# Graduate Courses

The faculty has approval to offer the following courses in the academic years 2019–2020 and 2020–2021; however, not all courses are taught each semester or summer session. Students should consult the *Course Schedule* to determine which courses and topics will be offered during a particular semester or summer session. The *Course Schedule* may also reflect changes made to the course inventory after the publication of this catalog.

### Mathematics: M

##### M 380C. Algebra.

A survey of algebraic structures, including groups, fields, rings, and modules. Three lecture hours a week for one semester. Prerequisite: Graduate standing and consent of instructor or the graduate adviser.

##### M 380D. Algebra.

Continuation of Mathematics 380C. Three lecture hours a week for one semester. Prerequisite: Graduate standing, consent of instructor or the graduate adviser, and Mathematics 380C.

##### M 381C. Real Analysis.

Same as Computational Science, Engineering, and Mathematics 385R. Measure and integration over abstract spaces; Lebesgue's theory of integration and differentiation on the real line. Three lecture hours a week for one semester. Computational Science, Engineering, and Mathematics 385R and Mathematics 381C may not both be counted. Prerequisite: Graduate standing and consent of instructor or the graduate adviser.

##### M 381D. Complex Analysis.

Same as Computational Science, Engineering, and Mathematics 385S. Introduction to complex analysis. Three lecture hours a week for one semester. Computational Science, Engineering, and Mathematics 385S and Mathematics 381D may not both be counted. Prerequisite: Graduate standing and consent of instructor or the graduate adviser.

##### M 381E. Functional Analysis.

Introduction to functional analysis. Three lecture hours a week for one semester. Prerequisite: Graduate standing; Computational Science, Engineering, and Mathematics 385R or Mathematics 381C; and consent of instructor or the graduate adviser.

##### M 382C. Algebraic Topology.

Surfaces, covering spaces, fundamental group, and homology. Three lecture hours a week for one semester. Prerequisite: Graduate standing, an undergraduate course in topology, and consent of instructor or the graduate adviser.

##### M 382D. Differential Topology.

Continuation of Mathematics 382C. Manifolds and maps, differential forms, transversality, and intersection theory. Three lecture hours a week for one semester. Prerequisite: Graduate standing, consent of instructor or the graduate adviser, and Mathematics 382C.

##### M 382E. Advanced Algebraic Topology.

Continuation of Mathematics 382C. Three lecture hours a week for one semester. Prerequisite: Graduate standing and consent of instructor or the graduate adviser.

##### M 382F. Algebraic Topology.

Continuation of Mathematics 382E. Three lecture hours a week for one semester. Prerequisite: Graduate standing, consent of instructor or the graduate adviser, and Mathematics 382E.

##### M 382G. Differential Geometry.

Continuation of Mathematics 382D. Three lecture hours a week for one semester. Prerequisite: Graduate standing and consent of instructor or the graduate adviser.

##### M 383C. Methods of Applied Mathematics.

Same as Computational Science, Engineering, and Mathematics 386C. Topics include basic normed linear space theory; fixed-point theorems and applications to differential and integral equations; Hilbert spaces and the spectral theorem; applications to Sturm-Liouville problems; approximation and computational methods such as the Galerkin, Rayleigh-Ritz, and Newton procedures. Three lecture hours a week for one semester. Computational Science, Engineering, and Mathematics 386C and Mathematics 383C may not both be counted. Prerequisite: Graduate standing.

##### M 383D. Methods of Applied Mathematics.

Same as Computational Science, Engineering, and Mathematics 386D. Topics include distributions, fundamental solutions of partial differential equations, the Schwartz space and tempered distributions, Fourier transform, Plancherel theorem, Green's functions, Sobolev spaces, weak solutions, differential calculus in normed spaces, implicit function theorems, applications to nonlinear equations, smooth variational problems, applications to classical mechanics, constrained variational problems. Three lecture hours a week for one semester. Computational Science, Engineering, and Mathematics 386D and Mathematics 383D may not both be counted. Prerequisite: Graduate standing; and Computational Science, Engineering, and Mathematics 386C or Mathematics 383C.

##### M 383E. Numerical Analysis: Linear Algebra.

Same as Computational Science, Engineering, and Mathematics 383C, Computer Science 383C, and Statistics and Data Sciences 393C. Survey of numerical methods in linear algebra: floating-point computation, solution of linear equations, least squares problems, algebraic eigenvalue problems. Three lecture hours a week for one semester. Only one of the following may be counted: Computational Science, Engineering, and Mathematics 383C, Computer Science 383C, Mathematics 383E, Statistics and Data Sciences 393C, Statistics and Scientific Computation 393C. Prerequisite: Graduate standing; Computer Science 367 or Mathematics 368K; and Mathematics 340L, 341, or consent of instructor.

##### M 383F. Numerical Analysis: Interpolation, Approximation, Quadrature, and Differential Equations.

Same as Computational Science, Engineering, and Mathematics 383D, Computer Science 383D, and Statistics and Data Sciences 393D. Survey of numerical methods for interpolation, functional approximation, integration, and solution of differential equations. Three lecture hours a week for one semester. Only one of the following may be counted: Computational Science, Engineering, and Mathematics 383D, Computer Science 383D, Mathematics 383F, Statistics and Data Sciences 393D, Statistics and Scientific Computation 393D. Prerequisite: Graduate standing; Computational Science, Engineering, and Mathematics 383C, Computer Science 383C, Mathematics 383E, or Statistics and Data Sciences 393C (or Statistics and Scientific Computation 393C); and Mathematics 427K and 365C, or consent of instructor.

##### M 384C. Mathematical Statistics I.

Same as Computational Science, Engineering, and Mathematics 384R and Statistics and Data Sciences 384 (Topic 2). The general theory of mathematical statistics. Includes distributions of functions of random variables, properties of a random sample, principles of data reduction, an overview of hierarchical models, decision theory, Bayesian statistics, and theoretical results relevant to point estimation, interval estimation, and hypothesis testing. Three lecture hours a week for one semester. Only one of the following may be counted: Computational Science, Engineering, and Mathematics 384R, Mathematics 384C, Statistics and Data Sciences 384 (Topic 2), Statistics and Scientific Computation 384 (Topic 2). Prerequisite: Graduate standing; and Mathematics 362K and 378K, Statistics and Scientific Computation 382, or consent of instructor.

##### M 384D. Mathematical Statistics II.

Same as Computational Science, Engineering, and Mathematics 384S and Statistics and Data Sciences 384 (Topic 3). Continuation of Computational Science, Engineering, and Mathematics 384R and Mathematics 384C. Three lecture hours a week for one semester. Only one of the following may be counted: Computational Science, Engineering, and Mathematics 384S, Mathematics 384D, Statistics and Data Sciences 384 (Topic 3), Statistics and Scientific Computation 384 (Topic 3). Prerequisite: Graduate standing; Computational Science, Engineering, and Mathematics 384R, Mathematics 384C, or Statistics and Scientific Computation 384 (Topic 2); and Mathematics 362K and 378K, Statistics and Data Sciences 382 (or Statistics and Scientific Computation 382), or consent of instructor.

##### M 384E. Design and Analysis of Experiments.

Same as Computational Science, Engineering, and Mathematics 384U and Statistics and Data Sciences 384 (Topic 6). Design and analysis of experiments, including one-way and two-way layouts; components of variance; factorial experiments; balanced incomplete block designs; crossed and nested classifications; fixed, random, and mixed models; and split plot designs. Three lecture hours a week for one semester. Only one of the following may be counted: Computational Science, Engineering, and Mathematics 384U, Mathematics 384E, Statistics and Data Sciences 384 (Topic 6), Statistics and Scientific Computation 384 (Topic 6). Prerequisite: Graduate standing; and Mathematics 362K and 378K, Statistics and Data Sciences 382 (or Statistics and Scientific Computation 382), or consent of instructor.

##### M 384F. Design of Experiments.

Design of experiments, including 2n and 3n factorial experiments, confounding, fractional factorials, sequential experimentation, orthogonal arrays, D-optimal experiments, and response surface methodology. Three lecture hours a week for one semester. Prerequisite: Graduate standing, and Mathematics 378K or the equivalent or consent of instructor.

##### M 384G. Regression Analysis.

Same as Computational Science, Engineering, and Mathematics 384T and Statistics and Data Sciences 384 (Topic 4). Simple and multiple linear regression, inference in regression, prediction of new observations, diagnosis and remedial measures, transformations, and model building. Emphasis on both understanding the theory and applying theory to analyze data. Three lecture hours a week for one semester. Only one of the following may be counted: Computational Science, Engineering, and Mathematics 384T, Mathematics 384G, Statistics and Data Sciences 384 (Topic 4), Statistics and Scientific Computation 384 (Topic 4). Prerequisite: Graduate standing; and Mathematics 362K and 378K, Statistics and Data Sciences 382, (or Statistics and Scientific Computation 382), or consent of instructor.

##### M 384H. Multivariate Statistical Analysis.

Introduction to the general multivariate linear model; a selection of techniques, such as principle component, factor, and discriminant analysis. Three lecture hours a week for one semester. Prerequisite: Graduate standing and consent of instructor.

##### M 385C. Theory of Probability.

Same as Computational Science, Engineering, and Mathematics 384K. Three lecture hours a week for one semester. Computational Science, Engineering, and Mathematics 384K and Mathematics 385C may not both be counted. Prerequisite: Graduate standing and consent of instructor.

##### M 385D. Theory of Probability.

Same as Computational Science, Engineering, and Mathematics 384L. Continuation of Computational Science, Engineering, and Mathematics 384K and Mathematics 385C. Three lecture hours a week for one semester. Only one of the following may be counted: Computational Science, Engineering, and Mathematics 384L, Mathematics 384L, 385D. Prerequisite: Graduate standing; Computational Science, Engineering, and Mathematics 384K or Mathematics 385C; and consent of instructor.

##### M 387C. Numerical Analysis: Algebra and Approximation.

Same as Computational Science, Engineering, and Mathematics 383K. Advanced introduction to scientific computing, theory and application of numerical linear algebra, solution of nonlinear equations, and numerical approximation of functions. Three lecture hours a week for one semester. Computational Science, Engineering, and Mathematics 383K and Mathematics 387C may not both be counted. Prerequisite: Graduate standing, and consent of instructor or the graduate adviser.

##### M 387D. Numerical Analysis: Differential Equations.

Same as Computational Science, Engineering, and Mathematics 383L. Advanced introduction to the theory and practice of commonly used numerical algorithms for the solution of ordinary differential equations, and elliptic, parabolic, and hyperbolic partial differential equations. Three lecture hours a week for one semester. Prerequisite: Graduate standing; and Computer Science 383C, Mathematics 387C, or consent of instructor.

##### M 389C. Actuarial Case Studies.

Explores aspects of basic ratemaking, reserving, catastrophe modeling, and rate classification in a property & casualty actuarial context. Covers loss & premium trending, loss triangles, loss development, loss ratios, on-level premium, accident year vs. calendar year vs. policy year data. Three lecture hours a week for one semester. Prerequisite: Graduate standing and either 389J or 389U with a grade of at least C.

##### M 389D. Introduction to Financial Mathematics for Actuaries.

Covers the financial derivative topics on the Society of Actuary FM/2 exam: general derivatives, options, hedging, investment strategies, forwards, futures, and swaps. Covers option pricing techniques in the MFE/3F exam: binomial option pricing, Monte Carlo Valuation using risk neutral probabilities, and Black-Scholes. Three lecture hours a week for one semester. Prerequisite: Mathematics 389F.

##### M 389F. Theory of Interest.

Measurement of interest, present and accumulated value, amortization, sinking funds, bonds, duration, and immunization. Covers the interest theory portion of an exam of the Society of Actuaries and the Casualty Actuarial Society. Three lecture hours a week for one semester. Only one of the following may be counted: Actuarial Foundations 329, Mathematics 329F, 389F. Prerequisite: Graduate standing and Mathematics 408D or 408L.

##### M 389J. Probability Models with Actuarial Applications.

Introductory actuarial models for life insurance, property insurance, and annuities. With Mathematics 389P, covers the syllabus for the professional actuarial exam on model construction. Three lecture hours a week for one semester. Prerequisite: Graduate standing, and Mathematics 358K or 378K with a grade of at least C.

##### M 389P. Actuarial Statistical Estimates.

Statistical estimation procedures for random variables and related quantities in actuarial models. With Mathematics 389J, covers the syllabus for the professional actuarial exam on model construction. Three lecture hours a week for one semester. Prerequisite: Graduate standing; and Mathematics 341 or 340L, and 389J with a grade of at least C in each.

##### M 189S. Seminar on Actuarial Practice.

Presentations by working actuaries on current issues in actuarial practice. One lecture hour a week for one semester. Offered on the credit/no credit basis only. Prerequisite: Graduate standing; and Actuarial Foundations 329 or Mathematics 329F or Mathematics 389F, and M389J or 389U with a grade of at least C in each.

##### M 389T. Time Series and Survival-Model Estimation.

Introduction to the probabilistic and statistical properties of time series; parameter estimation and hypothesis testing for survival models. Covers 30 percent of the syllabus for exam #4 of the Society of Actuaries and the Casualty Actuarial Society. Three lecture hours a week for one semester. Prerequisite: Graduate standing, Mathematics 341 or 340L, 358K or 378K, and 389U.

##### M 389U. Actuarial Contingent Payments I.

Intermediate actuarial models for life insurance, property insurance, and annuities. Three lecture hours a week for one semester. Prerequisite: Graduate standing; Mathematics 362K with a grade of at least C; credit with a grade of at least C or registration for Mathematics 340L (or 341); and credit with a grade of at least C or registration for Actuarial Foundations 329 or Mathematics 329F or 389F.

##### M 389V. Actuarial Contingent Payments II.

Advanced actuarial models for life insurance, property insurance, and annuities. Three lecture hours a week for one semester. Prerequisite: Graduate standing, and Mathematics 389F and 389U with a grade of at least C in each.

##### M 389W. Financial Mathematics for Actuarial Applications.

Subjects include pricing, stock price, and interest rate models for actuarial applications. Tools include lognormal distribution, Brownian motion, Black-Scholes, and delta hedging. Three lecture hours a week for one semester. Prerequisite: Mathematics 389D with a grade of at least C-.

##### M 390C. Topics in Algebra.

Recent topics have included algebraic geometry, number theory, algebraic curves, algebraic number theory, algebraic functions, rational curves on varieties, homological algebra. Three lecture hours a week for one semester. May be repeated for credit when the topics vary. Prerequisite: Graduate standing and consent of instructor.

##### M 391C. Topics in Analysis.

Recent topics have included measure and integration, real variables, complex analysis, functional analysis, ordinary differential equations, partial differential equations, integral transforms, operator theory, approximation theory, abstract harmonic analysis. Three lecture hours a week for one semester. May be repeated for credit when the topics vary. Prerequisite: Graduate standing and consent of instructor.

##### M 392C. Topics in Topology.

Recent topics have included algebraic topology, differential topology, geometric topology, Lie groups. Three lecture hours a week for one semester. May be repeated for credit when the topics vary. Prerequisite: Graduate standing and consent of instructor.

##### M 393C. Topics in Applied Mathematics.

Recent topics have included quantum mechanics, statistical physics, ergodic theory, group representations, statistical mechanics, quantum field theory, introductory partial differential equations, monotone operators and partial differential equations, Hilbert space methods for partial differential equations, Hamiltonian dynamics, nonlinear functional analysis, Euler and Navier-Stokes equations, microlocal calculus and spectral asymptotics, calculus of variations. Three lecture hours a week for one semester. May be repeated for credit when the topics vary. Prerequisite: Graduate standing and consent of instructor.

##### M 393N. Numerical Solution of Elliptic Partial Differential Equations.

Same as Computer Science 393N. The numerical solution of large systems of linear algebraic equations arising in the solution of elliptic partial differential equations by discretization methods. Three lecture hours a week for one semester. Computational Science, Engineering and Mathematics 393N and Mathematics 393N may not both be counted. Prerequisite: Graduate standing; and Computational Science, Engineering, and Mathematics 383K , Computer Science 386K, Mathematics 387C, or consent of instructor.

##### M 394C. Topics in Probability and Statistics.

Same as Computational and Applied Mathematics 394C. Recent topics have included nonparametric statistics, advanced probability. Three lecture hours a week for one semester. May be repeated for credit when the topics vary. Prerequisite: Graduate standing and consent of instructor.

##### M 395C. Topics in Logic and Foundations.

Recent topics have included set theory, model theory, proof theory, axiomatic theorem proving, automatic theorem proving, foundations of mathematics, recursion theory. Three lecture hours a week for one semester. May be repeated for credit when the topics vary. Prerequisite: Graduate standing and consent of instructor.

##### M 396C. Topics in Mathematics.

Recent topics have included set theory, history of mathematics. Three lecture hours a week for one semester. May be repeated for credit when the topics vary. Prerequisite: Graduate standing and consent of instructor.

##### M 396D. Conference Course.

Supervised study in mathematics. Conference course. May be repeated for credit. Offered on the credit/no credit basis only. Prerequisite: Graduate standing and consent of instructor.

##### M 397C. Topics in Numerical Analysis.

Recent developments and advanced topics in the field of numerical analysis. Three lecture hours a week for one semester. Mathematics 393D and 397C may not both be counted unless the topics vary. May be repeated for credit when the topics vary. Prerequisite: Graduate standing.

##### M 197S, 397S. Seminar in Mathematics.

One or three lecture hours a week for one semester. May be repeated for credit when the topics vary. Offered on the credit/no credit basis only. Prerequisite: Graduate standing and consent of instructor.

##### M 698. Thesis.

The equivalent of three lecture hours a week for two semesters. Offered on the credit/no credit basis only. Prerequisite: For 698A, graduate standing in mathematics and consent of the graduate adviser; for 698B, Mathematics 698A.

##### M 398R. Master's Report.

Preparation of a report to fulfill the requirement for the master's degree under the report option. The equivalent of three lecture hours a week for one semester. Offered on the credit/no credit basis only. Prerequisite: Graduate standing in mathematics and consent of the supervising professor and the graduate adviser.

##### M 398T. Supervised Teaching in Mathematics.

Three lecture hours a week for one semester. Offered on the letter-grade basis only. Prerequisite: Graduate standing and appointment as a teaching assistant.

##### M 399W, 699W, 999W. Dissertation.

May be repeated for credit. Offered on the credit/no credit basis only. Prerequisite: Admission to candidacy for the doctoral degree.