Higher-Level Mathematics Courses for Fall 2008
This is a list of higher level courses in the domains of pure and applied mathematics.
You can also look at the list
of higher level courses in actuarial science and financial mathematics.
Description:
This course, together with Math 248Q, develops an advanced perspective on and profound understanding of concepts, structures, and algorithms constituting the core of K-8 math curriculum. The topics of the course are chosen to support and extend the expectations set forth by the Mathematical Standards, K-8 (NCTM 2000). The course focuses on the abstract and theoretical structure of mathematics, emphasizing problem solving, multiple representations, and reasoning. Special attention is given to exploring and communicating the ideas and reasons behind the mathematical manipulations. This course is recommended for students in elementary education. This course does not count toward a major or minor in mathematics.
Taught by: Fabiana A. Cardetti. Meets: 2-3:15 TuTh
|
Description:
MATH 202 W: Pedagogical Seminar.
Description: The student will attend 15 seminars per semester, and will write a weekly one-page essay reflecting on the mathematical and pedagogical experiences in his/her section of Math 210 during the week. The student will finally integrate this material into a comprehensive fifteen-page paper on this topic, including a discussion of the positive and negative experiences of learning, how they feel the learning process might have been improved, and how the course affects their view of the teaching process. The student will also concentrate on one topic from the course, and using Internet and Library resources, discuss alternate ways of teaching (and learning) this topic. Concurrent registration in Math 210 is required. This course is primarily intended for students who see themselves as future teachers and will apply to the NEAG School of Education at the end of their sophomore year.
Taught by: Craig Miller.
Meets: 2-2:50 W
|
Description:
Systems of equations, matrices, determinants, linear transformations on vector spaces, characteristic values and vectors, from a computational point and theoretical point of view. The course is an introduction to the techniques of linear algebra with elementary applications.
-
Section 1: Taught by: Nung-Sing Sze. Meets:11-11:50 MWF
-
Section 2: Taught by: Sarah Glaz. Meets: 12:30-1:45 TTh
-
Section 3: Taught by: Matthew Cecil. Meets:2-3:15 TuTh
-
Section 4: Taught by: [Ralf Schiffler]. Meets: 12-1:15 TuTh
-
Section 5: Taught by: Erin T. Mullen. Meets: 9:30-10:45 TTh
-
Section 6: Taught by: [Marius Ionescu]. Meets: 12:30-1:45 TuThF
|
Description:
A fresh look at geometry, old and new. Euclidean and non-Euclidean geometries are examined from from different perspectives. Topics may include symmetries, the role of the parallel postulate and some topics from 19th and 20th century geometry, e.g. fractals and knots.
Taught by: Nirattaya Khamsemanan. Meets:
10:00-10:50 MWF
|
Description:
This course is essential preparation for theoretical upper division mathematics courses. It includes basic concepts, principles and techniques of mathematical proof. It will also cover concepts commonly assumed in some of the higher mathematics courses; these concepts include sets, set operations, indexed family of sets, mathematical induction, equivalence relations and partitions, functions, one-to-one functions, onto functions, and induced set functions. Students will write proofs and revise them according to instructor feedback. This is a required course for most mathematics majors.
Taught by: Kristen Sellke. Meets: 11:00-11:15 TuTh
|
Description:
Some aspects of the history of
mathematics from ancient times until the age of Newton and Leibniz.
Evaluation based on essays, mid-semester and final exams, and numerous
homework sets on mathematical methods.
|
Description:
To get exposure to various mathematical topics not met in other courses, students in this course attend talks once a week, by both local and outside speakers. Background for the talks will usually not go beyond calculus and some linear algebra. Attending at least 7 of the talks is required. The topic from one talk, at the student's choosing, will form the basis for a comprehensive written paper. The paper will give a self-contained introduction to the topic, include technical details going beyond the talk itself, and show a familiarity with relevant sources in the literature. It will be read by a member of the mathematics department and, following a discussion, be revised and re-submitted.
The course may be taken twice, first as Math 200 and then as Math 201W. Completion of the second course will fulfill a W requirement within the mathematics department.
Students enrolled, or planning to enroll, in this course are welcome to suggest topics for lectures in this course.
Taught by: Maria Gordina.
Meets: 5:30-6:20 Usually on Wednesday
|
Description: Analysis and Probability on Fractals
Taught by: Alexander Teplyaev. Meets: TuTh 11-12:15
|
Description:
Complex numbers and functions of a complex variable. Differentiation, Cauchy-Riemann equations, line integrals, Cauchy's theorem, Cauchy's integral formulas, series, residue calculus, fractional linear transformations, and conformal mapping. Applications will include Fourier and Laplace transforms. Prerequisite: Math 210 or 220, and Math 211 or 221
Taught by: Maria Gordina. Meets: 2-3:15 TuTh
|
Description:
This course provides a transition from calculus to real
analysis. To achieve this, we will revisit most of the results from the
one-variable calculus curriculum and prove them rigorously. The starting
point is a detailed examination of the real numbers, showing how they
arise naturally when we try to ensure that all sequences of rational
numbers that "should converge" actually have limits. This will involve
studying convergence and limits for sequences and series. Once we have a
solid understanding of the reals, we will give precise definitions of
limits and continuity for functions, and study one of the most important
limits - the derivative. We will prove all of the basic theorems of
differentiation, including the product, quotient and chain rules, and most
essentially, the mean value theorem. The mean value theorem is the first
step towards one of our main goals, the fundamental theorem of calculus.
In order to reach it we will first need to understand how local properties
such as continuity or bounds on the derivative of a function can be used
to control global properties of the function provided the underlying set
is compact; these ideas will lead us to both uniform continuity and a
proof of the extreme value theorem. After mastering the notation for
Riemann sums and proving the fundamental theorem of calculus we will spend
any time that remains (if there is any!) studying convergence of sequences
of functions.
This is a challenging course that emphasizes reading and understanding
proofs, as well as constructing and writing proofs of your own. However,
the rewards are commensurate with the challenge: the calculus is one of
the truly great achievements of the human intellect, and this is your
opportunity to really understand and appreciate it!
Taught by: Luke Rogers. Meets: 11:00-11:50 MWF
|
Description:
This is a thorough introduction to probability
theory that uses Calculus (Math 112-114 or Math 115-116, and Math
210).
We cover the following: combinatorial analysis (permutations,
combinations); basic set-up (sample space, events, axioms of
probability); conditional probability (Bayes rule), independence;
random variables (discrete and continuous); cumulative distributions,
densities; expectation, variance, moment generating functions; jointly
distributed random variables; limit theorems (Central Limit theorem,
weak law of large numbers).
|
Description:
This course studies fundamental algebraic systems in mathematics, selected from groups, rings, fields, and modules. Examples of groups include the invertible matrices with a fixed size and the roots of unity. Rings are illustrated by integers, polynomials, and modular arithmetic. Complex numbers, rational numbers, and rational functions are examples of fields. (There are also finite fields, which are used all the time in computer science.) Finally, ordinary vectors in space and any lattice in the plane are examples of modules. The concern with these algebraic systems is not simply the study of individual systems, but also of functions between systems which carry one operation into the other. For instance, the determinant not only converts matrices into numbers, but it sends a product of matrices into a product of numbers. The level of attention given to such operation-preserving transformations (putting them on an equal footing with the algebraic systems they transform) is one of the characteristic features of abstract algebra, and also one of the algebraic ideas which have reached into other areas of mathematics.
Math 213 and a linear algebra course are prerequisites.
Taught by: David Gross. Meets: 11:00-11:50 MWF
|
Description:
Number theory is the study of the integers, but this description hardly conveys the beauty of this part of
mathematics. One of the main goals of this course is pedagogical: to see that mathematics is a vibrant
intellectual activity and not a set of fixed rules developed by some higher authority. This viewpoint is
especially useful for future teachers. Students will carry out many numerical experiments, generate conjectures
based on patterns observed, and then prove or disprove these conjectures. The content will cover
the following topics: Euclid's algorithm, unique factorization, modular arithmetic, the distribution of primes,
Diophantine equations, cryptographic applications, quadratic reciprocity, and number theory in quadratic
rings and polynomial rings. The examples studied in this course will provide a lot of food for thought for anyone
who later takes abstract algebra.
Prerequisite: Math 213
Taught by: Alvaro Lozano-Robledo.
Meets: 2:00-3:15 TuTh
|
Description:
Combinatorics concerns itself with problems involving discrete
structures, generally on finite or countably infinite sets. Often we
want to count the number of ways something can be done: arranging 5
books on a shelf, partitioning a sports club into 5 disjoint
teams, or dividing a polygon into triangles using diagonals
which only intersect at a vertex. Sometimes we consider the
relationships among such objects, and the discrete structures involved,
yielding graphs (imagine an airline route map that connects some pairs
of cities, but not all) or partially ordered sets. In all of these we
look for elegant ways of understanding and proving our answers are
correct, avoiding simpleminded brute-force computations.
This course will give an overview of combinatorial techniques and
applications. We will count things using basic principles of
arithmetic, using infinite series, and using bijections that help us
translate objects we want to count into a different form that is easier
to count. We will see surprisingly deep applications of the obvious Pigeonhole
Principle. This course is an excellent way for students to
strengthen their proof writing in contexts which are more easily
accessible and concrete than many other areas of mathematics. These
ideas come up frequently in other areas of mathematics in computer
science, and in parts of chemistry and biology. Prerequisite: MATH 213
or 244.
Taught by: Thomas Roby. Meets:11-12:15 TuTh
|
Description: In differential geometry, one uses methods of differential and integral calculus to study problems in geometry. In this course we will look at one and two dimensional geometric objects and study both their intrinsic geometry, as well as how they sit naturally inside Euclidean space. Intrinsically, one thinks of the curve or surface as the entire world, with its own notions of space, distance and angle. By moving around this world, often taking little baby steps, we can come to understand how it compares to the geometric world we seem to live in. This is an introduction to the geometry of curves and surfaces, a very beautiful and classical subject.
Taught by: Andrew H. Haas. Meets: MWF 2-2:50
|
Description:
Description: Based on Math 211, Math 272 will cover several topics on
Ordinary Differential Equations and some basic Partial Differential
Equations. Linear systems; phase plane analysis; 2nd order ODEs;
Bessel's equation; Fourier series; the method of separation of variables
applied to the heat equation, Laplace equation, and wave equation.
Prerequisite: Math 211.
Taught By: [Ben Ari Iddo]. Meets: 9:30-10:45 TuTh
|
Description:
This is an introductory course to modern numerical techniques. It starts with
a review of relevant theorems of calculus and proceeds to issues related to
finite computer arithmetic: roundoff errors, algorithms, and convergence. Based on this knowledge efficient and numerically stable algorithms are developed for:
i) solution of equations in one variable, f(x)=0, most notably the Newton algorithm;
ii) interpolation and polynomial approximation, including cubic spline interpolation; iii) numerical integration; iv) direct methods for solving linear systems, Ax=b, based on Gaussian Elimination with pivoting.
Taught by: Vadim Olshevsky. Meets: 11:00-12:15 TuTh
|
Description:
We will look at the way three different areas use mathematics, mainly
linear algebra and differential equations to model aspects of real
life. Problems treated will be population modelling in various ways,
vibrations problems in simple and complex mechanical systems
(such as suspension bridges), and traffic flow problems, including how
differential equations can tell how people behave after a traffic
light turns green (and why you shouldn't honk or rubberneck!).
Reasonable background would be some differential equations and some
linear algebra.
Taught by: P. Joseph McKenna.
Meets: 11-12:15 TuTh
|
Description: Math 315
Taught by: Kyu-Hwan Lee. Meets: MWF 2-3:50
|
|
