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Higher-Level Mathematics Courses for Spring 2012
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: The development of the number system with applications to elementary number theory and analytic geometry. This course is intended only for students in elementary education, specifically those in pre-teaching elementary and the NEAG School of Education.
Prerequisites: PSYC 1100 and three credits of Mathematics other than MATH 1010(101). Not open for credit to students who have passed MATH 2110(210), 2410(211), 220 or 2130Q(230Q), or 2143Q(245Q).
Offered: Fall
Credits: 3
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Description: The development of the number system with applications to elementary number theory and analytic geometry. This course is intended only for students in elementary education, specifically those in pre-teaching elementary and the NEAG School of Education.
Prerequisites: PSYC 1100 and three credits of Mathematics other than MATH 1010(101). Not open for credit to students who have passed MATH 2110(210), 2410(211), 220 or 2130Q(230Q), or 2143Q(245Q).
Offered: Spring
Credits: 3
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Description: Weekly seminars and short essays reflecting on the learning experiences and content of MATH 2110(210).
Prerequisites: ENGL 1010 or 1011 or 3800. There is also a corequisite: MATH 2110Q(210Q).
Offered: Either semester
Credits: 1
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Description: Systems of equations, matrices, determinants, linear transformations on vector spaces, characteristic values and vectors, from a computational point of view. The course is an introduction to the techniques of linear algebra with elementary applications.
Prerequisites: MATH 1132(116), 1152(121 or 136), or 2142(244). Recommended Preparation: a grade of C- or better in MATH 1132(116). Not open for credit to students who have passed MATH 3210(215).
Offered: Either semester
Credits: 3
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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.
Prerequisites: MATH 1121(113), 1126, or 1131(115).
Offered: Either semester
Credits: 3
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Description: A course designed to prepare the serious student for the more 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, equivalence relations and partitions, functions, one-to-one functions, onto functions, induced set functions,... This is a required course for most mathematics majors.
Prerequisites: MATH 2110(210) or 220 or consent of instructor.
Offered: Either semester
Credits: 3
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Description: A historical study of the growth of the various fields of mathematics.
Prerequisites: (i) MATH 2110Q(210Q) or 2130Q(230Q), and 2210 (227Q) or 2410Q(211Q), or (ii) MATH 2144Q(246Q) or 2420(221Q); and ENGL 1010 or 1011 or 3800. This course may not be counted in any of the major groups described in the Mathematics Department listing.
Offered: Either semester
Credits: 3
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Description: The student will attend talks during the semester, and choose a mathematical topic from one of them to investigate in detail. The student will write a well-revised, comprehensive paper on this topic, including a literature review, description of technical details, and a summary and discussion.
Prerequisites: Either MATH 2110, 2130, or 2143; MATH 2410, 2420 or 2144; ENGL 1010 or 1011 or 3800.
Offered: Either semester
Credits: 2
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Description: The student will attend talks during the semester, and choose a mathematical topic from one of them to investigate in detail. The student will write a well-revised, comprehensive paper on this topic, including a literature review, description of technical details, and a summary and discussion, building upong the writing experience in MATH 2784.
Prerequisites: MATH 2784(200); ENGL 1010 or 1011 or 3800.
Offered: Either semester
Credits: 2
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Description: This course, with a change of topic, may be repeated for credit. Open only with consent of instructor.
Prerequisites: Open to juniors or higher.
Offered: Either semester
Credits: 3
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Description: Functions of a complex variable, integration in the complex plane, conformal mappings.
Prerequisites: MATH 2110(210) and 2410(211), or 2144 or 2420(221). Not open for credit to students who have passed MATH 5046(352).
Offered: Either semester
Credits: 3
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Description: Introduction to the theory of functions of one and several real variables.
Prerequisites: MATH 2142(244), a grade of C or better in 2710(213), or 214; MATH 2410(211) or 2420(221).
Offered: Either semester
Credits: 3
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Description: Introduction to the theory of probability. Discussion of some of the probability problems encountered in scientific and business fields.
Prerequisites: MATH 2110(210) or 220, which may be taken concurrently with the consent of the instructor. Not open if passed MATH 3610(283) or 3660(284).
Offered: Either semester
Credits: 3
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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.
Prerequisites: MATH 2142(244) or a grade of C or better in 2710(213). Recommended preparation: MATH 2210Q(227Q) or 2144Q(246Q).
Offered: Fall
Credits: 3
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Description: A course designed to prepare the serious student for the more 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, equivalence relations and partitions, functions, one-to-one functions, onto functions, induced set functions,... This is a required course for most mathematics majors.
Prerequisites: MATH 2110(210) or 220 or consent of instructor.
Offered: Either semester
Credits: 3
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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 focuses on those parts of classical number theory which still have modern relevance in the subject: the Euclidean algorithm, modular arithmetic, distribution of primes, diophantine equations, applications to cryptography, arithmetic in quadratic rings and polynomial rings, and quadratic reciprocity. The examples in this course will provide a lot of food for thought for anyone who later takes abstract algebra.
Prerequisites: MATH 2142(244), a grade of C or better in 2710(213), or 214.
Offered: Fall
Credits: 3
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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.
Prerequisites: MATH 2142(244) or a grade of C or better in 2710(213).
Offered: TBA
Credits: 3
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Description: The basic idea and of differential geometry is to say something about the geometry
of an object by moving a little bit on this object - for instance moving along a curve or on
a surface. Turning this approach into a questions it reads:
what kind of information can I get about my curve or my surface if I move
a little bit along them. It turns out that there is indeed a lot one can learn.
For a curve, one gets tangent directions, curvature and other geometric information
in this way.
For a surface, there are 2-d generalizations of these concepts.
One striking fact is that knowing this information everywhere allows you for
instance to discover that the earth is not flat. Furthermore it allows to explain
why there cannot be any maps of the earth which give the right distances and angles
at the same time. These types of considerations are also the basis for the theory of general relativity.
In this course, we will treat curves and surfaces from the above perspectives which lead us
to the results discussed above. We will provide a classical treatment, but the results and concepts
have applications in discretized versions for computer imaging and methods of finite elements.
Prerequisites: MATH 2142(244) or a grade of C or better in 2710(213), and either (i) MATH 2110(210) or 2130Q(230Q), and 2410Q(211Q), or (ii) MATH 2144Q(246Q).
Offered: Fall (even years)
Credits: 3
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Description: A historical study of the growth of the various fields of mathematics.
Prerequisites: (i) MATH 2110Q(210Q) or 2130Q(230Q), and 2210 (227Q) or 2410Q(211Q), or (ii) MATH 2144Q(246Q) or 2420(221Q); and ENGL 1010 or 1011 or 3800. This course may not be counted in any of the major groups described in the Mathematics Department listing.
Offered: Either semester
Credits: 3
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Description: Analysis of numerical methods associated with linear systems, eigenvalues, inverses of matrices, zeros of non-linear functions and polynomials. Roundoff error and computational speed.
Prerequisites: Either (i) MATH 2110(210) or 2130(230), 2410(211), and either 2210(227) or 3210(215) or (ii) MATH 2144; and knowledge of at least one programming language.
Offered: Fall
Credits: 3
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Description: Construction of mathematical models in the social, physical, life and management sciences. Linear programming, simplex algorithm, duality. Graphical and probabilistic modeling. Stochastic processes, Markov chains and matrices. Basic differential equations and modeling.
Prerequisites: MATH 2420(221); or MATH 2410(211) and 2210(227). Knowledge of a programming language is strongly recommended. Not open for credit to students who have passed MATH 5530(304) or 5540(305), CHEM 305, or PHYS 5350.
Offered: Fall
Credits: 3
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Description: Metric spaces, sequences and series, continuity, differentiation, the Riemann-Stieltjes integral, functions of several variables.
Prerequisites: Consent of Instructor
Offered: Fall
Credits: 3
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Description: Group theory, ring theory and modules, and universal mapping properties.
Prerequisites: Consent of Instructor
Offered: Fall
Credits: 3
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Description: Topological spaces, connectedness, compactness, separation axioms, Tychonoff theorem, compact-open topology, fundamental group, covering spaces, simplicial complexes, differentiable manifolds, homology theory and the De Rham theory, intrinsic Riemannian geometry of surfaces.
Prerequisites: Consent of Instructor
Offered: Fall
Credits: 3
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