6: College of Arts and Sciences

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Department of Mathematics

Course Descriptions | Faculty

Overview   In a world of increasing technological complexity, knowledge of mathematics is the gateway to the pursuit of many fields. Mathematics has long been the language of choice for expressing complex relationships and describing complicated patterns and processes. It is now true that many fields, in addition to mathematics and the sciences, rely on this in a fundamental way.

What was formerly "abstract" mathematics to many has become the concrete stuff of everyday life. "The unreasonable effectiveness of mathematics" manifests itself today in such familiar things as CAT scans, compact discs, satellite communications, and computer animation. These were all rendered possible by new discoveries made by mathematicians, within the last fifty years. Even the efficient operation of our financial markets is based, in part, on relatively recent theorems of mathematical analysis and probability theory.

Mathematics research today is a vibrant and dynamic enterprise. Thousands of mathematicians worldwide are at work on an unimaginably broad range of questions. Exciting recent advances include the solution of Fermat's Last Theorem, the classification of the finite simple groups, the solution of the Bieberbach conjecture, and the (controversial) proof of the four-color theorem. The discipline and creativity required by the study of mathematics can be a formidable preparation for later life. Past students of mathematics have had successful careers in almost every sphere, including all the professions. (In fact, presidents of Peru, Ireland, and France have been mathematicians.) The scope of mathematics courses offered at the University of Virginia allows majors to tailor a program of study to meet their needs. Students electing to major in mathematics should consult carefully with a faculty advisor to ensure the selection of a program of courses which provides a solid grounding in the fundamentals of higher mathematics and is appropriate to future goals.

Faculty   The faculty of the Department of Mathematics is committed to excellence in teaching and research. Its members carry out high-level research on diverse problems in algebra, analysis, topology, probability, and statistics, mathematical physics, and the history of the discipline. Their research has been widely published in prestigious research journals and is recognized internationally. Members of the department have won Sloan fellowships, Humboldt fellowships, and other scholarly honors, as well as numerous research grants. Many are currently supported by grants from the National Science Foundation and other federal agencies. Most have held visiting professorships abroad. In addition, the department offerings and ambiance are enhanced each year by the presence of several internationally recognized visiting faculty.

Students   There are currently about 75 students majoring in mathematics. Class sizes vary from a few large introductory classes to an average class size of twenty students for upper-level courses. This small class size affords students the opportunity to get individual attention.

Students who graduate with degrees in mathematics successfully pursue a variety of different careers. Many go directly into jobs in industry, insurance (as actuaries), government, finance, and other fields. Employers in the past have included Morgan Stanley, General Motors, MITRE Corp., the Census Bureau, the National Security Agency, and various consulting firms. Many find themselves well-equipped to go on to professional schools in law, medicine, and business. Some go directly into teaching. Others have gone on to graduate programs in mathematics, applied mathematics, statistics, engineering, systems engineering, economics, and computer science. Students who have combined the mathematics major with courses in computer programming, economics, and business have done exceptionally well in the job market.

Requirements for Major   Normally, the calculus sequence MATH 131, MATH 132, and MATH 221 or its equivalent must be completed before a student can declare a major in mathematics. At least a 2.2 average in the calculus sequence and a minimum grade of C in MATH 221 or its equivalent are required. However, the department may grant special permission to declare a major to a student who has only completed MATH 131 and 132, and at least one mathematics course (other than MATH 221 or its equivalent) which could be counted towards the major in mathematics, provided the student completes MATH 221 or its equivalent in the semester following the declaration of a mathematics major.

To graduate with a major in mathematics the student must show computer proficiency by completing CS 120, CS 101, or CS 182, or an approved equivalent course. This should be done as early as possible.

To help guide the student through the major, the mathematics department offers five options. Completion of one of these options is required. Each option contains a set of nine required courses (approximately 28 credit hours). To graduate, a student must obtain minimum grades of C in seven of these courses and C- in the other two.

Certain substitutions are allowed in all options, for example, MATH 531 for MATH 331 and MATH 551 for MATH 354.

  1. The Basic Concentration

    This is a traditional program for the mathematics major which provides an overview of key areas of mathematics:
    MATH 225Ordinary Differential Equations 4
    MATH 351Elementary Linear Algebra 3
    MATH 354Survey of Algebra 3
    Two from the following three:
    MATH 311Introduction to Mathematical Probability 3
    MATH 331Basic Real Analysis 3
    MATH 332Complex Variables with Applications 3
    Four electives at the 300 level or higher12
    Students fulfilling the requirements for this option have a wide range of career opportunities, from law to business to any field which requires deductive, logical reasoning skills.

  2. The Graduate Preparatory Concentration

    This concentration is for the student who plans to attend graduate school in mathematics or an allied field. The program emphasizes the fundamental ideas of mathematics with substantial work in proving and understanding the basic theorems. It consists of:
    MATH 225Ordinary Differential Equations 4
    MATH 332Complex Variables with Applications 3
    MATH 351Elementary Linear Algebra 3
    MATH 531Introduction to Real Analysis I 3
    MATH 551Introduction to Abstract Algebra I 3
    MATH 552Introduction to Abstract Algebra II 3
    Three electives at the 300 level or higher (taking MATH 354 and/or MATH 331 in preparation for MATH 551 and/or MATH 531 is advised)9
    This constitutes a minimum expected of an incoming graduate student in most programs nationwide. We strongly recommend MATH 532 (Real Analysis in Several Variables), as well as courses in differential geometry and topology (MATH 572 and MATH 577). Many of our graduate school bound students take additional courses including 700-level graduate courses.

  3. The Probability and Statistics Concentration
    This concentration is designed to give the student a good theoretical underpinning in probability and statistics, as well as the opportunity to go deeper in these fields. The program can lead to a Master of Science in Statistics with one additional year of course work, if additional courses in statistics are taken in the fourth year. (Those interested in the M.S. in Statistics should contact the graduate advisor in the Division of Statistics prior to the beginning of their fourth year.) The requirements for the option are the following:
    MATH 225Ordinary Differential Equations 4
    MATH 311Introduction to Mathematical Probability 3
    MATH 312Introduction to Mathematical Statistics 3
    MATH 331Basic Real Analysis or
    MATH 332Complex Variables with Applications 3
    MATH 351Elementary Linear Algebra 3
    MATH 354Survey of Algebra 3
    MATH 511Stochastic Processes 3
    STAT 512Applied Linear Models 3
    One additional course chosen from:
    MATH 430 Elementary Numerical Analysis3
    MATH 531 Introduction to Real Analysis I3
    STAT 313 Design and Analysis of Sample Surveys3
    STAT 513 Applied Multivariate Statistics3
    STAT 516 Experimental Design3
    STAT 517 Applied Time Series3
    STAT 519 Introduction to Mathematical Statistics 3

  4. The Financial Mathematics Concentration
    This program provides the student with a broad background of basic mathematics which is essential for an understanding of the mathematical models used in the financial markets. The mathematics of modern finance includes, but is not limited to, probability, statistics, regression, time series, partial differential equations, stochastic processes, stochastic calculus, numerical methods, and analysis. Probability and statistics and some acquaintance with numerical methods are essential as is some knowledge of economics/accounting and some computing experience. Additional background in statistics, optimization, and stochastic processes is also desirable. The program consists of:
    MATH 225 Ordinary Differential Equations 4
    MATH 311 Introduction to Mathematical Probability 3
    MATH 312 Introduction to Mathematical Statistics 3
    MATH 331 Basic Real Analysis or
    MATH 332 Complex Variables with Applications 3
    MATH 351 Elementary Linear Algebra 3
    MATH 354 Survey of Algebra 3
    MATH 414 Mathematics of Derivative Securities 3
    MATH 430 Elementary Numerical Analysis 3
    One additional course chosen from
    MATH 408 Operations Research 3
    MATH 511 Stochastic Processes 3
    STAT 512 Applied Linear Models 3
    STAT 517 Applied Time Series 3
    Two courses chosen from:[1]
    ECON 201 Principles of Economics: Microeconomics 3
    ECON 202 Principles of Economics: Macroeconomics 3
    COMM 201 Introduction to Financial Accounting 3
    COMM 202 Introduction to Management Accounting 3
    [1]Completing all four courses is recommended.

  5. Actuarial Concentration

    This concentration offers some of the basic mathematics and statistics necessary for a successful career in actuarial science, and it provides some of the academic background needed to pass the first few actuarial exams.
    Actuaries use mathematics, statistics, and financial theory to analyze future events, especially those related to insurance and pension programs. They may work for insurance companies, consulting firms, government, employee benefits departments of large organizations, banks, investment firms, or more generally, businesses that need to assess the financial consequences of risk.
    To become an actuary, you must pass a series of examinations administered by the professional actuarial societies: the Society of Actuaries (SOA) and the Casualty Actuarial Society (CAS). The first few exams are jointly administered. Exams which correspond to various courses are indicated below.
    The program consists of nine courses as follows:
    MATH 225[1] Ordinary Differential Equations 4
    MATH 311[2] Introduction to Mathematical Probability 3
    MATH 312[2] Introduction to Mathematical Statistics 3
    MATH 331 Basic Real Analysis 3
    MATH 351 Elementary Linear Algebra 3
    MATH 354 Survey of Algebra 3
    and either
    both STAT 540[3] Actuarial Statistics and
    STAT 541[4] Topics in Statistics: Actuarial Risk Theory 6
    and one elective from the list below3
    or one of either
    STAT 540[3]Actuarial Statistics or
    STAT 541[4]Topics in Statistics: Actuarial Risk Theory 3
    and two electives from the list below6
    Electives (all strongly recommended):
    MATH 408[5] Operations Research 3
    MATH 414 Mathematics of Derivative Securities 3
    MATH 430[6] Elementary Numerical Analysis 3
    MATH 511 Stochastic Processes 3
    STAT 512[7] Introduction to Stochastic Processes 3
    STAT 517[8] Applied Time Series 3

    [1]Part of exam 100, offered jointly by the SOA and CAS.
    [2]Exam 110, offered jointly by SOA and CAS.
    [3]Required SOA exams 140,150/CAS exam 4A.
    [4]Required SOA exam 151/CAS exam 5A.
    [5]Elective SOA exam 130.
    [6]Elective SOA exam 135/required CAS exam 3C.
    [7]One-third of required SOA exam 120/CAS exam 3A.
    [8]Two-thirds of required SOA exam 120/CAS exam 3A.

    It is highly advantageous for students interested in this concentration to take both MATH 311 and MATH 312 in their second year. Actuarial Statistics (STAT 540) and Actuarial Risk Theory (541) form the core of the actuarial program and should both be taken in the third year if possible. Both of these courses are offered every year if there is sufficient student interest, and otherwise in alternate years. With sufficient early course preparation, a summer internship after the third year has been an integral part of the program for those students who wished to intern.
    Other courses which are recommended but not required include:
    ECON 201 Principles of Economics: Microeconomics 3
    ECON 202 Principles of Economics: Macroeconomics 3
    ECON 301 Intermediate Microeconomics 3
    ECON 302 Intermediate Macroeconomics 3
    ECON 434 Theory of Financial Markets 3
    ECON 435[9] Corporate Finance 3
    STAT 514[10] Survival Analysis and Reliability Theory 3

    [9]Required SOA exam 230/required CAS exam 5A.
    [10]SOA exam 160.

  6. Five-year Teacher Education Program

    This option leads to both Bachelor of Arts and Master of Teaching degrees after five years. The program is for both elementary and secondary teachers; it is administered by the Curry School of Education. Required courses include:
    MATH 225 Ordinary Differential Equations 3
    MATH 311 Introduction to Mathematical Probability 3
    MATH 312 Introduction to Mathematical Statistics 3
    MATH 331 Basic Real Analysis or
    MATH 332 Complex Variables with Applications 3
    MATH 351 Elementary Linear Algebra 3
    MATH 354 Survey of Algebra 3
    MATH 501 History of Calculus or
    MATH 503 History of Mathematics 3
    MATH 570 Introduction to Geometry 3
    One elective at the 300 level or higher3

    The Curry School has additional requirements for this program.

    Distinguished Majors Program   The department offers a Distinguished Majors Program (DMP) to qualified majors in mathematics. Admission to the program is granted by the departmental committee for the DMP, usually at the end of the student's fourth semester. Criteria for acceptance into the program are based on the GPA in mathematics, letters of recommendation from mathematics instructors, and the cumulative GPA in the College (which should be near 3.4 or higher).

    The DMP is the same as the graduate school preparatory option, except that in the fourth year the students also take the seminar course MATH 583 in which they give an hour lecture and prepare a written exposition of their work in the seminar, under faculty guidance. Note that MATH 531 and MATH 551 are prerequisites for the seminar. As with the options, the DMP must consist of at least nine courses.

    Three levels of distinction are possible: distinction, high distinction, or highest distinction. The departmental recommendation for the level of distinction to be awarded is based on the quality of the students' seminar presentations, the overall work in the DMP and the entire major program of the students, as well as the students' College GPA.

    Requirements for Minor in Mathematics   Students not majoring in mathematics but who wish to declare a minor in mathematics must complete the calculus sequence through MATH 221 or its equivalent with at least a 2.0 average.

    To graduate with a minor in mathematics a student must complete five courses approved by the department of mathematics with minimum grades of C in three of these courses and minimum grades of C- in the other two. An approved course must carry at least three credits. Currently, the approved courses are those from the College department of mathematics with the MATH mnemonic numbered 225 or higher. Courses with the STAT mnemonic or from other departments or institutions can be offered if approved by the undergraduate committee.

    Courses which are being counted for a major or another minor cannot also be counted for the minor in mathematics.

    Requirements for Minor in Statistics   Before declaring a minor in statistics, students must complete the calculus sequence through MATH 221 or its equivalent with at least a 2.0 average. To graduate with a minor in statistics, students must complete five courses approved by the Department of Mathematics with minimum grades of C in three of these courses and minimum grades of C- in the other two.

    The program must include the following two core courses: MATH 311-312 or their equivalents. The other three courses must be chosen from MATH 511, MATH 531, STAT 512, STAT 513, STAT 514, STAT 516, STAT 517, STAT 518, or STAT 519. Students wishing approval for courses not listed above may file a petition with the undergraduate committee. Courses are ordinarily approved if they compare favorably in content and quality with the listed courses.

    Except for MATH 531, courses which are being counted for the mathematics major or minor may not also be counted for the minor in statistics.

    Echols Mathematics Club   is an undergraduate club for mathematics students which sponsors lectures, mathematics films, problem solving sessions for the Putnam Mathematical Competition and other similar activities.

    Additional Information   For more information, contact Loren Pitt, Lower Division Advisor, Room 204, (804) 924-4939, or Charles Dunkl, Upper Division Advisor, Room 223, (804) 924-4933, Department of Mathematics, Kerchof Hall, Charlottesville, VA 22903-3199.


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