# Courses

Fall 2024 Offerings Fall 2024 Course Booklet Course Info on SIS Course Advising for New Students Archives

## Course Descriptions

The courses below include descriptions of all undergraduate and graduate courses offered by the Department of Mathematics, though some courses may be taught more often than others. Descriptions for special topics seminars are updated each semester.

Visit the undergraduate and graduate pages for course requirements for specific programs. For up-to-date information on course offerings, schedules, room locations and registration, please visit the Student Information System (SIS).

### Undergraduate Courses

**MATH 0014 Introduction To Finite Mathematics.** Topics selected from financial mathematics, matrix algebra, linear inequalities and linear programming, counting arguments, and statistics and probability. Recommendations: High school geometry and algebra. (Math 30 is not a prerequisite.) Engineering students are not permitted to take MATH 14 for credit.

**MATH 15 Mathematics In Antiquity.** (Cross-listed as CLS 15) History of mathematics in Babylonian, Egyptian, Greek, and other ancient civilizations. Number systems and computational techniques; achievements in elementary algebra, geometry, and number theory; famous results, proofs and constructions. Emphasis on solving problems in the style and spirit of each culture. Engineering students are not permitted to take MATH 15 for credit.

**MATH 16 Symmetry. **A mathematical treatment of the symmetries of wallpaper patterns. The main goal is to prove that the symmetries of these patterns fall into seventeen distinct types. In addition, students will learn to identify the symmetries of given patterns (with special emphasis on the periodic drawings of M.C. Escher) and to draw such patterns. Three lectures, one section. Recommendations: High school geometry. Engineering students are not permitted to take MATH 16 for credit.

**MATH 19 The Mathematics Of Social Choice.** Introduction to mathematical methods for dealing with questions arising from social decision making. Topics vary but usually include ranking, determining the strength of, and choosing participants in multicandidate and two-candidate elections, and apportioning votes and rewards to candidates. Recommendations: High school algebra. Engineering students are not permitted to take MATH 19 for credit.

**MATH 21 Introductory Statistics. **Descriptive data analysis, sampling and experimentation, basic probability rules, binomial and normal distributions, estimation, regression analysis, one and two sample hypothesis tests for means and proportions. The course may also include contingency table analysis, and nonparametric estimation. Applications from a wide range of disciplines. Recommendations: High school algebra and geometry.

**MATH 30 Introduction to Calculus.** Functions and their graphs, limits, derivatives, techniques of differentiation. Applications of derivatives, curve sketching, extremal problems. Integration: indefinite and definite integrals, some techniques of integration, Fundamental Theorem of Calculus. Logarithmic and exponential functions with applications. Recommendations: High school geometry and algebra. MATH 30 is a one-semester calculus course and is not adequate preparation for MATH 34. Students will receive an additional two credits (with grade) for passing MATH 32 after receiving credit for MATH 30. MATH 32 must be taken at Tufts and for a grade.

**MATH 32 Calculus I.** Differential and integral calculus: limits and continuity, the derivative and techniques of differentiation, extremal problems, related rates, the definite integral, Fundamental Theorem of Calculus, derivatives and integrals of trigonometric functions, logarithmic and exponential functions. Recommendations: High school geometry, algebra, and trigonometry. Students will receive an additional two credits (with grade) for passing MATH 32 after receiving credit for MATH 30. MATH 32 must be taken at Tufts and for a grade.

**MATH 34 Calculus II. **Applications of the integral, techniques of integration, separable differential equations, improper integrals. Sequences, series, convergence tests, Taylor series. Polar coordinates, complex numbers. Students may count only one of MATH 34 and MATH 36 for credit. Recommendations: MATH 32.

**MATH 39 Honors Calculus I-ii.** (Formerly MATH 17). The first course of the two-semester sequence of honors calculus. Intended for students who have had at least the AB syllabus of advanced placement mathematics in secondary school. Stresses the theoretical aspects of the subject, including proofs of basic results. Topics include: convergence of sequences and series; continuous functions, Intermediate Value and Extreme Value Theorems; definition of the derivative, formal differentiation, finding extrema, curve-sketching, Mean Value Theorems; basic theory of the Riemann integral, Fundamental Theorem of Calculus and formal integration, improper integrals; Taylor series, power series and analytic functions. Recommendations: AB syllabus of advanced placement mathematics. Students who receive credit for MATH 39 (formerly MATH 17) cannot receive credit for MATH 30, 32, or 34 (formerly MATH 5, 11, or 12). Upon successful completion of MATH 39, all students receive two credits

**MATH 42 Calculus III.** Vectors in two and three dimensions, applications of the derivative of vector-valued functions of a single variable. Functions of several variables, continuity, partial derivatives, the gradient, directional derivatives. Multiple integrals and their applications. Line integrals, Green's theorem, divergence theorem, Stokes’ theorem. Prerequisite: MATH 34 or 39.

**MATH 44 Honors Calculus III.** Analysis of real- and vector-valued functions of one or more variables using tools from linear and multilinear algebra; stress is on theoretical aspects of the subject, including proofs of basic results. Topics include: geometry and algebra of vectors in 3-space, parametrized curves and arc length, linear transformations and matrices; Jacobian and gradient of a real-valued function, Implicit Function Theorem, extrema, Taylor's Theorem and Lagrange multipliers; multiple integrals, differential forms and vector fields, line integrals, parametrized surfaces and surface integrals, exact and closed forms, vector calculus.

Recommendations: MATH 39 or permission of instructor. Students who receive credit for MATH 44 cannot receive credit for MATH 42.

**MATH 51 Differential Equations. **An introduction to linear differential equations with constant coefficients, linear algebra, and Laplace transforms. Recommendations: MATH 42 or 44.

**MATH 65 Bridge to Higher Mathematics.** Introduction to rigorous reasoning, applicable across all areas of mathematics, to prepare for proof-based courses at the 100 level. Writing proofs with precise reasoning and clear exposition. Topics may include induction, functions and relations, combinatorics, modular arithmetic, graph theory, and convergence of sequences and series of real numbers. Recommendations: MATH 34 or permission.

**MATH 70 Linear Algebra.** Introduction to the theory of vector spaces and linear transformations over the real or complex numbers, including linear independence, dimension, matrix multiplication, similarity and change of basis, inner products, eigenvalues and eigenvectors, and some applications. Recommendations: MATH 34 or 39 or permission of instructor. Students may count only one of MATH 70 and MATH 72 for credit.

**MATH 87 Mathematical Modeling And Computation. **A survey of major techniques in the use of mathematics to model physical, biological, economic, and other systems; topics may include derivative-based optimization and sensitivity analysis, linear programming, graph algorithms, probabilistic modeling, Monte-Carlo methods, difference equations, and statistical data fitting. This course includes an introduction to computing using a high-level programming language, and studies the transformation of mathematical objects into computational algorithms. Prerequisites: (1) MATH 34 , 36, or 39, and (2) Math 70 or 72, or permission of instructor. Recommendations: MATH 34, MATH 36 or MATH 39, or consent.

**MATH 102 Math-Education: From Numbers to Functions. **An integrated presentation of mathematics and pedagogy with applications to science and real life situations. Focus on the mathematical concepts and the pedagogical insights behind the following topics: real numbers, fractions and their multiple representations, introduction to functions: the intuitive and formal definition of function, composition of functions, representations through tables, graphs, dynagraphs, algebraic and verbal expressions, the vertical line criterion, composition of functions, examples of functions coming from arithmetic operations as well as functions commonly used in mathematics and science, functional approach to division with remainder, decimals and decimal representation of rational numbers, divisibility for integers and decomposition into product of powers of primes. Teaching projects with school age students are an integral part of this course. Offered on line with a face-to -face component. Permission of instructor.

**MATH 103 Math-Education: Transformations and Equations. **An integrated presentation of mathematics and pedagogy with applications to science and real life situations. Focus on the mathematical concepts and the pedagogical insights behind the following topics: transformations of the plane with an emphasis on the comparison with arithmetic operations and the action of transformations on the graphs of functions. Geometric and algebraic interpretations of equations. The use of transformations in the solutions of linear and quadratic equations. Divisibility for integers and polynomials, the euclidean algorithm for the greatest common divisor, divisibility and factorization of polynomials and it solution in the solution of polynomial equations. Teaching projects with school age students are an integral part of the course. This course is offered on line with a face-to -face component. Permission of instructor.

**MATH 104 Math-Education: Change and Invariance. **An integrated presentation of mathematics and pedagogy with applications to science and real life situations. Focus on the mathematical concepts and the pedagogical insights behind the following topics: Helping students with word problems. Functions of several variables. Linear systems of equations and their solutions. Limits of sequences and of functions, limits at infinity. Slope and rate of change for non-linear functions. The derivative function and applications. Teaching projects with school age students are an integral part of the course. This course is offered on line with a face-to -face component. Permission of instructor.

**MATH 110 Special Topics in Mathematics Education. **Intended for education students. Meets with a mid-level mathematics course emphasizing proofs (such as Math 63, 70, and 72). Additional content connects the mathematics to the students' teaching. Students have extra pedagogical responsibilities to be determined with the mathematics instructor and the instructor in the Education Department. The grade in the mathematics course will count for 75% of the course grade, and to pass, the student must receive at least a B+ in the mathematics course. Does not count for any degree in the Mathematics Department nor for A&S Distribution Credit in Mathematical Sciences. Permission of Instructor.

**MATH 112 Topics In The History Of Mathematics.** The evolution of mathematical concepts and techniques from antiquity to modern times. Recommendations: MATH 34 or 39 or permission of instructor.

**MATH 123 Mathematical Aspects of Data Analysis.** Dimension reduction and data compression via principal component analysis, and the singular value decomposition; k-means clustering; clustering via diffusion on weighted graphs; support vector machines; tensor data analysis; kernel trick. Homework includes programming. Prerequisite: MATH 42, and MATH 70 or MATH 72. Some prior programming experience desirable, but not required.

**MATH 125 Numerical Analysis.** (Cross-listed as CS 125.) Analysis of algorithms involving computation with real numbers. Interpolation, methods for solving linear and nonlinear systems of equations, numerical integration, numerical methods for solving ordinary differential equations. Recommendations: MATH 51 and programming ability in a language such as C, C++, Fortran, or Matlab.

**MATH 126 Numerical Linear Algebra.** (Cross-listed as CS 126) The two basic computational problems of linear algebra: solution of linear systems and computation of eigenvalues and eigenvectors. Recommendations: MATH 70 or 72 and CS 11.

**MATH 133 Complex Variables. **Introduction to the theory of analytic functions of a single complex variable, analytic functions, Cauchy's integral theorem and formula, residues, series expansions of analytic functions, conformal representation, entire and meromorphic functions, multivalued functions. Recommendations: MATH 42 or 44, or permission of instructor.

**MATH 135 Real Analysis I.** An introduction to analysis. Metric spaces (with Euclidean spaces as the primary example), compactness, connectedness, continuity and uniform continuity, uniform convergence, the space of continuous functions on a compact set, contraction mapping lemma with applications. Recommendations: either MATH 0042 or 0044 and MATH 0065; or permission of instructor

**MATH 136 Real Analysis II. **Applications of ideas from MATH 135 to further, in-depth study of functions on Euclidean spaces. Derivatives as linear maps, differentiable mappings, inverse and implicit function theorems. Further topics such as theory of the Riemann and Lebesgue integral, Hilbert spaces, and Fourier series. Recommendations: either MATH 0070 or 0072 and MATH 0135; or permission of instructor.

**MATH 145 Abstract Algebra I. **An introduction to the basic concepts of abstract algebra, including groups and rings. Recommendations: MATH 0065 and either MATH 0070 or 0072; or permission of instructor.

**MATH 146 Abstract Algebra II. **Further topics in groups and rings. Field extensions and Galois theory. Recommendations: Either MATH 0070 or 0072 and either MATH 0145 or 0245; or permission of instructor.

**MATH 151 Mathematical Neuroscience. **(Cross listed w/ BIO 151) Mathematical and computational study of systems of differential equations modeling nerve cells (equilibria, limit cycles, bifurcations), neuronal networks (intrinsic rhythmic synchronization, entrainment by external inputs), and learning (synaptic plasticity), and of the potential function of rhythmic synchrony for signaling among neuronal networks and for plasticity. Prerequisite: Math 51 or instructor’s consent.

**MATH 153 Ordinary Differential Equations. **Upper-level course in ordinary differential equations from an applied point of view. Equilibria, limit cycles, and their stability. Saddle-node, pitchfork, transcritical, Hopf, and homoclinic bifurcations. Chaotic dynamics, strange attractors, and fractal dimension. Strong emphasis on examples from the natural sciences. Prerequisite: Math 42 or Math 44, and at least one of the following three: Math 51, Math 70, Math 72.

**MATH 155 Partial Differential Equations I.** Introduction to partial differential equations, with emphasis on linear first- and second-order wave equations, diffusion equations, and the Laplace and Poisson equations. Prerequisites: MATH 70 or MATH 72, and MATH 51 or MATH 153. MATH 155 and ME 150 cannot both be taken for credit.

**MATH 156 Partial Differential Equations II.** Introduction to the theory of nonlinear partial differential equations, including the method of characteristics, weak solutions, shocks and jump conditions, nonlinear wave equations, nonlinear diffusion and reaction-diffusion equations, applications to fluid dynamics. Prerequisite: MATH 155 or permission of instructor.

**MATH 164 The Mathematics of Poverty and Inequality.** Mathematical description of wealth distribution (some distribution theory, Lorenz curves), and the quantification of inequality (Hoover index, Gini coefficient, Theil indices, Sen's properties of inequality metrics). Agent-based models of wealth distribution, random walks, Wiener processes, Boltzmann and Fokker-Planck equations, and their application to models of wealth distribution. Wealth condensation and weak solutions. Upward mobility and first-passage times. Methods of mathematical modeling and comparison with empirical observations are emphasized throughout. Prerequisites: MATH 42: Calculus III or equivalent; and MATH 51: Differential Equations or equivalent; or instructor permission. Recommended but not strictly necessary: MATH 135: Real Analysis or equivalent; and MATH 165: Probability or equivalent.

**MATH 165 Probability. **Probability, conditional probability, random variables and distributions, expectation, special distributions, joint distributions, laws of large numbers, and the central limit theorem.

Recommendations: MATH 42 or 44, or permission of instructor.

**MATH 166 Statistics.** Statistical estimation, sampling distributions of estimators, hypothesis testing, regression, analysis of variance, and nonparametric methods. Recommendations: MATH 165 or permission of instructor.

**MATH 171 Point-set Topology.** Introduction to the basic definitions and constructions of topology, with a goal of providing ideas and tools that are essential for further study of many branches of modern mathematics. The definition of a topological space, examples of topological spaces, continuous functions, compactness, connectedness, and separability. Other topics may include homeomorphisms, subspaces, the quotient topology, and countability axioms. Prerequisite: MATH 0065 or permission of instructor.

**MATH 175 Algebraic Topology.** Applications of algebra to the study of topological objects, with emphasis on surfaces. Surfaces as manifolds, homotopy of curves, fundamental group, simple connectedness, covering spaces, genus, Euler Characteristic, orientability, and the classification of compact surfaces. Recommendations: MATH 135 and 145.

**MATH 181 Computational Geometry.** (Cross-listed as CS 163.) Design and analysis of algorithms for geometric problems. Topics include proof of lower bounds, convex hulls, searching and point location, plane sweep and arrangements of lines, Voronoi diagrams, intersection problems, decomposition and partitioning, farthest-pairs and closest-pairs, rectilinear computational geometry. Recommendations: CS 160 or permission of instructor.

**MATH 185 Differential Geometry. **Study of basic notations of differential geometry in the context of curves and surfaces. Curvature and torsion, implicit function theorem, coordinate systems, first and second fundamental forms, geodesics, Gauss-Bonnet theorem. Recommendations: MATH 70 or 72, and 135, or permission of instructor.

**MATH 190 Advanced Special Topics. **Content and prerequisites vary from semester to semester. Topics covered in recent years have included mathematical neuroscience, Lie algebras, and nonlinear dynamics and chaos.

**MATH 191 Computation Theory.** (Cross-listed as CS 170). Models of computation: Turing machines, pushdown automata, and finite automata. Grammars and formal languages, including context-free languages and regular sets. Important problems, including the halting problem and language equivalence theorems.

Recommendations: CS 15 and MATH 61.

**MATH 192 Seminars In Mathematics.** Attendance at department seminars and colloquia. May include research, teaching-based, and/or student-run seminars with significant math content and/or an outside speaker. Attendance at 10 seminars required for passing grade with up to two outside Tufts allowed if approved by instructor. Prerequisites: Graduate Standing or consent.

**MATH 195 Senior Honors Thesis A. **Thesis course for thesis honors candidates; see Thesis Honors Program for details. Open to seniors. This is a yearlong course. Each semester counts as 4 credits towards a student’s credit load. Students will earn 8 credits at the end of the second semester.

**MATH 196 Senior Honors Thesis B. **Thesis course for thesis honors candidates; see Thesis Honors Program for details. Open to seniors. This is a yearlong course. Each semester counts as 4 credits towards a student’s credit load. Students will earn 8 credits at the end of the second semester.

### Graduate Courses

**MATH 220 Special Topics in Numerical Analysis. **A special topics course in the field of Numerical Analysis or Numerical Linear Algebra. Topics change from year to year, and the course may be taken more than once for credit.

**MATH 225 Numerical Analysis.** (Cross-list w/ CS 226) Analysis of algorithms involving computation with real numbers. Interpolation, approximation, orthogonal polynomials, methods for solving linear and nonlinear systems of equations, integration including Gaussian quadrature, ordinary differential equations including A-stability, introduction to methods for hyperbolic partial differential equations: upwinding, Lax-Friedrichs, Lax-Wendroff. Strong theoretical emphasis. Prerequisites: Math 51 or 153, Math 70 or 72, and graduate standing; or permission of instructor.

**MATH 226 Numerical Linear Algebra. **(Cross-list w/ CS 228) Design and analysis of algorithms for solving linear systems of equation, least squares problems, and eigenvalue problems, with a strong emphasis on matrix theory. Unitary matrices (including projections, rotations, and reflections). Matrix factorizations (including LU, Cholesky, QR, and the singular value decomposition). Conditioning; stability; perturbation analysis; operation counts. Krylov subspace methods (including theoretical analysis, preconditioning, and the connection to Gaussian quadrature). Applications. Prerequisites: Math 70 or 72 and graduate standing; or permission of instructor.

**MATH 229 Graph Algorithms.** Mathematical theory and implementation of graph algorithms and their applications. Topics include basic spectral graph theory, shortest path, spanning trees, coloring, maximal independent set, matching, aggregations, sparsifiers, randomized algorithms, and multilevel methods. Prerequisites: Math 125 or 225, Math 126 or 226, and graduate standing; or permission of instructor.

**MATH 230 Special Topics in Analysis.** A special topics course in the field of Analysis. Topics change from year to year, and the course may be taken more than once for credit.

**MATH 233 Complex Analysis.** Analytic functions, power series. Integration in the complex plane, Cauchy's integral theorem and formulas. Entire functions. Singularities. Conformal mapping, Riemann mapping theorem. Prerequisites: Math 135 and graduate standing; or permission of instructor.

**MATH 235 Analysis.** Measure and integration: sigma-algebras, measurable sets and functions, Lebesgue measure and integration, Monotone/Dominated Convergence, Lp-spaces, Fubini-Tonelli theorem, bounded variation, absolute continuity, Radon Nikodym theorem, Carathéodory extension and abstract measure. Real functions and functionals: Banach spaces, Hilbert spaces, and topological vector spaces, linear functionals and representation theorems. Prerequisites: Math 135, Math 136, and graduate standing; or permission of instructor.

**MATH 237 Functional Analysis. **Topological vector spaces, seminorms and local convexity, Banach Steinhaus theorem, open mapping theorem, Hahn-Banach theorem, duality. Test functions and distributions, localization and supports of distributions. Fourier transforms, inversion, tempered distributions, Paley-Wiener theorem, Sobolev's lemma. Banach algebras, Gelfand transforms, spectral theory of bounded linear operators on Hilbert spaces. Prerequisites: Math 235 and graduate standing, or permission of instructor.

**MATH 245 Algebra I. **General properties of groups, rings, modules over a principal ideal ring, and field extensions. Prerequisites: Math 145 and graduate standing; or permission of instructor.

**MATH 246 Algebra II.** Foundational results in commutative algebra, algebraic geometry, and homological algebra. Prerequisites: Math 245; or permission of instructor.

**MATH 250 Special Topics in Differential Equations.** A special topics course in the field of Differential Equations (either Ordinary or Partial). Topics change from year to year, and the course may be taken more than once for credit.

**MATH 255 Partial Differential Equations I.** The theory of the Laplace, heat, and wave equations: Fundamental solutions, mean-value formulas, properties of solutions, Green's functions, energy methods, Duhamel's principle. Quasilinear first-order PDEs, including the Hamilton-Jacobi equation, the method of characteristics, hyperbolic conservation laws and systems thereof, shocks and entropy conditions. Other selected topics. Prerequisites: Math 51 or 153, Math 135, and graduate standing; or permission of instructor.

**MATH 256 Partial Differential Equations II.** Boundary-value problems of Sturm-Liouville type, separation of variables, special functions. Similarity solutions, transform methods, power-series solutions, the Cauchy-Kovalevskaya Theorem. Topics in functional analysis, including L^p spaces and derivatives and existence of weak solutions to second-order elliptic equations and linear evolution equations. Interior and boundary regularity. Topics in the calculus of variations. Other selected topics. Prerequisites: Math 255; or permission of instructor.

**MATH 257 Numerical Partial Differential Equations.** Mathematical theory and implementation of computational methods for the solution of partial differential equations (PDEs). Topics include finite-difference methods for hyperbolic PDEs, finite element methods for elliptic PDEs, and iterative linear solvers for large systems of linear equations. Analysis of consistency, stability, and accuracy using variational formulations and functional analysis. Prerequisites: Math 135 and 151/251 or consent

**MATH 260 Special Topics in Probability and Statistics. **A special topics course in the field of Probability and Statistics. Topics change from year to year, and the course may be taken more than once for credit.

**MATH 270 Special Topics in Topology.** A special topics course in the field of Topology. Topics change from year to year, and the course may be taken more than once for credit.

**MATH 275 Algebraic Topology I.** An introduction to the algebraic invariants assigned to topological spaces. Topics include topological manifolds, classification of surfaces, homotopy type, fundamental group, covering spaces, CW and/or simplicial complexes, introduction to homology. Prerequisites: graduate standing; or permission of instructor.

**MATH 276 Algebraic Topology II.** Homology, homological algebra, cohomology and Poincare duality. Group cohomology if time permits. Prerequisites: Math 275; or permission of instructor.

**MATH 280 Special Topics in Differential Geometry. **A special topics course in the field of Differential Geometry and/or Manifolds. Topics change from year to year, and the course may be taken more than once for credit.

**MATH 281 Advanced Computational Geometry.** (Cross-listed as CS 263.) Design and analysis of sequential, parallel, probabilistic, and approximation algorithms for geometry problems. Geometric data structures, complexity, searching, computation, and applications. Selected advanced topics.

Recommendations: CS 163 or permission of instructor.

**MATH 285 Manifolds. **Key examples of manifolds such as spheres, tori, projective spaces. Topics in manifolds, including quotients, submanifolds, regular level sets, Lie groups, and smooth maps between manifolds. Topics in tangent spaces, including differential and rank of a smooth map, regular level set theorem (implicit function theorem), vector fields, integral curves, and the Lie algebra of a Lie group. Topics in differential forms and integration, including: wedge product, pullback of forms, exterior derivatives, orientation, integral of an n-form, and Stokes' theorem. Prerequisites: Math 135, Math 136, Math 145, and graduate standing; or permission of instructor.

**MATH 286 Differential Geometry. **Topics on Riemannian Manifolds, including Riemannian metric, curves

and surfaces in three dimensions, affine connections, and Theorema Egregium. Connections and curvature using differential forms, geodesics, the exponential map, distance and volume, Gauss–Bonnet Theorem, and the De Rham Cohmology. Additional topics as time permits. Prerequisites: Math 285; or permission of instructor.

**MATH 287 Lie Groups.** Real and complex Lie groups, relations between Lie groups and Lie algebras, exponential map, the adjoint representation, homogeneous manifolds, semisimplicity, maximal tori, root space decompositions, compact forms, Cartan decompositions, and the classification of simple Lie algebras. Prerequisites: Math 285; or permission of instructor.

**MATH 290 Graduate Special Topics.** A special topics course in any generic field of Mathematics. Topics change

from year to year, and the course may be taken more than once for credit.

**MATH 291 Graduate Development Seminar. **Graduate-student-led training for teaching and public speaking about mathematics, and other professional skills. May involve group discussions with faculty about pedagogy and communication of research level mathematics to a broader audience. Intended for first-year PhD students in Mathematics. Meets once a week for 75 minutes. Students are also expected to attend either a research seminar or a colloquium each week, when offered. Math 291 is offered in the Fall and Math 292 in the Spring. Recommendations: PhD standing or consent of department.

**MATH 292 Graduate Development Seminar.** Graduate-student-led training for teaching and public speaking about mathematics, and other professional skills. May involve group discussions with faculty about pedagogy and communication of research level mathematics to a broader audience. Intended for first-year PhD students in Mathematics. Meets once a week for 75 minutes. Students are also expected to attend either a research seminar or a colloquium each week, when offered. Math 291 is offered in the Fall and Math 292 in the Spring. Recommendations: PhD standing or consent of department.

**MATH 293 One-on-One Course.** Guided individual study of an approved topic.

**MATH 294 Internship in Mathematics.** Study of approved topics in Mathematics in concert with an internship in a related outside the University. Prerequisites: Graduate Standing and Permission of Instructor

**MATH 295 Master’s Thesis I. **Guided research on a topic that has been approved as suitable for a master's thesis.

**MATH 296 Master’s Thesis II.** Guided research on a topic that has been approved as suitable for a master's thesis.

**MATH 297 PhD Thesis I.** Guided research on a topic suitable for a doctoral dissertation.

**MATH 298 PhD Thesis II. **Guided research on a topic suitable for a doctoral dissertation.

**MATH 401 Master's Continuation, Part-time.** No description at this time.

**MATH 402 Master's Continuation, Full-time. **No description at this time.

**MATH 405 Grad Teaching Assistant.** No description at this time.

**MATH 406 Grad Research Assistant. **No description at this time.

**MATH 501 Doctoral Continuation, Part-time. **No description at this time.

**MATH 502 Doctoral Continuation, Full-time. **No description at this time.