### Mathematics

**School/College**: College of Letters and Science

**Degrees Conferred:**

- M.S. in Mathematics
- Ph.D. in Mathematics

#### Contents

- Graduate Faculty
- Master of Science in Mathematics
- Doctor of Philosophy in Mathematics
- Courses - Mathematical Sciences
- Courses - Mathematical Statistics
- Courses - Atmospheric Sciences

#### Overview

The Department of Mathematical Sciences offers graduate programs of study in mathematics with specializations in the fields of algebra, analysis, topology, applied mathematics, probability and statistics, actuarial science, atmospheric science, and industrial mathematics.

The programs of study at the master's level are designed to suit both the student intending to continue toward a Ph.D. as well as the student who wishes to begin a professional career upon completion of the master's program.

The student may prepare for a career in teaching at the secondary or college level and for a career in research in the academic, industrial, government, or business communities.

#### Graduate Faculty

- Distinguished Professor
- Roebber, Paul, Ph.D., McGill University
- Tsonis, Anastasios, Ph.D., McGill University

- Professors
- Ancel, Fredric D., Ph.D., University of Wisconsin-Madison
- Beder, Jay, Ph.D., George Washington University
- Brazauskas, Vytaras, Ph.D., University of Texas-Dallas
- Fan, Dashan, Ph.D., Washington University
- Ghorai, Jugal, Ph.D., Purdue University
- Guilbault, Craig R., Ph.D., University of Tennessee-Knoxville
- Kahl, Jonathan, Ph.D., University of Michigan
- Key, Eric, Ph.D., Cornell University
- Larson, Vincent, Ph.D., Massachusetts Institute of Technology
- Musson, Ian M., Ph.D., University of Warwick, England
- Okun, Boris, Ph.D., SUNY at Binghamton
- Stockbridge, Richard, Ph.D., University of Wisconsin-Madison
- Swanson, Kyle, Ph.D., University of Chicago
- Volkmer, Hans, Ph.D., University of Konstanz
- Wade, Bruce A., Ph.D., University of Wisconsin-Madison
- Willenbring, Jeb, Ph.D., University of California-San Diego
- Xie, Dexuan, Ph.D., University of Houston
- Zou, Yi Ming, Ph.D., Indiana University

- Associate Professors
- Bell, Allen D., Ph.D., University of Washington, Chair
- Boyd, Suzanne L., Ph.D., Cornell University
- Brucks, Karen, Ph.D., North Texas State University
- Gervini, Daniel, Ph.D., University of Buenos Aires
- Hruska, G. Christopher, Ph.D., Cornell University
- Kravtsov, Sergey, Ph.D., Florida State University
- Lauko, Istvan, Ph.D., Texas Tech University
- Lin, Tzu-Chu, Ph.D., University of Iowa
- McLeod, Kevin, Ph.D., University of Minnesota
- Pinter, Gabriella, Ph.D., Texas Tech University
- Sun, Lijing, Ph.D., Wayne State University
- Zhu, Chao, Ph.D., Wayne State Unviersity

- Assistant Professors
- Evans, A. Clark, Ph.D., Florida State University
- Hinow, Peter, Ph.D., Vanderbilt University
- Wang, Lei, Ph.D., University of Michigan
- Wei, Wei, Ph.D., University of Waterloo

#### Master of Science in Mathematics

Five options for the master's degree are offered: the standard mathematics/statistics option (A), the industrial mathematics option (B), the atmospheric sciences option (C), the applied statistics option (D), and the actuarial science option (E). Students who plan to continue for a Ph.D. degree with a focus on mathematics/statistics should elect the standard mathematics/statistics option; those who seek the Ph.D. with a focus on atmospheric sciences should elect the atmospheric sciences option.

##### A. Standard Mathematics/Statistics Option

###### Admission

An applicant must meet Graduate School requirements plus these departmental requirements to be considered for admission to the program:

- Completion of three semesters of undergraduate calculus.
- At least 18 credits of acceptable undergraduate preparation beyond calculus.

Applicants may be admitted with specific program-defined course deficiencies provided that the deficiencies amount to no more than two courses.

The student is expected to satisfy deficiency requirements within three enrolled semesters. The deficiencies are monitored by the Graduate School and the individual graduate program unit. No course credits earned in making up deficiencies may be counted as program credits required for the degree.

###### Major Professor as Advisor

The student must have a major professor to advise and supervise the student's studies as specified in Graduate School regulations. The entering graduate student is assigned a temporary advisor by the Associate Chair for Graduate Programs.

###### Credits and Courses

Minimum degree requirement is 24 to 36 credits, depending upon which option the student chooses: either 24 credits from mathematics courses, at least 18 credits of which are numbered 700 or above; or 30 credits from mathematics courses, at least 12 credits of which are numbered 700 or above; or at least 36 credits in mathematics courses open for graduate credit.

Under the 30-credit option, up to 9 credits may be in approved courses from outside the Department. Under the 36 credit option, up to 12 credits may be taken in approved courses from outside the Department.

###### Thesis

A thesis is optional. A student choosing the thesis option must enroll in Math 790. A maximum of 3 credits of thesis may be counted toward the degree requirements. An acceptable thesis will represent an original contribution and may involve applications, a novel exposition, or computational aspects of a mathematical problem or theory. The student must pass an oral defense of the thesis.

###### Examination or Project

Each student who does not elect the thesis option must satisfy one of the following requirements:

- Pass a written comprehensive examination.
- Present a satisfactory oral and written report on a comprehensive project done under the supervision of a faculty advisor.

The project option is open only to students who complete the 36 credit graduation requirement. Students electing the project should register for 1 to 3 credits of the M.S. seminar 791. Students planning to continue for a Ph.D. should select the written comprehensive examination option.

###### Time Limit

Under the 24 or 30 credit option, the student must complete all degree requirements within five years of initial enrollment. Under the 36 credit option, the student must complete all degree requirements within seven years of initial enrollment.

##### B. Industrial Mathematics Option

###### Objective

The objective of the master's program in industrial mathematics is to enable students to acquire the fundamentals of applied mathematics in areas of classical and numerical analysis, differential equations and dynamical systems, and probability and statistics. At the same time, the connection of these fields to modeling of physical, biological, and engineering phenomena will be stressed by requiring credits outside of the Department of Mathematical Sciences. Students are to obtain practical experience in mathematical modeling and analysis during an internship or industrial project that will culminate in a thesis.

###### Admission

An applicant must meet the Graduate School requirements as well as the following departmental requirements to be considered for admission to the program:

- A bachelor's degree in an area of mathematical science (applied or pure mathematics, actuarial science, statistics, etc.), computer science, economics or finance, physics, engineering, or a related field.
- Completion of at least three semesters of undergraduate calculus plus at least 6 credits of acceptable mathematics courses requiring calculus.
- Knowledge of a high-level programming language.

Students satisfying only the minimum mathematics requirements will be expected to take courses that do not count toward the degree.

###### Major Professor as Advisor

The student must have a major professor to advise and supervise the student's studies. The entering graduate student is assigned an advisor by the chair of the Industrial Mathematics Committee. Depending on the thesis topic, the student may later change advisors.

###### General Requirements

A student must have completed, either prior to entering the program or by the time of graduation, courses in advanced calculus, numerical analysis, and ordinary differential equations. In addition, students must complete courses involving Fourier series, linear algebra, linear programming, mathematical modeling, partial differential equations, probability, and calculus-based statistics.

###### Credits and Course Work

At least 36 graduate credits in G or U/G courses at UWM are required, subject to the following regulations. A student must have:

- At least 18 credits from the list of approved industrial mathematics courses, including Math 701, 715, and at least 6 additional credits at or above the 600 level.
- At least 6 upper level (300 or above) credits of a coherent set of courses, approved by the advisor, in an application area (e.g., physics, engineering, business) outside of the Department. Students already proficient in an application area are expected to substitute mathematics courses.
- Not more than 6 credits in any combination of independent study or seminar or thesis (Math 790, 791, 792, 799, or 990);
- Not more than 12 credits below the 500 level from within the Department of Mathematical Sciences;
- Demonstrated knowledge of an advanced scientific programming language approved by the Industrial Mathematics Committee; and
- Advisor's prior written approval for every course.

###### Thesis

A thesis in which the student solves a mathematical problem with an industrial source is required. The student must work with the advisor/major professor from the start of the thesis through its completion, receiving his/her approval. The student must pass an oral defense before three faculty members.

###### Time Limit

Full-time students, without deficiencies, could be expected to complete the program in two years. All degree requirements must be completed within seven years of initial enrollment.

###### Special Recommendation

It is recommended that, by the time of graduation, students master the material presented in the following courses, either prior to enrolling or through course work: 313, 314, 564, 571, 601, 602, 701, 702, and 715. Students must work closely with their advisors to ensure satisfaction of the General, Course Work, and Thesis requirements for timely graduation.

###### Approved Industrial Mathematics Courses

- Applied Mathematics
- Math 307/308 Theoretical Mechanics
- Math 320 Introduction to Differential Equations
- Math 321 Vector Analysis
- Math 322 Introduction to Partial Differential Equations
- Math 371 Introduction to Stochastic Models in Finance
- Math 405 Mathematical Models and Applications
- Math 520 Non-Linear Differential Equations
- Math 521/522 Advanced Calculus
- Math 525 Introductory Theory of Differential Equations
- Math 535 Linear Algebra
- Math 581 Introduction to the Theory of Chaotic Dynamical Systems
- Math 601/602 Advanced Engineering Mathematics I/II
- Math 621/622 Introduction to Analysis
- Math 623 Complex Analysis
- Math 701/702 Industrial Mathematics I/II
- Math 703 Boundary Value Problems
- Math 705 Mathematical Fluid Dynamics
- Math 709 Differential Geometry
- Math 716 Ordinary Differential Equations
- Math 719 Partial Differential Equations
- Math 726 Introduction to Functional Analysis
- Math 727 Calculus of Variations
- Math 728 Integral Equations
- Math 801 Topics in Applied Mathematics: (Subtitle)
- Math 816/817 Advanced Ordinary Differential Equations I/II
- Math 819/820 Advanced Partial Differential Equations I/II
- Math 827 Fourier Analysis

- Numerical Analysis
- Math 313 Linear Programming and Optimization
- Math 314 Mathematical Programming and Optimization
- Math 413 Introduction to Numerical Analysis
- Math 414 Numerical Analysis
- Math 416 Computational Linear Algebra
- Math 715 Numerical Analysis
- Math 793 Scientific Computational Laboratory: (Subtitle)
- Math 813 Numerical Solution of Ordinary Differential Equations
- Math 814 Numerical Solution of Partial Differential Equations
- Math 815 Topics in Numerical Analysis: (Subtitle)

- Probability and Statistics
- MthStat 361/362 Introduction to Mathematical Statistics I/II
- MthStat 461/462 Data Analysis and Graphing Using SAS-I/II
- Math 471 Introduction to the Theory of Probability
- MthStat 561 Analysis of Variance
- MthStat 562 Design of Experiments
- MthStat 563 Regression Analysis
- MthStat 564 Time Series Analysis
- MthStat 565 Nonparametric Statistics
- MthStat 567 Statistical Methods in Reliability
- MthStat 568 Multivariate Statistical Analysis
- MthStat 569 Advanced Biostatistics
- Math 571 Introduction to Probability Models
- MthStat 761/762 Mathematical Statistics I/II
- Math 768 Applied Stochastic Processes
- MthStat 861/862 Decision Theory I/II
- MthStat 863 Hypothesis Testing
- MthStat 869 Advanced Topics in Mathematical Statistics
- Classes in Biostatistics at the Medical College of Wisconsin

##### C. Atmospheric Sciences Option

###### Admission

An applicant must meet Graduate School requirements to be considered for admission to the program. Entering graduate students should have a general background in both mathematics and physics; given the intrinsic multi-disciplinary nature of the atmospheric sciences, no specific undergraduate course work is required. However, applicants should have an adequate mathematical background that includes calculus, vector analysis, ordinary and partial differential equations, and linear algebra. Students lacking this background may be admitted provided that the deficiencies amount to no more than two courses. Deficiencies must be made up within three enrolled semesters of graduate study.

###### Major Professor as Advisor

The student must have a major professor to advise and supervise the student's studies as specified in Graduate School regulations. This advisor will be assigned upon enrollment in the program by the chair of the Atmospheric Sciences Committee.

###### Credits and Courses

The minimum degree requirement is 30 graduate credits, 12 of which must be in the atmospheric sciences 700 number sequence, 6 of which will be approved graduate elective credits, and 6 of which will be from either the mathematics sequences 521/522 or 601/602, or from two of the following courses: Math 703, Math 705, Math 801, and AtmSci 750.

###### Thesis

A thesis is optional. Students who choose the thesis option must enroll in Atm Sci 990 for the final 6 credits of the required total. An acceptable thesis will represent an original contribution to knowledge in the atmospheric sciences. Upon completion of the thesis, students must pass an oral examination to defend the thesis.

###### Comprehensive Examination

In the non-thesis option, students must pass a written comprehensive examination that tests basic knowledge of the atmospheric sciences.

###### Time Limit

Full-time students, without deficiencies, can be expected to complete the program within two years. All degree requirements must be completed within five years of initial enrollment.

##### D. Applied Statistics Option

The M.S. Option in Applied Statistics is designed for students who will seek employment as statisticians in non-academic settings after obtaining a M.S. degree. Students desiring to obtain a Ph.D. with a concentration in statistics should follow Option A: Standard Mathematics/Statistics Option.

###### Admission

An applicant must meet Graduate School requirements plus the following departmental requirements to be considered for admission to the program: completion of three semesters of undergraduate calculus and at least 18 credits of acceptable undergraduate preparation beyond calculus; these credits should include a one year sequence in mathematical statistics.

###### Major Professor as Advisor

The student must have a major professor to advise and supervise the student's studies as specified in Graduate School regulations. The entering graduate student is assigned a temporary advisor by the Department Graduate Program Coordinator.

###### Credit and Courses

The minimum requirement is 32 credits in Mathematics or Mathematical Statistics. All students must complete Math 535, 571, 621, and 622 and MthStat 462, 761, and 762. In addition, each student must complete 12 approved credits of applied statistics from UWM or the Medical College of Wisconsin.

###### Thesis Option

Students with grade point averages of 3.7 and no grades below B+ after their first 18 credits completed in the program will be eligible to write a thesis and earn 3 credits toward the degree. In addition, students who write a thesis are exempt from the Master's Proficiency exam.

###### Master's Proficiency Exam

Students who are not eligible for or who do not complete the thesis option are required to pass a written comprehensive examination.

###### Time Limit

Students must complete all degree requirements with 5 years of initial enrollment.

##### E. Actuarial Science Option

###### Objective

The program provides a mathematically rigorous education in actuarial science, prepares students for actuarial professional exams, and develops their economics and business reasoning skills. Students obtain thorough knowledge in the fundamentals of actuarial science such as applied probability models, applied statistics, credibility, financial economics, life contingencies, loss models, and risk theory. Emphasis is placed on developing skills that are highly valued by employers and thus are essential for a successful career as actuary. This program is intended for students who will seek employment as an actuary upon completion of the degree. Those interested in entering the department's Ph.D. program should consider a different Master's option.

###### Admission

Students with undergraduate degrees in mathematics, statistics, actuarial science, economics or a related area are eligible for admission. Applicants should have a strong mathematical background that includes three semesters of calculus, linear algebra, probability, and mathematical statistics. Students lacking this background may be admitted provided that the deficiencies amount to no more than two courses. Although not required, having one actuarial professional exam passed would be an asset.

###### Major Professor as Advisor

The student must have a major professor, selected from the members of the Actuarial Science Committee, to advise and supervise the student's studies. The entering student is assigned an advisor by the chair of the committee. Before the start of studies, each student in the program must develop a plan of study in consultation with the Committee.

###### Credits and Course Work

The minimum degree requirement is 27 or 30 credits. In order to qualify for the 27 credit option, a student must enter the program with two actuarial exams passed and a proficiency in applied statistics, economics, and finance, as determined by the Actuarial Committee. In order to graduate, the following requirements must be completed:

- Eighteen credits among MthStat 596, 597, 691, 692, 795, and either Math 571 or 768. Students already proficient in some of these areas may substitute up to six credits of other courses in actuarial science, probability or statistics at the 700 level or above. (All substitutions have to be approved by the Actuarial Science Committee and the Graduate Program Coordinator.)
- At least nine credits from the following list: Math 311, 790, 792, 799, MthStat 563, 564, Econ 701, 702, BusMgmt 705. Students already proficient in these areas but having less than two actuarial exams passed must substitute at least six credits of other courses in probability or statistics at the 700 level or above. Credits for Math 790, 792 and 799 can be counted toward the degree requirement only for students who have passed two actuarial professional exams and only when these courses cover topics in actuarial science, probability, or statistics.

Students who have completed program courses for undergraduate credit should discuss alternative graduate-level courses to substitute for those courses in their programs of study.

###### Thesis

A thesis is not required for the actuarial science option. Rather, students must pass three departmental written proficiency exams, which are based on the learning objectives of the actuarial professional exams P/1, FM/2, and one of MFE/3F, MLC, C/4. Waivers for departmental exams are granted for students who have passed the corresponding professional exams.

###### Professional Development

For future advancement in the field of actuarial science, "Validation by Educational Experience" (VEE) credits are required. VEE credits may be earned from the Society of Actuaries (SOA) and Casualty Actuarial Society (CAS) with a grade of B- or better in the following courses: MthStat 563, 564 (VEE-Applied Statistics); Econ 701, 702 (VEE-Economics); BusMgmt 705 (VEE-Corporate Finance). Courses taken at other universities may be used to meet the VEE requirement of the SOA/CAS.

###### Time Limit

Students who are eligible to complete the program with 27 credits may earn the degree in three semesters. Other full time students are expected to complete the program in two years. Students must complete all degree requirements within five years of initial enrollment.

#### Doctor of Philosophy in Mathematics

##### Admission

Applicant must meet Graduate School requirements plus departmental requirements as given for admission to the master's program. A master's degree is not a prerequisite for admission to this Ph.D. program.

##### Reapplication

A student who receives the master's degree must formally reapply for admission to the Graduate School before continuing studies toward the Ph.D.

##### Major Professor as Advisor

The student must have a major professor to advise and supervise the student's studies as specified in Graduate School regulations. The entering graduate student is assigned a temporary advisor by the Department Graduate Program Coordinator.

##### Course of Study

Minimum degree requirement is 54 graduate credits beyond the bachelor's degree, at least 27 of which must be earned in residence at UWM. The student, in consultation with the major professor, must select both a primary and a secondary area of specialization. The primary area may be chosen from one of the following seven fields with minimum credit requirements as shown. The secondary area may be chosen from another of these fields or may be chosen from another appropriate department. Minimum course requirements for all work in both areas of specialization require approximately two full years of study.

- Algebra field
- 12 credits in algebra
- 3 in complex analysis
- 3 in real analysis
- 3 in topology
- 3 in applied mathematics
- 3 outside the field

- Analysis field
- 3 credits in algebra
- 6 in complex analysis
- 6 in real analysis
- 3 in topology
- 3 in applied mathematics
- 3 outside the field

- Applied mathematics field
- 3 credits in algebra
- 6 in complex analysis
- 3 in real analysis
- 12 in applied mathematics
- 3 outside the field

- Probability and statistics
- 3 credits in complex analysis
- 6 in real analysis
- 12 in probability and statistics
- 3 in applied mathematics

- Topology field
- 3 credits in algebra
- 3 in complex analysis
- 3 in real analysis
- 12 in topology
- 3 in applied mathematics
- 3 outside the field

- Industrial mathematics field
- 3 credits in algebra or topology
- 3 in complex analysis
- 3 in real analysis
- 9 in applied mathematics
- 6 in probability and statistics
- 6 in approved credits outside the 600 and 601 curricular codes

- Atmospheric sciences field
- 12 credits in atmospheric sciences (synoptic, dynamic, cloud/radiative, air pollution)
- 9 in applied mathematics or probability and statistics
- 3 in analysis

##### Foreign Language

Except for students in the atmospheric sciences field, each student must pass a written examination in one foreign language; the examination is administered by the Department's Language Committee. Acceptable languages are French, German, and Russian; exceptions may be permitted upon written request of advisor.

##### Computer Proficiency

The student shall pass an examination on a higher programming language and/or other appropriate advanced computer skills; the examinations administered by the Department's Computer Committee. The Computer Committee may accept advanced computer science course work in lieu of the examination.

##### Residence

The student must meet minimum Graduate School residence requirements.

##### Doctoral Preliminary Examination

When the student is sufficiently prepared — normally when the student has earned 24 credits in specified areas above the 700 level — a doctoral preliminary examination to determine the student's knowledge and achievement is taken. For students in mathematics, the exam evaluates the student's general knowledge of mathematics, as well as the student's knowledge of the major area of concentration. Students in atmospheric sciences are examined in three areas: at least one from the 190 curricular area code and at least one from the 600/601 curricular area codes. Students must pass this examination to continue in the program. With permission of the examination committee, the student may repeat this examination once. If the student does not have a master's degree in mathematics before this examination, the committee will determine whether the student's performance is sufficient to qualify for the master's degree.

##### Doctoral Dissertation Proposal Hearing

After passing the language requirements and the doctoral preliminary examination, the student participates in a doctoral dissertation proposal hearing. At this hearing, the student is examined on the student's chosen area of research and a dissertation topic is approved.

##### Dissertation

The primary requirement for the Ph.D. in mathematics is the candidate's completion, under the supervision of the Department advisor, of an original and significant mathematical investigation presented in the form of a dissertation. The investigation is to be in the field of algebra, analysis, applied mathematics, probability and statistics, topology, or atmospheric sciences. A dissertation for the industrial mathematics field must involve an industrial problem requiring a mathematical solution.

##### Dissertation Defense

The candidate must, as the final step toward the degree, present a colloquium based on the dissertation and must pass an oral examination in defense of the dissertation. If the candidate does not successfully defend a thesis within five years of admission to candidacy, the candidate may be required to take another doctoral preliminary examination and be readmitted to candidacy.

##### Time Limit

All degree requirements must be completed within ten years from the date of initial enrollment in the doctoral program.

##### Minor Area for Other Ph.D. Majors

A doctoral student planning a physical science major other than mathematics may fulfill requirements for mathematics as the minor area of concentration by completing 12 credits of approved mathematics courses with a grade of B or better, at least 6 credits of which must be in courses 700 or above.

A doctoral student planning a non-physical science major may fulfill requirements for mathematics as the minor area of concentration by completing 12 credits with a grade of B or better in approved mathematics courses 300 or above.

For additional information on the Ph.D., see the Graduate School Doctoral Requirements page.

#### Courses - Mathematical Sciences

*Courses numbered 300-699 are Undergraduate/Graduate. Courses numbered 700 and above are Graduate only.*

- 311 Theory of Interest. 4 cr. U/G.
- Interest theory: annuities, ammortization schedules, portfolio immunization. Financial derivatives: bonds, forwards, options. Topics are consistent with syllabus of SOA Financial Mathematics Exam. Does not carry grad cr in Math Dept. Prereq: jr st & Math 232(P); or grad st.

- 314 Mathematical Programming and Optimization. 3 cr. U/G.
- Introduction to operations research. Network analysis; integer programming; game theory; nonlinear programming; dynamic programming. Prereq: jr st; Math 313(P) or cons instr; or grad st.

- 320 Introduction to Differential Equations. 3 cr. U/G.
- Elementary types and systems of differential equations, series solutions, numerical methods, Laplace transforms, selected applications. No grad cr in Math Sci. Prereq: jr st, Math 233(P) & Math 234(P) or ElecEng 234(P); or grad st.

- 321 Vector Analysis. 3 cr. U/G.
- Topics selected from vector algebra; scalar and vector fields; line, surface, and volume integrals; theorems of Green, Gauss, and Stokes; vector differential calculus. Prereq: jr st, Math 233(P) & Math 234(P) or ElecEng 234(P); or grad st.

- 322 Introduction to Partial Differential Equations. 3 cr. U/G.
- Partial differential equations of mathematical physics, boundary value problems in heat flow, vibrations, potentials, etc. Solved by Fourier series; Bessel functions and Legendre polynomials. Prereq: jr st, Math 320(P); or grad st.

- 371 Introduction to Stochastic Models in Finance. 3 cr. U/G.
- Elementary modeling of financial instruments for students in mathematics, economics, business, etc. Statistical and stochastic tools leading to the Black-Scholes model. Real data parameter fitting. Prereq: jr st & one of the following pairs; Econ 413(431)(P) & 506(P), Bus Adm 210(P) & 350(P), Bus Adm 701(P) & 702(P), or Math 234(P) & MthStat 361(P), or cons instr; or grad st.

- 405 Mathematical Models and Applications. 3 cr. U/G.
- Construction and mathematical models with applications to the social and life sciences. Models may involve Markov chains, linear programming, game theory, graph theory and growth processes. Prereq: jr st & either Math 234(P) or Math 205(P) & 211(P); or grad st.

- 413 Introduction to Numerical Analysis. 3 cr. U/G.
- Root finding and solution of nonlinear systems; direct solution of linear systems; interpolation & approximation of functions; least squares; fast Fourier transform; quadrature. Prereq: jr st, Math 233(C), & Math 234(C) or ElecEng 234(C); or grad st.

- 415 (414) Introduction to Scientific Computing. 3 cr. U/G.
- Nonlinear systems; iterative solution of linear systems; initial value problems in ordinary differential equations; boundary value problems in ordinary and partial differential equations. Prereq: jr st, Math 233(P), 234(P) or ElecEng 234(P); or grad st.

- 417 (416) Computational Linear Algebra. 3 cr. U/G.
- Direct solution of linear systems; iterative solution of linear systems; least squares; eigenvalue problems. Prereq: jr st & Math/ElecEng 234(P); or grad st.

- 431 Modern Algebra with Applications. 3 cr. U/G.
- Groups, rings, fields, Boolean algebras with emphasis on their applications to computer science and other areas. Does not carry grad cr in math sci. Prereq: jr st & Math 232(P); or grad st.

- 451 Axiomatic Geometry. 3 cr. U/G.
- An axiomatic approach to Euclidean and non-Euclidean geometry (historic role of the parallel postulate and models). Dept permission necessary for grad cr in math sci. Prereq: jr st, Math 341(241)(P), & Math 232(C); or grad st.

- 453 Transformations in Geometry. 3 cr. U/G.
- Selected topics from vector geometry and geometric transformations such as the study of invariants and conics. Recom for secondary school teachers. Departmental permission necessary for grad cr in math sci. Prereq: jr st, Math 341(241)(P), & Math 232(C); or grad st.

- 497 Study Abroad: (Subtitled). 1-12 cr. U/G.
- Designed to enroll students in UWM sponsored programs before course work level, content and credits are determined and/or in specially prepared program course work. Retakable w/chg in topic. Prereq: jr st; acceptance for Study Abroad Prog.

- 511 Symbolic Logic. 3 cr. U/G.
- First-order predicate calculus; formal properties of theoretical systems; chief results of modern mathematical logic; advanced topics such as completeness and computability. CompSci 511, Math 511 & Philos 511 are jointly offered; they count as repeats of one another. Prereq: jr st & either Philos 212(P) or 6 cr in math at the 300-level or above.

- 521 Advanced Calculus. 3 cr. U/G.
- Fundamental notions of sets and functions; limits, continuity; Riemann integral, improper integral; infinite series; uniform convergence; power series; improper integrals with a parameter. Prereq: jr st, Math 232(P), & 341(241)(P); or grad st. Math 233(R) & 234(R).

- 522 Advanced Calculus. 3 cr. U/G.
- Linear functions; differentiation of functions of several variables (implicit functions, Jacobians); change of variable in multiple integrals; integrals over curves, surfaces; Green, Gauss, Stokes theorems. Prereq: jr st, Math 233(P), 234(P) & 521(P); or grad st.

- 531 Modern Algebra. 3 cr. U/G.
- Integers; groups; rings; fields; emphasis on proofs. Prereq: jr st; Math 234(P) & 341(P).

- 535 Linear Algebra. 3 cr. U/G.
- Vector spaces; linear transformations and matrices; characteristic values and vectors; canonical forms; bilinear, quadratic, and Hermitian forms; selected applications. Prereq: jr st, Math 234(P) or 240(P), & Math 341(241)(P); or grad st.

- 537 Number Theory. 3 cr. U/G.
- Number theoretic functions; distribution of primes; Diophantine approximation; partitions; additive number theory; quadratic reciprocity. Prereq: jr st, Math 232(P) & 341(241)(P); or grad st.

- 551 Elementary Topology. 3 cr. U/G.
- General theory of point sets in Euclidean spaces, with emphasis on topology of two-dimensional and three-dimensional spaces; elementary notions of metric spaces; applications. Prereq: jr st & either Math 521(P) or 529(P); or grad st.

- 553 Differential Geometry. 3 cr. U/G.
- The theory of curves and surfaces by differential methods. Prereq: jr st, Math 233(P), 234(P) & 341(241)(P); or grad st.

- 571 (472) Introduction to Probability Models. 3 cr. U/G.
- Probability review, Markov chains in discrete and continuous time. Random walks, branching processes, birth and death processes. Queuing theory. Applications to physical sciences, engineering, mathematics. Prereq: jr st, Math 233(P), Math 234(P) or ElecEng 234(P), & one college-level course in statistics or probability; or grad st.

- 581 Introduction to the Theory of Chaotic Dynamical Systems. 3 cr. U/G.
- Iterated mappings, one parameter families, attracting and repelling periodic orbits, topological transitivity, Sarkovski's theorem, chaos, bifurcation theory, period doubling route to chaos, horseshoe maps, attractors. Prereq: jr st & Math 521(P), 529(P) or 621(P), or cons instr; or grad st.

- 601 Advanced Engineering Mathematics I. 3 cr. U/G.
- Sequences and series, elementary complex analysis; Fourier series; linear and nonlinear ordinary differential equations; matrix theory, elementary functional analysis; elementary solution of partial differential equations. Prereq: jr st; Math 234(P) or ElecEng 234(P); 3 cr Math at 300-level or above; or cons instr.

- 602 Advanced Engineering Mathematics II. 3 cr. U/G.
- Continuation of Math 601. Partial differential equations, Fourier and Laplace transforms, convolutions, special functions, mathematical modeling. Prereq: jr st; Math 601(P).

- 615 Numerical Solution of Partial Differential Equations. 3 cr. U/G.
- Finite difference solution of elliptic boundary value problems and of evolution problems; solution of hyperbolic conservation laws; finite volume methods; finite element methods. Prereq: jr st; Math 413(P), 415(414)(P), or 417(416)(P); Math 322(P) or 602(P); or cons instr.

- 617 Optimization. 3 cr. U/G.
- Unconstrained and constrained optimization: linear, nonlinear, and dynamic programming; barrier, penalty, and Lagrangian methods; Karush-Kuhn-Tucker theory, quadratic, and sequential quadratic programming; evolutionary algorithms. Prereq: jr st; Math 321(P) or 602(P); or grad st or cons instr.

- 621 Introduction to Analysis. 3 cr. U/G.
- Topology of Euclidean space; continuity; differentiation of real and vector-valued functions; Riemann-Stieltjes integration. Prereq: jr st, Math 233(P) 234(P), & 341(241)(P); or grad st.

- 622 Introduction to Analysis. 3 cr. U/G.
- Continues Math 621. Sequences and series of functions; uniform convergence; power series; functions of several variables; inverse and implicit function theorems; differential forms; Stokes' theorem. Prereq: jr st & Math 621(P); or grad st.

- 623 Complex Analysis. 3 cr. U/G.
- Complex numbers; definition and properties of analytic functions of a complex variable; conformal mapping; calculus of residues; applications to mathematics and physics. See also Math 713. Prereq: jr st & either Math 321(P) OR 521(P); or grad st.

- 631 Modern Algebra. 3 cr. U/G.
- Group theory, including normal subgroups, quotients, permutation groups, Sylow's theorems, Abelian groups; field theory; linear algebra over general fields. Prereq: jr st, Math 531(P) or cons instr; or grad st.

- 632 Modern Algebra. 3 cr. U/G.
- Continuation of Math 631. Ring theory, including ideals, quotient rings, Euclidean rings, polynomial rings, unique factorization; modules, including vector spaces, linear transformations, canonical forms; bilinear forms. Prereq: jr st & Math 631(P) or cons instr; or grad st.

- 675 Topics in Modern Mathematics: (Subtitled). 3 cr. U/G.
- Instructor and student presentations of modern topics. Specific topics and any additional prerequisites announced in Schedule of Classes each time course is offered. Retakable w/chg in topic to 9 cr max. Prereq: jr st, Math 233(P) & cons instr; or grad st.

- 701 Industrial Mathematics I. 3 cr. G.
- Elementary functional analysis, wavelets, control theory. Use of mathematical software emphasized throughout. Prereq: grad st in nat sci discipline; Math 522(P) or 602(P) or 622(P).

- 702 Industrial Mathematics II. 3 cr. G.
- Optimal control theory, digital signal processing, image processing, linear programming, nonlinear optimation, artificial neural networks. Use of mathematical software emphasized throughout. Prereq: grad st in nat sci discipline; Math 701(P).

- 703 Boundary Value Problems. 3 cr. G.
- Analytic methods for PDE's in mathematical physics, emphasis on green's functions. Theory of distributions, fundamental solutions, generalized eigenfunction expansions, generalized fourier and laplace transforms. Prereq: grad st; Math 322(P) & 623(P).

- 709 Differential Geometry. 3 cr. G.
- The theory of curves, surfaces, and manifolds in modern terminology. Global results on closed surfaces, geodesics, differential forms and tensor calculus.introduction to riemanniam geometry. Prereq: grad st; Math 522(P) or 622(P).

- 711 Theory of Functions of a Real Variable. 3 cr. G.
- Equivalence relations; cardinal and ordinal numbers; topology of real line; cantor and borel sets; lebesgue measure on real line; baire and measurable functions; lebesgue integral. Prereq: grad st; Math 522(P) & 551(P); or Math 622(P).

- 712 Theory of Functions of a Real Variable. 3 cr. G.
- Lebesgue integration; modes of convergence; lp spaces; vitali covering and lebesgue density theorems; dini derivates; differentiation; fundamental theorem of the lebesgue integral calculus; fubini's theorem. Prereq: grad st; Math 711(P).

- 713 Theory of Functions of a Complex Variable. 3 cr. G.
- Complex numbers; linear transformations; elementary functions; conformal mapping; complex integration; infinite sequences; dirichlet problem; multivalued functions. Prereq: grad st; Math 522(P) or 621(P).

- 714 Theory of Functions of a Complex Variable. 3 cr. G.
- Continuation of Math 713. Prereq: grad st; Math 713(P).

- 715 Numerical Analysis. 3 cr. G.
- Interpolation and approximation; differentiation and quadrature; numerical solution of ordinary differential equations; solution of linear and nonlinear algebraic equations. Prereq: grad st; Math 413(P); Math 521(P) or 621(P).

- 716 Ordinary Differential Equations. 3 cr. G.
- Existence and uniqueness theorems for systems of ode; qualitative properties of solutions, including stability and asymptotic behavior; general theory of linear systems; sturm-liouville problems. Prereq: grad st; Math 522(P) or 622(P).

- 719 Partial Differential Equations. 3 cr. G.
- First and second order equations; characteristics, cauchy problem; classical solutions of linear elliptic, parabolic and hyperbolic equations. Prereq: grad st; Math 522(P) or 622(P); math 320(P).

- 721 Abstract Measure and Integration. 3 cr. G.
- General theory of measures and integration; differentiation of set functions; relation to stochastic variables; atomic measures; haar measure and integral applications to probability theory. Prereq: grad st; Math 712(P).

- 726 Introduction to Functional Analysis. 3 cr. G.
- Basic notions of functional analysis in hilbert space will be introduced. The concepts will be illustrated by applications to elementary differential and integral equation problems. Prereq: grad st; Math 522(P) or 622(P).

- 731 Abstract Algebra. 3 cr. G.
- Basic course which is prerequisite for all other 700-799 level courses in algebra; groups, rings, fields, galois theory, modules, and categories. Prereq: grad st; Math 632(P); cons instr.

- 732 Abstract Algebra. 3 cr. G.
- Continuation of Math 731. Prereq: grad st; Math 731(P).

- 735 Theory of Groups. 3 cr. G.
- Topics selected from permutation groups; representations of groups and algebras; group algebras; group characters; extension problems; simple groups; solvable and nilpotent groups. Prereq: grad st; Math 732(P).

- 736 Theory of Rings and Modules. 3 cr. G.
- Noetherian and artinian rings and modules; primitive, prime and simple rings and ideals; radicals; localization; morita theory; construction and study of special classes of rings. Prereq: grad st; Math 732(P).

- 737 Theory of Rings and Modules. 3 cr. G.
- Continuation of Math 736. Prereq: grad st; Math 736(P) or cons instr.

- 751 Introductory Topology. 3 cr. G.
- Fundamental properties and examples of topological spaces and continuous functions, including compactness, connectedness, metrizability, completeness, product and quotient spaces, homeomorphisms, embedding, extension, and euclidean spaces. Prereq: grad st; Math 522(P) or 621(P).

- 752 Introductory Topology. 3 cr. G.
- Continuation of Math 751. Prereq: grad st; Math 751(P).

- 753 Introduction to Algebraic Topology. 3 cr. G.
- Homology theory; complexes and simplicial homology theory; general homology theories; cohomology rings; applications to manifolds, fixed point theorems, etc. Prereq: grad st; Math 632(P); Math 551(P) or 751(P) or cons instr.

- 754 Introduction to Algebraic Topology. 3 cr. G.
- Continuation of Math 753. Prereq: grad st; Math 753(P).

- 767 Statistical Methods for Engineers and Scientists. 3 cr. G.
- Elementary baysian decision theory; prior posterior and predictive distributions; posterior and pre-posterior analysis of two action decision problems; concept of likelihood functions for binomial, poisson, exponential and normal distributions; simple and multiple regression analysis; introduction to autoregressive models. Not open to students who have cr in ElecEng 767, which is identical to Math 767. Prereq: grad st; Math 362(P) or math 467(P).

- 768 Applied Stochastic Processes. 3 cr. G.
- Concepts in queueing theory; exponential channels; applications of markov chains to queueing problems; queue disciplines with priorities. Not open to students who have cr in ElecEng 768, which is identical to Math 768. Prereq: grad st; Math 361(P) or math 467(P).

- 771 Theory of Probability. 3 cr. G.
- Measure-theoretic foundations; limit-law theorems; weak and strong laws of large numbers; central limit problem; conditional expectations, martingales; stochastic processes. Prereq: grad st; Math 471(C) or 712(C).

- 781 Iterated Maps as Dynamical Systems. 3 cr. G.
- Periodic, recurrent and non-wandering points, kneading theory, unstable manifolds, unimodal mappings, turbulent and chaotic maps, symbolic dynamics, structural stability, topological conjugacy, topological dynamics. Prereq: grad st; Math 711(P) or cons instr.

- 790 Master's Thesis. 1-3 cr. G.
- Cr count toward masters degree only if student completes thesis option. Prereq: grad st; cons instr.

- 791 Master's Seminar. 1-3 cr. G.
- May not be taken for cr more than once. Prereq: grad st; cons instr.

- 792 Industrial Internship. 1-3 cr. G.
- Students earn credits for serving in an industrial internship that involves work of an advanced mathematical nature. They must prepare a report based on the internship. Retakable w/chg in topic to 6 cr max. Prereq: grad st; cons instr.

- 793 Scientific Computational Laboratory: (Subtitled). 1-2 cr. G.
- Retakable w/chg in topic to 6 cr max. Prereq: grad st; Math 715(C).

- 799 Seminar in Mathematics: (Subtitled). 1-3 cr. G.
- Specific topics and any additional prerequisites announced in Timetable each time course is offered. Retakable w/chg in topic to 9 cr max. Prereq: grad st & cons instr.

- 801 Topics in Applied Mathematics: (Subtitled). 3 cr. G.
- Specific topics and any additional prerequisites will be announced in the Timetable each time the course is offered. Retakable w/chg in topic to 9 cr max. Prereq: grad st; cons instr.

- 807 Group Theory and Its Applications to Physics. 3 cr. G.
- Representations of discrete and continuous groups, including rotation groups, unitary groups and crystal point and space groups. Symmetries of elementary particles. Molecular obitals, energy bands. Counts as a repeat of Physics 807. Prereq: grad st; Physics 532(P).

- 809 Topics in Differential Geometry: (Subtitled). 1-3 cr. G.
- Specific topics and any additional prerequisites will be announced in the Timetable each time the course is offered. Topics may be selected from Riemannian geometry, minimal surfaces and surfaces of prescribed mean curvature, geometric partial differential equations, or related areas of geometry. Retakable w/chg in topic to 9 cr max. Prereq: grad st; cons instr.

- 813 Numerical Solution of Ordinary Differential Equations. 3 cr. G.
- Methods for initial value and boundary value problems; stiff equations, singular points and bifurcation. Prereq: grad st; Math 715(P).

- 814 Numercal Solution of Partial Differential Equations. 3 cr. G.
- Finite difference and finite element methods for linear elliptic, parabolic and hyperbolic equations; nonlinear equations. Prereq: grad st; Math 715(P).

- 815 Topics in Numerical Analysis: (Subtitled). 3 cr. G.
- Retakable w/chg in topic to 9 cr max. Prereq: grad st; Math 715(P).

- 816 Advanced Ordinary Differential Equations. 3 cr. G.
- Existence and uniqueness theorems; singularity of solutions; oscillation and comparison theorems; poincare-bendixon theory. Prereq: grad st; Math 716(P).

- 817 Advanced Ordinary Differential Equations II. 3 cr. G.
- Continuation of Math 816; dynamical systems, bifurcation theory, topological methods. Prereq: grad st; Math 816(P).

- 821 Advanced Topics in Real Analysis: (Subtitled). 3 cr. G.
- Specific topics and any additional prerequisites will be announced in the Timetable each time the course is offered. Retakable w/chg in topic to 9 cr max. Prereq: grad st; Math 712(P).

- 825 Functional Analysis. 3 cr. G.
- Basic theorems of b-spaces and f-spaces including the closed graph; Hahn-Banach and Banach-Steinhaus theorems; Banach algebras; generalized functions; spectral theory. Prereq: grad st; Math 712(P).

- 841 Advanced Topics in Algebra: (Subtitled). 3 cr. G.
- Specific topics and any additional prerequisites will be announced in the Timetable each time the course is offered. Retakable w/chg in topic to 9 cr max. Prereq: grad st; Math 732(P); cons instr.

- 843 Homology. 3 cr. G.
- Modules; diagrams; categories; functors; complexes; cohomology; extensions; resolutions; injective and projective systems; graded modules; homological dimension; spectral sequences; derived functors. Prereq: grad st; Math 731(P).

- 844 Homology. 3 cr. G.
- Continuation of Math 843. Prereq: grad st; Math 843(P).

- 851 Advanced Topics in Topology: (Subtitled). 3 cr. G.
- Specific topics and any additional prerequisites will be announced in the Timetable each time the course is offered. Retakable w/chg in topic to 9 cr max. Prereq: grad st; Math 752(P); cons instr.

- 873 Advanced Topics in Probability: (Subtitled). 3 cr. G.
- Specific topics and any additional prerequisites will be announced in the Timetable each time the course is offered. Retakable w/chg in topic to 9 cr max. Prereq: grad st; cons instr.

- 881 Topics in Nonlinear Dynamics: (Subtitled). 3 cr. G.
- Specific topics and any additional prerequisites will be announced in the Timetable each time the course is offered. Retakable w/chg in topic to 9 cr max. Prereq: grad st; Math 711(P); cons instr.

- 888 Candidate for Degree. 0 cr. G.
- Available for graduate students who must meet minimum credit load requirement. Fee for 1 cr assessed. Prereq: grad st.

- 990 Reading and Research. 1-6 cr. G.
- To be arranged with your instructor and department chair. Retakable. Prereq: grad st.

#### Courses - Mathematical Statistics

*Courses numbered 300-699 are Undergraduate/Graduate. Courses numbered 700 and above are Graduate only.*

- 361 Introduction to Mathematical Statistics I. 3 cr. U/G.
- Probability spaces; discrete and continuous, univariate and multivariate distributions; moments; independence, random sampling, sampling distributions; normal and related distributions; point and interval estimation. Not recom for grad students in math, or students not planning to take MthStat 362. Prereq: jr st; Math 233(P).

- 362 Introduction to Mathematical Statistics II. 3 cr. U/G.
- Testing statistical hypothesis; linear hypothesis; regression; analysis of variance and experimental designs; distribution-free methods; sequential methods. Not recom for grad students in math. Prereq: jr st; MthStat 361(P).

- 462 Data Analysis and Graphing Using SAS-II. 2 cr. U/G.
- Continuation of MthStat 461. Procedures GLM, LIFEREG, LIFETEST, LOGISTIC, PROBIT and advanced GRAPHING. Offered second half of sem. U cr does not count toward math sci major. Prereq: jr st; MthStat 461(P) or cons instr.

- 465 Introductory Mathematical Statistics for Social Sciences and Education. 3 cr. U/G.
- Probability distributions; parameter estimation and confidence intervals; hypothesis testing; applications. Not open for cr to students w/cr in MthStat 467, 362, or for grad cr in math. Not open for cr toward major in math except in School of Education. Prereq: jr st; Math 211(P) or 232(P).

- 467 Introductory Statistics for Physical Sciences and Engineering Students. 3 cr. U/G.
- Concepts of probability and statistics; probability distributions of engineering applications; sampling distributions; hypothesis testing, parameter estimation; experimental design; regression analysis. Not open for cr for Math majors or students with cr in MthStat 362 or 465. Ind Eng 467 & MthStat 467 are jointly offered & count as repeats of one another. Prereq: jr st; Math 233(P).

- 469 Biostatistics. 3 cr. U/G.
- Simple distributions, estimation and hypothesis testing, simple regression, analysis of variance, nonparametric methods in biology. Demography and vital statistics and bioassay and clinical trials. Not allowed as part of core curric for Math majors. Not open for cr to students with cr in MthStat 569 & not open for grad cr in Math. Prereq: jr st; an elementary stats course.

- 562 Design of Experiments. 3 cr. U/G.
- Latin squares; incomplete block designs; factorial experiments; confounding; partial confounding; split-plot experiments; fractional replication. Prereq: jr st; MthStat 362(P); Math 234(P) or 240(P).

- 563 Regression Analysis. 3 cr. U/G.
- Straight line, polynomial and multiple regression; multiple and partial correlation; testing hypotheses in regression; residual analysis. Prereq: jr st; MthStat 467(P) or 362(P).

- 564 Time Series Analysis. 3 cr. U/G.
- Autocorrelation; spectral density; linear models; forecasting; model identification and estimation. Prereq: jr st; MthStat 362(P).

- 565 Nonparametric Statistics. 3 cr. U/G.
- Sign, rank and permutation tests; tests of randomness and independence; methods for discrete data and zeroes and ties; power and efficiency of nonparametric tests. Prereq: jr st; MthStat 362(P).

- 566 Computational Statistics. 3 cr. U/G.
- Basics of programming and optimization techniques; resampling, bootstrap, and Monte Carlo methods; design and analysis of simulation studies. Prereq: jr st; MthStat 362(P) or cons instr.

- 568 Multivariate Statistical Analysis. 3 cr. U/G.
- Multivariate normal distribution; Wishart distribution; Hotelling's T2; multivariate normal distribution; multivariate analysis of variance; classification problems. Prereq: jr st; MthStat 362(P); Math 535(P).

- 596 Actuarial Statistics I: Fitting of Loss Models. 3 cr. U/G.
- Statistical modeling of insurance data. Model specification, fitting and validation. Measures of confidence for model-based decisions. Prereq: jr st; B- or better in each Math 234(P) and MthStat 362(P); CompSci 151(P) or 201(P); or cons instr.

- 597 Actuarial Statistics II: Credibility, Risk Measures and Related Topics. 3 cr. U/G.
- Statistical techniques for insurance data. Credibility and ratemaking. Risk measures. Dependent risks and copulas. Simulations. Prereq: jr st; B- or better in each Math 234(P) and MthStat 362(P); CompSci 151(P) or 201(P); or cons instr.

- 691 Actuarial Models I: Life Contingencies. 3 cr. U/G.
- Modeling and valuation of cash flows dependent on death, survival and other random events. Survival models for single and multiple risks. Life insurances and annuities. Prereq: jr st; B- or better in each Math 571(P) and Math 311(P); or cons instr.

- 692 Actuarial Models II: Financial Economics. 3 cr. U/G.
- Modeling and managing of financial risks. Interest rate models. Valuation of derivatives securities. Risk management. Prereq: jr st; B- or better in each Math 571(P) and Math 311(P); or cons instr.

- 761 Mathematical Statistics. 3 cr. G.
- Probability and distribution theory; point and interval estimation; testing hypotheses; large sample inference; nonparametric inference; sequential analysis. Prereq: grad st; Math 522(C) or 622(C).

- 762 Mathematical Statistics. 3 cr. G.
- Continuation of MthStat 761. Prereq: grad st; MthStat 761(P).

- 795 Actuarial Risk Theory. 3 cr. G.
- Risk models; premium principles; reinsurance contracts; ruin theory; ordering of risks; bonus-malus systems; IBNR techniques. Prereq: grad st; Math 571(P) & MthStat 596(P), or cons instr

- 863 Hypothesis Testing. 3 cr. G.
- Exponential families; uniformly most--powerful tests; least favorable priors; unbiased tests; invariant tests; applications to exponential families and the general linear hypothesis. Prereq: grad st; MthStat 762(P).

- 869 Advanced Topics in Mathematical Statistics: (Subtitled). 3 cr. G.
- Specific topics and any additional prerequisites will be announced in the Timetable each time the course is offered. Retakable w/chg in topic to 9 cr max. Prereq: grad st; MthStat 762(P).

#### Courses - Atmospheric Sciences

*Courses numbered 300-699 are Undergraduate/Graduate. Courses numbered 700 and above are Graduate only.*

- 330 Air-Pollution Meteorology. 3 cr. U/G.
- Pollutant sources and sinks, fundamental pollutant chemistry, monitoring techniques, averaging boundary layers and turbulence, diffusion theories, diffusion models, regional and global-scale pollution problems. Prereq: Atm Sci 240(P); Chem 102(P); stats course recom.

- 350 Atmospheric Thermodynamics. 3 cr. U/G.
- Radiant energy, sensible heat, and atmospheric thermodynamics; the gas laws; hydrostatic and psychrometric equations; dry and moist convection; clouds and their physical and energy relations. Optional field exercise. Prereq: jr st; Physics 220(P); Math 232(P); Atm Sci 240(P).

- 351 Dynamic Meteorology I. 3 cr. U/G.
- The role of dynamics in atmospheric physics; equations of motion; symmetric circulation models; gravity waves; Rossby waves, quasi-geostrophy; introduction to instability of atmospheric flows. Prereq: jr st; Atm Sci 240(P); Math 233(P).

- 352 Dynamic Meteorology II. 3 cr. U/G.
- Circulation, vorticity, potential vorticity; shallow water equations: Poincare, Kelvin, and Rossby waves, energy and enstrophy; quasi-geostrophy for a stratified atmosphere; barotropic and baroclinic instability. Prereq: jr st; Atm Sci 351(P); Math 234(P).

- 360 Synoptic Meteorology I. 4 cr. U/G.
- Fundamental principles; synoptic-scale structure and dynamics; equivalent barotropic model; vertical motions; introduction to and application of quasi-geostrophic theory. Prereq: jr st; Math 232(P); Physics 210(P); Atm Sci 240(P).

- 361 Synoptic Meteorology II. 4 cr. U/G.
- Extension of quasi-geostrophic theory to Q-vectors; isentropic potential voracity applied to mid-latitude weather systems; fronts and jets. Prereq: jr st; Atm Sci 360(P).

- 405 Atmospheric Science for in-Service Teachers: 1-3 cr. U/G.
- Basic, advanced or new topics in atmospheric sciences for in-service teachers. Retakable w/chg in topic to 9 cr max. Prereq: in-service teacher; add'l prereqs depending on topic.

- 460 Mesoscale Circulations. 3 cr. U/G.
- Theory, analysis and forecasting of mesoscale flows, including convective systems, polar lows, terrain and surface-forced flows, jet streams and hurricanes. Prereq: jr st; Atm Sci 360(R) or cons instr.

- 464 Cloud Physics. 3 cr. U/G.
- Formation of cloud droplets, droplet growth by condensation, formation of ice crystals, precipitation processes, weather radars, cloud models. Prereq: jr st; Physics 210(P); Math 232(P); Atm Sci 350(P).

- 470 Tropical Meteorology. 3 cr. U/G.
- Dynamics and energetics of tropical circulations. Origins and evolution of equatorial disturbances and easterly waves. Structure and dynamics of tropical cyclones. Hurricane modeling and prediction. Prereq: Atm Sci 351(P) or 360(P).

- 480 The General Circulation and Climate Dynamics. 3 cr. U/G.
- Historical overview, the zonally symmetric circulation, momentum, heat and water budgets, stationary waves, the El Nino Southern oscillation, global warming, interpentadal variability in the North Atlantic. Prereq: jr st; Atm Sci 351(P).

- 497 Study Abroad: (Subtitled). 1-12 cr. U/G.
- Designed to enroll students in UWM sponsored program before course work level, content, and credits are determined and/or in specially prepared course work. Retakable w/chg in topic. Prereq: jr st; acceptance for Study Abroad Prog.

- 500 Statistical Methods in Atmospheric Sciences. 3 cr. U/G.
- Mathematical and statistical tools applicable to the investigation of atmospheric problems; the nature and treatment of atmospheric data. Prereq: jr st; Atm Sci 240(P) or 350(P), & Math 231(P), 232(P) or cons instr.

- 505 Micrometeorology. 3 cr. U/G.
- Surface energy budget; radiation balance and heat transfer; boundary-layer profiles of wind, temperature and moisture; turbulence and boundary-layer fluxes; evapotranspiration; special topics. Prereq: jr st; Atm Sci 351(P) & 330(P).

- 511 Seminar in Atmospheric Radiation and Remote Sensing. 3 cr. U/G.
- Basic laws of radiation, absorption and scattering, weather radar, retrieval of soundings, remote sensing and climate, weather satellites. Prereq: jr st; Math 232(P); Atm Sci 350(P) & Physics 210(P).

- 520 Advanced Dynamic Meteorology. 3 cr. U/G.
- Properties of atmospheric sound, gravity, Rossby waves. Baroclinic instability, cyclogenesis, frontogenesis, and the general circulation. Introduction to numerical prediction. Prereq: jr st; Math 234(P), Atm Sci 350(P) & 351(P) or equiv.

- 690 Seminar in Atmospheric Sciences: (Subtitled). 1-3 cr. U/G.
- Intensive topical studies of currently active problem areas. Retakable w/chg in topic to 9 cr max. Satisfies L&S Seminar req. Prereq: jr st; cons instr.

- 705 Air Pollution Modeling. 3 cr. G.
- Computational techniques for determining surface fluxes of heat and momentum. Numerical methods for solving advection and diffusion problems; statistical diffusion modeling. Prereq: grad st; cons instr.

- 711 Cloud Dynamics. 3 cr. G.
- Atmospheric applications of turbulent flow theory. Nonprecipitating clouds: structure of individual cumulus clouds, stratocumulus and cumulus boundary layers. Precipitating clouds: thunderstorms, squall lines, hurricanes. Prereq: grad st; cons instr.

- 725 Remote Sensing of the Environment. 3 cr. G.
- Remote sensing technology, data processing, and analysis in meteorology, with application to oceanography and geology. Radar and acoustic sounding. Erts, sms/goes, thermal scanner, conventional weather satellites. Prereq: grad st in Physics, Math, Geog, Geo Sci, Engr, or Atm Sci.

- 750 Nonlinear Time Series Analysis. 3 cr. G.
- Phase space reconstruction; singular spectrum analysis; prediction; dimension estimation; application of nonlinear time series analysis techniques to selected data sets. Prereq: grad st; cons instr.

- 751 Geophysical Fluid Dynamics. 3 cr. G.
- Waves and instabilities in the atmosphere and ocean; wave-mean flow interactions; geophysical turbulence; ageostrophic circulations. Prereq: grad st.

- 760 Advanced Cloud, Aerosol & Precipitation Principles, Processes & Interactions. 4 cr. G.
- (3 hr lc, 2 hr la). Theoretical & experimental look at cloud & precipitation formation, interaction & dissipation microphysics & chemistry aerosol physics & chemistry, & their application. Prereq: grad st; Atm Sci 464(C) or cons instr.

- 761 Advanced Synoptic/Mesoscale Meteorology. 3 cr. G.
- Advanced analysis techniques for snyoptic/mesoscale diagnoses, case studies of relevant circulation systems; role of planetary, synoptic, and mesoscale flows in system development. Prereq: grad st; cons instr.

- 888 Candidate for Degree. 0 cr. G.
- Available for graduate students who must meet minimum credit load requirement. Fee for 1 cr assessed. Prereq: grad st.

- 943 Seminar: Hydrology: 3 cr. G.
- Retakable w/chg in topic to 9 cr max. Prereq: grad st.

- 950 Seminar on Topics in Atmospheric Sciences. 1-3 cr. G.
- Selected topics in atmospheric dynamics, satellite meteorology, atmospheric & oceanic convection, air & water pollution, numerical prediction remote sensing, & others. Prereq: grad st in physical sciences or engineering.

- 990 Master's Thesis. 1-8 cr. G.
- Prereq: grad st; cons instr & completed thesis proposal.

- 997 Doctoral Externship. 1-12 cr. G.
- Prereq: grad st; admis to candidacy for the PhD.

- 998 Doctoral Dissertation. 1-12 cr. G.
- Prereq: grad st; admis to candidacy for PhD.

- 999 Advanced Independent Reading. 1-4 cr. G.
- Independent meteorological study. Retakable to 4 cr max. Prereq: grad st & cons instr.