean) harmonic analysis. Topics usually include convergence of Fourier series, harmonic functions and their conjugates, Hilbert transform, Calderon-Zygmund theory, Littlewood-Paley theory, pseudo-differential operators, restriction theory of the Fourier transform, connections to PDE. Applications to number theory and/or probability theory may also be discussed, as well as Fourier analysis on groups.

Full details for MATH 7150 - Fourier Analysis

Selection of advanced topics from number theory. Course content varies.

Full details for MATH 7370 - Topics in Number Theory

Selection of advanced topics from modern algebraic, differential, and geometric topology. Content varies.

Full details for MATH 7580 - Topics in Topology

Selection of topics from algebraic geometry. Content varies.

Full details for MATH 7670 - Topics in Algebraic Geometry

A twice weekly seminar in logic. Typically, a topic is selected for each semester, and at least half of the meetings of the course are devoted to this topic with presentations primarily by students. Opportunities are also provided for students and others to present their own work and other topics of interest.

Full details for MATH 7820 - Seminar in Logic

Supervised research for the doctoral dissertation.

Full details for MATH 7900 - Supervised Reading and Research