## Mathematics

(Mathematics, Statistics, and Computer Science)

###### http://wp.stolaf.edu/math/

**Chair, 2014-15:** Jill Dietz, algebraic topology, group theory

**Faculty, 2014-15:** Richard J. Allen, logic programming, intelligent tutoring systems; Joseph Benson, differential geometry; Adam H. Berliner, matrix theory, graph theory, linear algebra; Kosmas Diveris, commutative algebra; Kristina C. Garrett, partition theory, enumerative combinatorics; Bruce Hanson, complex analysis; Paul D. Humke, real analysis, dynamical systems; Ryota Matsuura, mathematics education, number theory; Steven McKelvey, operations research, wildlife modeling; Marju Purin, homological algebra, representation theory; Matthew Richey, mathematical physics, computational mathematics (on leave); Paul Roback, statistics; Thomas Rogers, mathematics; Kay E. Smith, logic, math history; Rebecca Vandiver, mathematical biology; Paul Zorn, complex analysis, mathematical exposition

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 242, Math 244, and Math 252) 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.

Students should consult the mathematics program web page (wp.stolaf.edu/math) for lists of courses that satisfy each perspective, lists of sequences, and other useful information. Early consultation with a mathematics faculty member about planning an appropriate IMaP is encouraged.

##### 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: Mathematics 390:
*Mathematics Practicum*, internships, independent studies, and other courses provide valuable opportunities to apply mathematical knowledge beyond the classroom. - Mathematical biology concentration: Mathematics students with an interest in biology may want to consider a concentration in this exciting field.
- 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 problem-solving activities and sponsors student participation in regional and national competitions. St. Olaf also periodically hosts its own mathematics competition, the Carlson Contests.
- Pi Mu Epsilon and Mathematical Association of America: The department has active student chapters of these national organizations.

##### 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 mathematics teaching licensure

Mathematics majors who intend to teach grades 5-12 mathematics must meet all of the normal requirements for the major while including specific courses required for State of Minnesota licensure. Their IMaPs must include the basic requirements for all majors and Mathematics 232, 244, 252, 262, and 356, and a course in statistics. In addition, students must take Education 350: *Teaching of Mathematics, 5-12* as well as several Education courses, as listed in the EDUCATION description. Contact the mathematics licensure advisor for advice for planning mathematics and education course sequences.

##### 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. Fifty percent of the GRE subject test includes topics from single and multivariable calculus, and differential equations; 25% of the test includes topics from linear and abstract algebra, and elementary number theory; and 25% of the test includes topics from real analysis, topology, combinatorics, complex analysis, probability, statistics, and more. Taking many level III courses will help students prepare for graduate study. Real Analysis II, Abstract Algebra II, Topology, Combinatorics, and Complex Analysis are especially recommended for students going into "pure" mathematics; Differential Equations II, Complex Analysis, Real Analysis II, and Applied Mathematics Seminars are especially recommended for students going into applied mathematics. Research experiences (on- or off-campus) and independent studies will also help students assess and explore their interest in further mathematical study. Students thinking about going to graduate school should consult early on with a mathematics faculty member about planning an appropriate IMaP.

##### 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 Mathematics Placement Director.

This course explores the nature of mathematics and its role in contemporary society. The content and format of the course vary depending on the instructor's interests. In particular, the course may focus on one or two mathematical topics in depth or survey a wider range of topics. Recent topics have included mathematics of voting, probability, game theory, finance, and connections between mathematics and the arts. The course requires no prerequisites beyond high school algebra and geometry. Offered annually.

This course combines a full semester of differential calculus with reviews of algebra and pre-calculus topics for students who did not complete a pre-calculus course in high school, or are otherwise not prepared to enroll in Mathematics 120. Review of algebraic techniques and the language of functions are integrated with topics from the calculus of functions of a single real variable, including trigonometric, exponential, and logarithmic functions. Topics are explored graphically, symbolically, and numerically. Class meets four times per week, including one laboratory meeting. Prerequisite: mathematics placement recommendation. Counts toward mathematics major. Offered during fall semester.

This course introduces differential and integral calculus of functions of a single real variable, including trigonometric, exponential, and logarithmic functions. Derivatives, integrals, and differential equations are explored graphically, symbolically, and numerically. Applications of these topics are emphasized throughout the course. Prerequisite: mathematics placement recommendation. Offered each semester.

This course covers methods and applications of integration, geometric and Taylor series, and introduces partial derivatives and double integrals. Prerequisite: MATH 119 or MATH 120 or equivalent, or mathematics placement recommendation. Credit may be earned for either MATH 126 or MATH 128, but not both. Offered each semester.

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

218 Geometry and Decorative Art in Morocco

Islamic art is decorative and based on 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. Students stay with Moroccan families while in Fes. 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 periodically during 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 study of matrix algebra, determinants, vector spaces, bases and dimension, linear transformations, and eigenvalues. Prerequisite: MATH 119 or MATH 120. Offered each semester and periodically during 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: MATH 126 or MATH 128, and MATH 220. Offered each semester.

This course introduces differential equations and analytical, numerical, and graphical techniques for the analysis of their solutions. First- and second-order differential equations and linear systems are studied. Applications are selected from areas such as biology, chemistry, economics, ecology, and physics. Laplace transforms or nonlinear systems may be covered as time permits. Students use computers extensively to calculate and visualize results. Prerequisite: MATH 126 or MATH 128 and MATH 220. Offered each semester. Counts toward neuroscience and mathematical biology concentrations.

Using problem solving techniques, students study topics from non-continuous mathematics, including basic counting principles, mathematical induction, recursion, efficiency of algorithms, and graph theory. Prerequisite: MATH 119 or MATH 120 or CSCI 121 or CSCI 125 or permission of instructor. Offered in 2012-13 and alternate years. Students may not take this course after MATH 244, MATH 252, or MATH 282 without 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., MATH 244 and MATH 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: MATH 220 or permission of instructor. Offered peridically during Interim.

This course introduces students to the essential modeling techniques of formulation, implentation, validation, and analysis. Students engage in these areas by combining experiment, mathematical theory, statistics, and computation to better understand a wide variety of biological systems. Prerequisite: MATH 126 or MATH 128, and MATH 220. Offered annually in the spring semester. Counts toward neuroscience and mathematical biology concentrations.

This course introduces the study of patterns and relationships satisfied by natural numbers. Topics include divisibility, modular arithmetic, prime numbers, congruences, primitive roots, and quadratic residues. The course is offered in Budapest, Hungary, a world center for research mathematics. The course includes several hours of Hungarian language instruction and lectures by Hungarian mathematicians and scholars. Prerequisite: MATH 220 or permission of instructor. Offered most Interims.

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: MATH 220. Offered annually. Counts toward mathematical biology concentration.

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. Emphasis is on theory and on developing the ability to write proofs. Prerequisite: MATH 126 or MATH 128, and MATH 220. Permission of instructor required for first year students. Offered each semester.

Algebra is concerned with sets of objects and operations that satisfy a few basic properties. Using the properties we study axiomatic systems such as groups, rings, and fields, covering topics such as homomorphisms, cosets, quotient structures, polynomial rings, and finite fields. Emphasis is on theory and on developing the ability to write proofs. Prerequisite: MATH 220. Permission of instructor required for first-year students. Offered each semester.

This course introduces the mathematics of randomness. Topics include probabilities on discrete and continuous sample spaces, conditional probability and Bayes' Theorem, random variables, expectation and variance, distributions (including binomial, Poisson, geometric, normal, exponential, and gamma) and the Central Limit Theorem. Students use computers to explore these topics. Prerequisite: MATH 126 or MATH 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). Students use computers to explore these topics, but prior computer experience is not assumed. Prerequisites: MATH 126 or MATH 128 and MATH 220; MATH 226 and/or MATH 262 recommended. Offered annually.

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. Offered periodically.

294 Internship

298 Independent Study

This course covers partial differential equations from an applied perspective and emphasizes simple models involving phenomena such as wave motion and diffusion. Topics and techniques such as separation of variables, boundary value problems, Fourier series, and orthogonal functions are developed carefully. Mathematical computing is used freely. Prerequisite: MATH 230. Offered in 2013-14 and alternate years. Counts toward neuroscience and mathematical biology concentrations.

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: MATH 220, and MATH 226 or MATH 244. Offered annually.

The main topics of this course are measure theory on the real line, the Lebesgue integral and its relation to the Riemann integral, and convergence theorems for the Lebesgue integral. Applications to probability and harmonic analysis may be included. Prerequisite: MATH 244. Offered in 2013-14 and alternate years.

This course is an introduction to topological spaces and their structures mainly from the point-set perspective. Standard topics include separation axioms, compactness, and connectedness. Other topics from geometric and algebraic viewpoints may be included. Prerequisite: MATH 244. Offered in 2012-13 and alternate years.

This course is a continuation of the study of the theory of groups, rings, and fields. Topics include group actions, Sylow theory, and Galois theory. Other topics may include representation theory, module theory, and others. Prerequisite: MATH 252. Offered in 2012-13 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 hyperbolic geometry and 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: MATH 220, and MATH 244 or MATH 252. Offered annually during Interim.

This course covers basic enumeration, including generating functions, recursion, and the inclusion-exclusion principle. Basic combinatorial objects such as set partitions, permutations, integer partitions, and posets are discussed. Making conjectures and proving theorems combinatorially are emphasized. Students also explore topics in graph theory, matrix theory, and representation theory. Prerequisite: MATH 252; some previous exposure to counting methods (e.g., counting permutations and combinations) is helpful but not required. Offered in 2013-14 and alternate years.

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

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 most years.

Students work in groups on substantial 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 annually 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. Topics are determined by the instructor and may relate to his/her research interests. Prerequisite: determined by instructor. Offered most years in the fall semester. May be offered as a 1.00 credit course or .50 credit course.

398 Independent Research

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