Mathematics

Professors: Priscilla Bremser, David Dorman, Michael Olinick (on leave academic year 2008-09), William Peterson (chair),  John C. Baldwin Professor of Mathematics and Natural Philosophy: Peter Schumer; Charles A. Dana Professor: John Emerson (on leave academic year 2008-09); Associate Professor: Stephen Abbott, Frank Swenton; Assistant Professors: Emily Proctor, John Schmitt;  Lecturer: Janine Clookey; Department Coordinator: Naomi Neff

The Department of Mathematics offers courses in a variety of areas within the mathematical sciences. These areas include foundations, analysis, algebra and number theory, topology, geometry, applied mathematics, probability and statistics, and mathematical modeling. Accompanying the growth of applications in many fields has been the enrichment of our own programs by the addition and reorganization of courses offered to accommodate such applications.
     Our programs are designed to be carefully integrated into the liberal arts traditions of Middlebury College. They serve the needs of four groups of Middlebury College students:
     1. Those who are interested in mathematics, but who will not have a major or a minor in a field that requires mathematics courses.
     2. Those who need mathematics as an integral part of their chosen major or minor.
     3. Those who choose to major or minor in mathematics, emphasizing a traditional study of mathematics and its modern applications.
     4. Those who choose the mathematical sciences option in the mathematics major, emphasizing the connections between mathematics and allied disciplines.
     Each student will choose several electives, with the help of an adviser, in order to best fulfill the student's particular mathematical needs and goals.
     Required for the Major in Mathematics: (Ten courses total)
     I.  Core courses: MATH 0122, MATH 0200, MATH 0223,  MATH 0302, and MATH 0323;
     II.  Electives: four MATH electives at the 0200-level or above;
     III. Senior thesis: MATH 0704 in the senior year.

Note:  Students are strongly encouraged to include a proof-based course such as MATH 0241, or MATH 0247 early in their programs.  This is especially helpful prior to taking MATH 0302 or MATH 0323. 
     Required for the Mathematical Sciences Option in the Mathematics Major: (Ten courses total)
     I.  Core courses: MATH 0122, MATH 0200, and MATH 0223;
     II.  Electives. Six courses from categories A and B.  At least four of the six courses must have the MATH designation, and at least two must be from category B. 
     A. Courses in applied specialization: CSCI 0102, MATH 0225, MATH 0310, MATH 0315, MATH 0318, ECON 0380, PHYS 0212, CSCI 0201, CSCI 0463; 
     B. Advanced Electives: MATH 0302, MATH 0311, MATH 0323, MATH 0325, MATH 0410,  CSCI 0302, ECON 0390, ECON 0411, MATH 0500 (with prerequisite: at least one course from categories A or B);
     III. Senior thesis: MATH 0704 in the senior year.

Note:  Students should consult the mathematics department for examples of course sequences in the mathematical sciences option recommended for emphases in Mathematical Economics, Computer Science, or Physical Sciences/Engineering. For students completing double majors, electives used towards a major in another department cannot also be counted as electives in the mathematical sciences option.
     Honors Program: A student who wishes to be considered for departmental honors in mathematics must submit a proposed plan of study during his or her junior year. Candidates for departmental honors should include two additional electives in their programs (12 courses total). For the mathematical sciences option, an honors program must include one of MATH 0302/0323 and an elective sequence such as MATH 0310-0410 or MATH 0310-0311. Students should consult their advisers as they develop proposals for honors study.
     Required for the Minor in Mathematics: MATH 0121, MATH 0122, MATH 0200, and three courses at the 0200-level or above.
    Joint Majors: Mathematics is frequently elected as a joint or double major with other disciplines. The mathematics component of a joint major requires the following courses: MATH 0122, MATH 0200, MATH 0223; three MATH electives at the 0200-level or higher; one course from MATH 0302 and MATH 0323;  MATH 0704. Joint majors are required to have faculty advisors in both departments and approval by both chairs. See the Economics Department catalog listing for specific recommendations regarding a joint Economics/Mathematics major. The senior thesis project (MATH 0704) must combine mathematics with the other discipline and be approved by both departments. 
    Mathematics majors interested in obtaining secondary school teaching certification need to notify the Teacher Education Program, preferably by the middle of their sophomore year. They should also contact Peter Schumer, the department's teacher education representative.  Students who plan to pursue study abroad during their junior year should also plan their course of study in the major by the end of their first year.     
     Advanced Placement: Advanced placement in the department is offered to first-year students whose secondary training indicates they can commonly bypass one or more of the beginning courses in mathematics. Majors typically begin their study of mathematics in MATH 0122 or MATH 0200. Mathematics majors who need to begin the study of calculus with MATH 0121 may arrange with their advisers to use this course as one of the required electives. Credits for MATH 0121 and 0122 may be earned through the College Board AP exams or international exams such as the A-Levels or IB. At the discretion of the chair, additional courses may be waived in recognition of exceptional secondary school preparation. However, in all cases the major must include at least 7 Middlebury College or approved transfer courses, and the minor must include at least 4. Students who have earned grades on advanced placement calculus exams that are eligible for credit may not register for the equivalent course at Middlebury College. Thus students who have earned 4 or 5 on the Calculus AB exam or a 3 on the Calculus BC exam may not register for MATH 0121, students who have earned 4 or 5 on the Calculus BC exam may not register for MATH 0121 or MATH 0122, and students who have earned 4 or 5 on the Statistics exam may not register for MATH 0116. This policy applies irrespective of whether students choose to use their AP credits toward meeting Middlebury's graduation requirements. The following international credentials carry the same credit as a 4 or 5 on the Calculus BC Exam: A-level exam with a mathematics grade of A, B, or C; or IB Higher Level Mathematics with a grade of 6 or 7.
    Other Credits: Because of the wide variation in course offerings at other institutions, students wishing to substitute a course from another college for any course in mathematics must seek approval from the department before registering for the course. In addition, students seeking MATH 0121 credit for a summer course taken elsewhere must pass a written examination given by the department in the fall. Check with the department early in the first week of classes for details.
    
Interdepartmental Courses

INTD 0206 Science as Art in Contemporary Theatre (Fall)
See Interdepartmental Courses for course description. LIT DED (C. Faraone, S. Abbott)

Departmental Courses

MATH 0116 Introduction to Statistical Science (Not offered 2008-09)
A practical introduction to statistical methods and the examination of data sets. Computer software will play a central role in analyzing a variety of real data sets from the natural and social sciences. Topics include descriptive statistics, elementary distributions for data, hypothesis tests, confidence intervals, correlation, regression, contingency tables, and analysis of variance. The course has no formal mathematics prerequisite, and is especially suited to students in the physical, social, environmental, and life sciences who seek an applied orientation to data analysis. (Credit is not given for MATH 0116 if the student has taken ECON 0210 or PSYC 0201 previously or concurrently.) 3 hrs. lect., 1 hr. computer lab. DED

MATH 0120 Extended Calculus I (Not offered 2008-09)
A course for students with only two or three years of secondary school mathematics. This course integrates the material prerequisite to calculus with calculus itself. Algebra, graphs, and properties of continuous functions are studied as these topics relate to the calculus. Trigonometric functions, analytic geometry, logarithms, and exponential functions will be integrated into the course as the need arises. Extended Calculus is the conclusion of a Winter-Spring course sequence that begins in January term. 4 hrs. lect./disc. DED 

MATH 0121 Calculus I (Fall, Spring)
Introductory analytic geometry and calculus. Topics include limits, continuity, differential calculus of algebraic and trigonometric functions with applications to curve sketching, optimization problems and related rates, the indefinite and definite integral, area under a curve, and the fundamental theorem of calculus. Inverse functions and the logarithmic and exponential functions are also introduced along with applications to exponential growth and decay. 4 hrs. lect./disc. DED (P. Bremser, J. Schmitt)

MATH 0122 Calculus II (Fall, Spring)
A continuation of MATH 0121, may be elected by first-year students who have had an introduction to analytic geometry and calculus in secondary school. Topics include a brief review of natural logarithm and exponential functions, calculus of the elementary transcendental functions, techniques of integration, improper integrals, applications of integrals including problems of finding volumes, infinite series and Taylor's theorem, polar coordinates, ordinary differential equations. (MATH 0121 or by waiver) 4 hrs. lect./disc. DED (fall: D. Dorman, F. Swenton; spring: P. Schumer, F. Swenton)

MATH 0200 Linear Algebra (Fall, Spring)
Matrices and systems of linear equations, the Euclidean space of three dimensions and other real vector spaces, independence and dimensions, scalar products and orthogonality, linear transformations and matrix representations, eigenvalues and similarity, determinants, the inverse of a matrix and Cramer's rule. (MATH 0121 or by waiver) 3 hrs. lect./disc. DED (fall: J. Schmitt, P. Schumer; spring: E. Proctor)

MATH 0223 Multivariable Calculus (Fall, Spring)
The calculus of functions of more than one variable. Introductory vector analysis, analytic geometry of three dimensions, partial differentiation, multiple integration, line integrals, elementary vector field theory, and applications. (MATH 0122 and MATH 0200 or by waiver) 3 hrs. lect./disc. DED (fall: P. Bremser; spring: S. Abbott)

MATH 0225 Topics in Linear Algebra and Differential Equations (Spring) 
Topics may include diagonalization of matrices, quadratic forms, inner product spaces, canonical forms, the spectral theorem, positive matrices, the Cayley-Hamilton theorem, ordinary differential equations of arbitrary order, systems of first-order differential equations, power series, and eigenvalue methods of solution, applications. (MATH 0122 or by waiver and MATH 0200) 3 hrs. lect./disc. DED (D. Dorman)

MATH 0241 Elementary Number Theory (Not offered 2008-09)
Divisibility and prime factorization. Congruences; the theorems of Lagrange, Fermat, Wilson, and Euler; residue theory; quadratic reciprocity. Diophantine equations. Arithmetic functions and Mobius inversion. Representation as a sum of squares. (MATH 0122 or by waiver) DED

MATH 0247 Graph Theory (Not offered 2008-09)
Topics from graph theory and combinatorial mathematics including graphs, enumeration problems, planar graphs, Euler paths and the Konigsberg Bridge Problem, Hamiltonian graphs, the four color conjecture, tilings and tesselations of the plane, Fibonacci sequences, applications. (MATH 0122) 3 hrs. lect./disc. DED

MATH 0261 History of Mathematics (CW 5) (Not offered 2008-09)
This course studies the history of mathematics chronologically beginning with its ancient origins in Babylonian arithmetic and Egyptian geometry. The works of Euclid, Apollonius, and Archimedes and the development of ancient Greek deductive mathematics is covered. The mathematics from China, India, and the Arab world is analyzed and compared. Special emphasis is given to the role of mathematics in the growth and development of science, especially astronomy. European mathematics from the Renaissance through the 19th Century is studied in detail including the development of analytic geometry, calculus, probability, number theory, and modern algebra and analysis. (MATH 0122 or waiver) DED CMP

MATH 0302 Abstract Algebra (Fall, Spring)
Groups, subgroups, Lagrange's theorem, homomorphisms, normal subgroups and quotient groups, rings and ideals, integral domains and fields, the field of quotients of a domain, the ring of polynomials over a domain, Euclidean domains, principal ideal domains, unique factorization, factorization in a polynomial ring. (MATH 0200 or by waiver) 3 hrs. lect./disc. DED (fall: P. Schumer; spring: E. Proctor)

MATH 0310 Probability (Fall)
An introduction to the concepts of probability and their applications, covering both discrete and continuous random variables. Probability spaces, elementary combinatorial analysis, densities and distributions, conditional probabilities, independence, expectation, variance, weak law of large numbers, central limit theorem, and numerous applications. (concurrent or prior MATH 0223 or by waiver) 3 hrs. lect./disc. DED (W. Peterson)

MATH 0311 Statistics (Not offered 2008-09)
An introduction to the mathematical methods and applications of statistical inference. Topics will include: survey sampling, parametric and nonparametric problems, estimation, efficiency and the Neyman-Pearsons lemma. Classical tests within the normal theory such as F-test, t-test, and chi-square test will also be considered. Methods of linear least squares are used for the study of analysis of variance and regression. There will be some emphasis on applications to other disciplines. (MATH 0310) 3 hrs. lect./disc. DED

MATH 0315 Mathematical Models in the Social and Life Sciences (Fall)
An introduction to the role of mathematics as a modeling tool and an examination of some mathematical models of proven usefulness in problems arising in the social and life sciences. Topics will be selected from the following: axiom systems as used in model building, optimization techniques, linear and integer programming, theory of games, systems of differential equations, computer simulation, stochastic process. Specific models in political science, ecology, sociology, anthropology, psychology, and economics will be explored. (MATH 0200 or waiver) DED (D. Dorman)

MATH 0318 Operations Research (Not offered 2008-09)
Operations research is the utilization of quantitative methods as an aid to managerial decisions. In the course, several of these methods will be introduced and studied in both a mathematical context and a physical context. Topics included will be selected from the following: classification of problems and the formulation of models, linear programming, network optimization, transportation problems, assignment problems, integer programming, nonlinear programming, inventory theory, and game theory. (MATH 0200 or waiver) DED

MATH 0323 Real Analysis (Fall)
An axiomatic treatment of the topology of the real line, real analysis, and calculus. Topics include neighborhoods, compactness, limits, continuity, differentiation, Riemann integration, and uniform convergence. (MATH 0223) 3 hrs. lect./disc. DED (S. Abbott)

MATH 0325 Complex Analysis (Fall)
An introduction to functions of a complex variable. Mappings of the complex plane, analytic functions, Cauchy Integral Theorem and related topics. (MATH 0223 or by waiver) 3 hrs. lect./disc. DED (F. Swenton)

MATH 0330 Approaches to Geometry (Not offered 2008-09)
Euclid and axiomatics. The parallel Postulate and non-euclidean geometries. Desargues's Theorem and projective geometry. Convex bodies in the plane. Affine geometry in n dimensions. Arrangements of lines. Finite geometry and coordinatization of planes. The Erlangen Program. (MATH 0200) DED 

MATH 0335 Differential Geometry (Not offered 2008-09)
This course will be an introduction to the concepts of differential geometry.  For curves in space, we will discuss arclength parameterizations, Frenet formulas, curvature, and torsion.  On surfaces, we will explore the Gauss map, the shape operator, and various types of curvature.  We will apply our knowledge to understand geodesics, metrics, and isometries of general geometric spaces.  If time permits, we will consider topics such as minimal surfaces, constant curvature spaces, and the Gauss-Bonnet theorem. (MATH 0200 and MATH 0223) 3 hr. lect./disc. DED

MATH 0341 Topics in Number Theory (Not offered 2008-09)
Drawing on the tools gained in MATH 0241 this course will begin the study of advanced topics in number theory. Topics may include, but won't be limited to, the study of algebraic numbers and number fields, quadratic and cyclotomic extensions of the rational numbers, primary decomposition in number fields, the prime number theorem, Dirichlet's theorem on primes in arithmetic progressions, binary quadratic forms and Gauss's class number problem, p-adic analysis, and the Hasse-Minkowski principle. (MATH 0241 or by waiver) DED

MATH 0345 Combinatorics (Spring)
Special topics in combinatorics including Latin squares, theory of partitions, Kirkman's triple system, permutations and derangements, binomial coefficients, and design theory. (MATH 0200 or by waiver) 3 hrs. lect./disc. DED (J. Schmitt)

MATH 0351 Set Theory (Not offered 2008-09)
This course will begin with a review of the basic facts of union, intersection and complementation of sets. We will also study functions, equivalence relations, and orderings in the context of set theory. The main effort will be the study of ordinal numbers, cardinal numbers, the axiom of choice and transfinite induction. (MATH 0223 or MATH 0241 or MATH 0302) 3 hrs. lect./disc. DED 

MATH 0402 Topics in Algebra (Not offered 2008-09)
A further study of topics from MATH 0302. These may include field theory, algebraic extension fields, Galois theory, solvability of polynomial equations by radicals, finite fields, elementary algebraic number theory, solution of the classic geometric construction problems, or the classical groups. (MATH 0302 or by waiver) 3 hrs. lect./disc. DED

MATH 0410 Stochastic Processes (Spring)
Stochastic processes are mathematical models for random phenomena evolving in time or space. This course will introduce important examples of such models, including random walk, branching processes, the Poisson process and Brownian motion. The theory of Markov chains in discrete and continuous time will be developed as a unifying theme. Depending on time available and interests of the class, applications will be selected from the following areas: queuing systems, mathematical finance (Black-Scholes options pricing), probabilistic algorithms, and Monte Carlo simulation. (MATH 0310) DED (W. Peterson)

MATH 0423 Topics in Analysis (Spring)
In this course we will study advanced topics in real analysis, starting from the fundamentals established in MA401. Topics may include: basic measure theory; Lebesgue measure on Euclidean space; the Lebesgue integral; total variation and absolute continuity; basic functional analysis; fractal measures. (MATH 0323 or by waiver) 3 hrs. lect./disc. DED (F. Swenton)

MATH 0432 Elementary Topology (CW 15) (Not offered 2008-09)
An introduction to the concepts of topology. Theory of sets, general topological spaces, topology of the real line, continuous functions and homomorphisms, compactness, connectedness, metric spaces, selected topics from the topology of Euclidean spaces including the Jordan curve theorem. (MATH 0122 or by waiver) 3 hrs. lect./disc. DED 

MATH 0451 Mathematical Logic (Not offered 2008-09)
This course begins with a look at sentential logic, truth tables and their applications. It also explores first order logic and natural deduction systems, and considers how to decide whether a statement is true or provable. Important theorems covered include the Completeness Theorem and the Compactness Theorem. The course concludes with a discussion of the Incompleteness Theorem, the monumental work of Kurt Gödel. (MATH 0302 or by waiver) DED

MATH 0480 Research Experience in Mathematics (CW) (Not offered 2008-09)
The object of this course is active participation in mathematical investigation and exposition. Students work collaboratively on current research questions provided by the instructor. The course includes a review of relevant literature and research methods. Students are required to present their findings both in writing (consistent with the standards of the discipline) and in public presentations. (MATH 0302 or MATH 0323 or waiver) DED

MATH 0500 Advanced Study (Fall, Winter, Spring)
Individual study for qualified students in more advanced topics in algebra, number theory, real or complex analysis, topology. Particularly suited for those who enter with advanced standing. (Approval required) 3 hrs. lect./disc. (Staff)

MATH 0704 Senior Seminar (Fall, Spring)
Each student is required to complete and present a major paper on a topic chosen with the advice of a faculty member. In addition, during the academic year, each student is expected to attend a series of lectures designed to introduce and integrate ideas of mathematics not covered in the previous three years. 3 hrs. lect./disc. DED (fall: E. Proctor; spring: D. Dorman)