Please note: This is NOT the most current catalog.

## Mathematics

(Mathematics, Statistics, and Computer Science)

###### http://www.stolaf.edu/depts/math

**Chair, 2009-10:** Paul Zorn, complex analysis, mathematical exposition

**Faculty, 2009-10:** Richard J. Allen, logic programming, intelligent tutoring systems; Adam Berliner, matrix theory, graph theory, combinatorics; Peder A. Bolstad, precalculus, graph theory; Jill M. Dietz,m algebraic topology; Kristina Garrett, combinatorics; Rosemary Gundacker, mathematics education; Bruce H. Hanson, complex analysis; Paul D. Humke, real analysis, dynamical systems (on leave); Delia Letang, analytic number theory, automorphic forms; Urmila Malvadkar, mathematical biology; Steven McKelvey, operations research, wildlife modeling; Matthew Richey, mathematical physics, computational mathematics; Paul Roback, statistics (on leave); Kay E. Smith, logic, discrete mathematics; Martha Tibbetts Wallace, mathematics education; Mike Weimerskirch, combinatorics, probability, mathematics education; Alexander Woo, algebraic geometry, commutative algebra, discrete mathematics

Mathematics, as the study of patterns and order, is a creative art, a language, and a science. The practice of mathematics combines the aesthetic appeal of creating patterns of ideas with the utilitarian appeal of applications of these same ideas. As the language of physical science, mathematics is also used increasingly to model phenomena in the biological and social sciences. Mathematical literacy is indispensable in today’s society. As part of the Department of Mathematics, Statistics, and Computer Science (MSCS), members of the mathematics faculty strive to help students understand natural connections among these related but distinct disciplines.

Mathematics at St. Olaf is interesting, exciting, accessible, and an appropriate area of study for all students. Each year, seven to ten percent of graduating seniors complete mathematics majors. The department offers courses representing various mathematical perspectives: theoretical and applied, discrete and continuous, algebraic and geometric, and more. Our faculty also teach courses in statistics, computer science, and mathematics education.

A concentration in statistics and a major in computer science are also available. Courses in these areas are taught by faculty from the Department of Mathematics, Statistics, and Computer Science. For further information on these, consult the separate listings under STATISTICS and COMPUTER SCIENCE.

overview of the major

Students arrange a major in mathematics by developing an Individualized Mathematics Proposal (IMaP). An IMaP outlines a complete, coherent program of study consistent with the goals of the individual student. The courses included in a student’s IMaP are determined after consultation with an MSCS faculty member and approved by the department chair. About ten courses are normally required.

##### INTENDED LEARNING OUTCOMES FOR THE MAJOR

REQUIREMENTS FOR THE MAJOR

Students arrange a major in mathematics by developing an Individualized Mathematics Proposal (IMaP). An IMaP outlines a complete, coherent program of study consistent with the goals of the individual student. The courses included in a student’s IMaP are determined after consultation with an MSCS faculty member and approved by the department chair.

A path through the major as described by a student’s IMaP normally includes two semesters of calculus, one semester of linear algebra, and at least seven intermediate or advanced mathematics courses. The intermediate courses should include two transition courses (from among Math 244, Math 252, and Math 242) and courses from at least three different mathematical perspectives (computation/modeling, continuous/analytic, discrete/combinatorial, axiomatic/algebraic). Students must take at least two Level III courses, at least one of which must be part of a designated Level II–Level III sequence.

An IMaP may include up to two related courses from Statistics or Computer Science; a current listing of such courses is available on the mathematics web page. A student may also find a course outside of MSCS that contributes significantly to a mathematical path of study and may petition to have the course included in his or her IMaP.

Mathematics majors who intend to teach grades 5-12 mathematics must meet the above requirements (see also EDUCATION description and the Mathematics Licensure Adviser). Their IMaPs must include Mathematics 232, 244, 252, 262, and 356, a course in statistics, and Education 350 in order to meet the State of Minnesota licensure requirements. Course requirements include calculus, linear algebra, discrete mathematics, statistics, and geometry, as well as Education 350.

Students should consult the mathematics program web page (www.stolaf.edu/depts/math) for lists of courses that satisfy each perspective, lists of sequence courses, and other useful information.

##### SPECIAL PROGRAMS AND OPPORTUNITIES

Mathematical experiences inside and outside the classroom are important parts of an IMaP. Following are some of the many possibilities. For more information consult the mathematics program web page or a faculty member.

- Research: An invigorating way to explore mathematics, research opportunities exist both on and off campus.
- Experiential learning: The Mathematics Practicum (Mathematics 390), internships, independent studies and other courses provide valuable opportunities to apply mathematical knowledge beyond the classroom.
- Study abroad: The IMaP’s flexibility allows some study abroad programs to fit into a student’s mathematics major. Students interested in a program focused on upper-level mathematics should consider the Budapest Semesters in Mathematics.
- Problem solving and competitions: The department organizes a problem-solving seminar and sponsors student participation in regional and national competitions. St. Olaf also hosts its own annual mathematics competition, the Carlson Contests.
- Mathematical Association of America: The department has an active student chapter of this national organization.

##### DISTINCTION

Information about distinction, awarded for distinguished work that goes beyond the minimum requirements for the major, is available in the MSCS department and on the mathematics website.

##### RECOMMENDATIONS FOR GRADUATE STUDY

Students planning graduate work in the mathematical sciences should pursue opportunities that add both depth and breadth to their majors. Courses across a broad range of the curriculum will help students prepare for the Graduate Record Exam subject test in mathematics. Taking many Level III courses will help students prepare for graduate study. Research experiences (on- or off-campus) and independent studies will also help students assess and explore their interest in further mathematical study.

##### COURSES

This course is designed for students who need additional preparation before taking calculus. The course emphasizes functions, including polynomial, exponential, logarithmic, and trigonometric. Enrollment requires permission of the Director of Mathematics Placement. Offered annually in the Fall Semester.

This course explores the nature of mathematics and its role in the modern world. The content and format of the course vary, depending on the instructor. The course may focus on one topic or survey a range of topics. Recent topics include environmental modeling, social choice, routes and networks, scheduling, and symmetry. The course is intended for all students. Offered each semester.

118 Gateways: Geometric Patterns in Islamic Culture

In this mathematical exploration of the geometry underlying the patterns and images of Islamic art and architecture, students encounter the origins of patterns found in Islamic religious beliefs and the development over time of this expression of mathematics through culture. They study and analyze examples occurring in the architecture of buildings and monuments found in the Islamic world. Students apply the acquired geometry and Islamic culture by creating new original patterns and defending them as appropriate representations of Islamic decoration. Offered annually in the Spring Semester.

This course introduces differential and integral calculus of functions of a single real variable, including trigonometric, exponential, and logarithmic functions. Derivatives and integrals are explored graphically, symbolically, and numerically. Applications of the derivative are included. Prerequisite: Mathematics Placement Recommendation. Offered each semester.

121 Calculus I with Laboratory

Similar to Mathematics 120, but includes a 1-hour weekly laboratory session. Offered periodically.

This continuation of Mathematics 120 covers methods and applications of integration and introduces multivariable calculus. May include geometric and Taylor series. Prerequisite: Mathematics 120 or equivalent, or Mathematics Placement Recommendation. Credit may be earned for either Mathematics 126 or 128, but not both. Offered each semester.

This course covers the material in Mathematics 126 in greater depth and includes supplementary material. Prerequisite: 4 or 5 on AP Calculus AB exam or permission of Mathematics Placement Director. Credit may be earned for either Mathematics 126 or 128, but not both. Offered each semester.

This course is intended for non-majors. Students encounter selected mathematical topics demonstrating the scope of mathematical inquiry, its "unreasonable effectiveness," and its connections with other disciplines. Not open to first-year students. Offered in Interim.

218 Geometry and Decorate Art in Morocco

Islamic art is decorative and based on place geometry. Students study this art, its origins, and its significance, along with cultural topics related to Moroccan life, in the imperial city of Fes. Field work includes identification and analysis of distinct geometrical patterns found on buildings, monuments, and artifacts. Students also use geometry to create their own art. Mosaic designs are still created in Fes, a center for Islamic geometric patterns. Home stays with Moroccan families are an option. Field trips visit sites in and around Fes, with day-long visits to Meknčs, Moulay Idriss, and Volubis, and a longer excursion to Marrakech and Casablanca. Offered in Interim.

This course beautifully illustrates the nature of mathematics as a blend of technique, theory, abstraction, and applications. The important problem of solving systems of linear equations leads to the algebra of matrices, determinants, vector spaces, bases and dimension, linear transformations, and eigenvalues. Prerequisite: Mathematics 120 or 121. Offered each semester.

Mathematicians make discoveries only after computing many examples, noticing patterns, and then inventing tools and language to describe what they see. Using computers, students conceptualize and prove theorems in a variety of mathematical areas. Closed to students who have taken courses beyond Mathematics 232. Prerequisite: Mathematics 126 or 128. Offered periodically in Interim.

This course extends important ideas of single-variable calculus (derivatives, integrals, graphs, approximation, optimization, fundamental theorems, etc.) to higher-dimensional settings. These extensions make calculus tools far more powerful in modeling the (multi-dimensional) real world. Topics include partial derivatives, multiple integrals, transformations, Jacobians, line and surface integrals, and the fundamental theorems of Green, Stokes, and Gauss. Prerequisites: Mathematics 126 or 128, and 220. Offered each semester.

This course introduces differential equations and analytical, numerical, and graphical techniques for their analysis. First- and second-order differential equations and linear systems are studied. Applications are selected from areas such as biology, chemistry, economics, ecology, and physics. Students use computers extensively to calculate and visualize results. Prerequisite: Mathematics 126 or 128 and 220. Offered each semester.

Discrete (noncontinuous) mathematics has become increasingly important as more phenomena are investigated, represented and solved using computers (essentially, discrete machines). Students explore finite graphs, recurrence relations, and combinatorial optimization using problem solving techniques and algorithm design strategies. Prerequisite: Mathematics 120 or 121, or permission of the instructor. Offered alternate years.

234 The Structure of Higher Mathematics (abroad)

This course provides students with a transition from calculus and linear algebra to more advanced courses in theoretical mathematics. The course is taught with a cultural context in Budapest, Hungary. This course not only supplies a bridge from beginning to advanced mathematics, but also allows participants to encounter an important worldwide center of mathematics. Prerequisite: Math 220 or permission of instructor.

235 Structures in Higher Mathematics

This course introduces students to key concepts and forms of proof found in theory courses (e.g., Mathematics 244 and Mathematics 252). Topics include basic logic and set theory, mathematical induction, primes, congruences, and infinite sets. Students learn to read, write, and understand mathematical proofs. Prerequisites: Mathematics 220 or permission of instructor. Offered in Interim.

This course introduces students to the mathematics of complex systems, as applied to problems from biology. Topics include discrete and continuous models of single species and multiple species populations, age structure of populations, disease spread, evolution and game theory, and competition. Prerequisite: Mathematics 126 or 128, and Mathematics 220. Offered annually in the Spring Semester.

This course introduces number theory -- the study of patterns and relationships satisfied by natural numbers. Topics include prime numbers, congruences, primitive roots, quadratic residues, and the design and breaking of codes. Prerequisite: Mathematics 220 or permission of instructor. Offered periodically.

This course introduces number theory -- the study of patterns and relationships satisfied by natural numbers. Topics include prime numbers, congruences, primitive roots, quadratic residues, and the design and breaking of codes. The course is taught with a cultural context in Budapest, Hungary. Prerequisite: Mathematics 220 or permission of instructor. Offered during Interim.

242 Modern Computational Mathematics

Modern mathematics is characterized by the interaction of theoretical and computational techniques. In this course, students study topics from pure and applied mathematics with the aid of computation. Symbolic, graphical, and numerical computational techniques are introduced. Students develop computational skills sufficient to investigate mathematical questions independently. No previous programming experience is required. Prerequisite: Mathematics 220. Offered annually in the Fall Semester.

Students encounter the theory of calculus and develop tools for communicating mathematical ideas with technical accuracy and sophistication. The goal is mastery of the concepts (e.g., limit, continuity, derivatives, and integrals) necessary to verify such important results as the Fundamental Theorem of Calculus, the Mean Value Theorem, and the Bolzano-Weierstrass Theorem. Prerequisite: Mathematics 126 or 128, and Mathematics 220. Offered each semester.

Algebra is concerned with sets of objects and operations on these sets. In an axiomatic or abstract treatment one assumes basic properties and then deduces many other properties. Using this method we study structures known as groups, rings, and fields. Prerequisite: Mathematics 220. Offered each semester.

This course is an introduction to the mathematics of randomness and games of chance. Topics include combinatorial analysis, elementary probability measures, conditional probability, random variables, special distributions, expectations, generating functions, and limit theorems. Prerequisite: Mathematics 126 or 128. Offered each semester.

Students are introduced to modeling and mathematical optimization techniques (e.g., linear programming, network flows, discrete optimization, constrained and unconstrained nonlinear programming, queuing theory). The course emphasizes applications, but prior computer experience is not assumed. Prerequisites: Mathematics 126 or 128 and 220. Recommended: Mathematics 226 and/or 262. Offered annually in the Spring Semester.

Students work intensively on a special topic in mathematics or its applications. Topics vary from year to year. May be repeated if topics are different.

294 Internship

298 Independent Study

A sequel to Mathematics 230, this course studies differential equations from a more rigorous mathematical perspective and extends their use as modeling tools. Students examine dynamical systems and their use in the study of chaos and fractals and partial differential equations and their use to describe complex physical phenomena such as wave motion and diffusion. Mathematical computing plays an important role. Prerequisite: Mathematics 230. Offered periodically.

Complex analysis treats the calculus of complex-valued functions of a complex variable. Familiar words and ideas from ordinary calculus (limit, derivative, integral, maximum and minimum, infinite series) reappear in the complex setting. Topics include complex mappings, derivatives, and integrals; applications focus especially on the physical sciences. Prerequisite: Mathematics 220. Recommended: Mathematics 226.

The main topics are measure theory on the real line and the Lebesgue integral, up to and including the convergence theorems. Applications to probability and harmonic analysis may be included. Prerequisite: Mathematics 244, or permission of instructor. Offered in 2009-10 and alternate years.

This course is an introduction to topological spaces and their structure from point-set, differential and algebraic points-of-view. Topics may include separation axioms, compactness, connectedness, classification of surfaces, homology, fundamental group, and others. Prerequisite: Math 244 or 252. Offered in 2008-09 and alternate years.

This course offers a continuation of group theory and field theory, including group actions, Sylow theory, and Galois theory. Other topics may include representation theory, module theory and more, depending on the instructor. Prerequisite: Math 252. Offered in 2008-09 and alternate years.

Properties of axiomatic systems are illustrated with finite geometries and applied in a synthetic examination of Euclid's original postulates, well-known Euclidean theorems, and non-Euclidean geometries. Euclidean, similarity, and affine transformations are studied analytically. These transformations are generalized to obtain results in projective geometry or used to generate fractals in an exploration of fractal geometry. Dynamic geometry software and hands-on labs are used to explore both the transformations and properties of these geometries. Prerequisite: Mathematics 220, and Mathematics 244 or 252. Offered during Interim.

This course covers basic enumeration, including generating functions, recursion, inclusion-exclusion, Polya theory, etc. Students also explore topics in graph theory and constructive combinatorics, time permitting. Prerequisite: Mathematics 252; some previous exposure to counting methods (e.g., Mathematics 232 or Mathematics 262) is helpful. Offered in 2009-10 and alternate years.

Mathematical logic uses mathematical methods to analyze reasoning and to examine what mathematics can and cannot do. It also provides the underlying paradigm on which intelligent computer systems are based. Initially, students study the language and rules of inference of predicate logic and investigate the relationships between provability and truth and between a mathematical theory and its models. Later, they explore applications of logic to computer science. Prerequisite: Mathematics 244 or 252. Offered periodically.

Students work intensively on a special topic in mathematics. Topics vary from year to year. May be repeated if topics are different. Offered periodically.

384 Topics in Applied Mathematics

Students work intensively on a special topic in applied mathematics. Topics vary from year to year. May be repeated if topics are different. Offered periodically.

Students work in groups on significant problems posed by and of current interest to area businesses and government agencies. The student groups decide on promising approaches to their problem and carry out the necessary investigations with minimal faculty involvement. Each group reports the results of its investigations with a paper and an hour-long presentation to the sponsoring organization. Prerequisite: Permission of instructor. Offered during Interim.

394 Internship

396 Directed Undergraduate Research

This course provides a comprehensive research opportunity, including an introduction to relevant background material, technical instruction, identification of a meaningful project, and data collection. The topic is determined by the faculty member in charge of the course and may relate to his/her research interests. Prerequisite: Determined by individual instructor. Offered based on department decision.

398 Independent Research

related courses

Computer Science 231 Mathematical Foundations of Computing

Students learn mathematical topics that form an essential background for the study of computer science, including functions, relations, basic logic, predicate calculus and formal reasoning, verification of programs, proof techniques, basics of counting, graphs and trees, discrete probability, and introduction to computability. Prerequisites: Computer Science 121 or 125 or permission of instructor. Students with especially weak or especially strong mathematics backgrounds should consult with the program director.

Computer Science 253 Algorithms and Data Structures

This course surveys standard data structures and algorithms with emphasis on implementation experience and complexity analysis. Topics include algorithmic strategies, fundamental computer algorithms, stacks, queues, lists, trees, hash tables, specialized trees (e.g., binary, AVL, B-trees), heaps and priority queues, compression, and decompression. Prerequisites: Completion of BTS-T; Computer Science 231 and either Computer Science 125 or 251, or consent of the instructor.

Computer Science 333 Theory of Computation

Students learn about formal languages, automata, and other topics concerned with the theoretical basis and limitations of computation. The course covers automata theory including regular languages and context-free languages, computability theory, complexity theory including classes P and NP, and cryptographic algorithms. Prerequisite: Computer Science 231 or permission of instructor.

Statistics 272 Statistical Modeling

This course takes a case-study approach to the fitting and assessment of statistical models with application to real data. Specific topics include two-sample comparisons, simple linear regression, multiple regression, model diagnostics, logistic regression for focus on problem-solving tools, interpretation, mathematical models underlying analysis methods, and written statistical reports. Prerequisite: Statistics 110 or 212 or 263, or permission of instructor. Offered each semester.

Statistics 316 Advanced Statistical Modeling

This course extends and generalizes methods introduced in Statistics 272. Topics include generalized linear models, including logistic and Poisson regression. Correlated data methods including longitudinal data analysis and multilevel models are covered. Applications are drawn from across the disciplines. Prerequisite: Statistics 272. Offered annually in the Spring Semester.

Statistics 322 Statistical Theory

This course is an investigation of modern statistical theory along with classical mathematical statistics topics such as properties of estimators, likelihood ratio tests, and distribution theory. Additional topics may include Bayesian analysis, censored data methods, missing data, and other computationally intensive methods. Prerequisite: Statistics 272 and Mathematics 262. Offered annually in the Fall Semester.