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Higher-Level Mathematics Courses for Spring 2009
This is a list of high level courses in the domains of pure and applied mathematics.
You can also look at the list
of high level courses in actuarial science and financial mathematics.
Description:
The purpose of Math 2011 (248) is to promote a thorough understanding of the mathematics students should know to be qualified to teach at the elementary level. These courses are not intended to correct deficiencies in basic mathematical background or issues such as "the mathematics kids need to know" and "the methods for teaching elementary school mathematics". However, the student-centered focus, the variety of instructional techniques, and the ties to the standards help students begin building an advanced perspective on the teaching of elementary mathematics. The course focuses on the abstract and theoretical structure of mathematics, emphasizing problem solving, communication, and reasoning. Special attention is given to exploring and communicating the ideas and reasons behind the mathematical manipulations. The topics of the courses are chosen to support and extend the expectations set forth by the Mathematical Standards, K-8 (NCTM 2000). This course is a successor to Math 2010 (247).
Taught by: Fabiana A. Cardetti. Meets: 2-3:15 TuTh
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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.
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Section 1: Taught by: $Unique empty. Meets: 2-2:50 MWF
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Section 2: Taught by: $Unique empty. Meets: 9:00-9:50 MWF
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Section 3: Taught by: Andrew H. Haas. Meets:11-12:15 TuTh
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Section 4: Taught by: David Gross. Meets: 9:30-10:45 TuTh
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Section 5: Taught by: Sarah Glaz. Meets: 2-4:15 TTh
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Section 6: Taught by: $Unique empty. Meets: 2-2:50 MWF
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Section 7: Taught by: $Unique empty. Meets: 3-4:15 MW
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Description:
Foundations and essential results of Euclidean and non-Euclidean geometries. Groups of isometries acting on the Euclidean plane, the Real plane and the extended Complex plane. Additional topics may include symmetry groups and fractal geometry.
Section 1: Taught by: $Unique empty.. Meets:
10:00-10:50 MWF
Section 2: Taught by: $Unique empty. Meets: 11:00-12:15 TuTh
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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.
Section 1:
Taught by: $Unique empty. Meets: 10:00-10:50 MWF
Section 2:
Taught by: Taught by: $Unique empty.
Meets: 12:000-12:50 MWF
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Description:
A historical study of the growth of the various fields of mathematics.
Taught by: Gerald Leibowitz. Meets: 12-12:50 MWF
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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 2784 (200) and then as Math 2794W (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: $Unique empty.
Meets: 5:30-6:20 Usually on Wednesday
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Description:
The famous mathematician Paul Erdos
often spoke about The Book, where he imagined God wrote down the best proof of every theorem. About 10 years ago two mathematicians wrote a real book called "Proofs From The Book". Each chapter is a self-contained morsel of beauty, while certain themes permeate the whole. This class aims to explore the beauty of mathematics through various examples of particularly elegant proofs.
For example, we will see 6 proofs that there are infinitely many primes, proofs that π2 and e are irrational, which integers can be represented as the sum of 2 squares, the Cauchy-Schwarz inequality, and Cayley's formula for the number of labelled trees. Some of the topics require nothing more than a familiarity with calculus, while others involve number theory, combinatorics, or graph theory. We will develop the requisite concepts in the course.
Topics include:
- Six proofs of the infinitude of primes
- Bertrand's postulate
- Sum of 2 squares
- π2 and er are irrational
- Sum of 1/(n2) is (π2 )/6 (including Euler's method of divergent sums)
- Relative sizes of rationals, reals, naturals; Schroeder-Bernstein theorem
- Cauchy-Schwarz, Arithmetic/Geometric/Harmonic means
- Cotangent and the Herglotz trick
- Pigeonhole principle
- Applications of Euler's Formula
- Cayley's formula for labeled trees
- Bijections: Partition numbers
- How to guard a museum
- Friendship Theorem
Prerequisites: A linear algebra course and one course which has Math 2710 as a prerequisite. Consent of instructor required.
Taught by: $Unique empty. Meets: 12:30-1:45 TuTh
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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 may include Laplace and Fourier transforms and other areas of current interest.
Taught by: William Abikoff. Meets: 9:30-10:45 TuTh
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Description:
The fundamental concepts of calculus are established with full mathematical rigor and proofs. The course covers: the topology of the real line; sequences, continuity, differentation and integration of functions of one variable; convergence of series of real numbers and functions. Emphasis will be placed on understanding, constructing and writing mathematical proofs.
Taught by: $Unique empty. Meets: 11:00-11:50 MWF
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Description:
This is the second semester of a year long transitional course in Analysis. The subject is the theory of functions of several variables. As in Math 3150 (273), the emphasis is on understanding, constructing and writing mathematical proofs. Topics include: rigorous treatment of fundamental concepts in calculus, including limits and convergence of sequences and series, continuity and differentiability of functions in a Euclidean space.
Taught by: Ron Blei. Meets: 9:30-10:45 TuTh
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Description:
This is a thorough introduction to probability
theory that uses Calculus (Math 1120-1122 or Math 1131-1132, and Math
2110).
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).
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Section 1: Taught by: $Unique empty. Meets: 11:00-11:50 MWF
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Section 2: Taught by: $Unique empty. Meets:10:00-10:50 MWF
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Section 3: Taught by: Richard Bass. Meets: 9:30-10:45 TuTh
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Section 4: Taught by: Gerald Leibowitz. Meets: 9:00-9:50 MWF
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Description:
A sequel to Math 3160 (231). The course covers conditional probability and conditional expectation, Markov Chains in discrete time and continuous time, renewal theory, the Poisson process, and the Brownian Motion process.
Taught by: Evarist Gine-Masdeu.
Meets: 11-11:50 MWF
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Description:
This course is an introduction to abstract linear algebra with an emphasis on the development of careful mathematical reasoning. Among the topics covered by the course are vector spaces, linear maps, matrices, bases, orthogonality, dual spaces, scalar products, eigenvalues, triangulation and diagonalization. Writing proofs will be an integral part of the class.
Taught by: Ralf Schiffler. Meets: 9:30-10:45 TuTh
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Description:
The main goal of this course is to discuss Galois theory, which is the study of relationships among roots of polynomials. For example, we will use Galois theory to prove that there is no formula analogous to the quadratic formula for the roots of xn - x -1 when n is at least 5, or in fact for the roots of most polynomials of degree at least 5. More generally, Galois theory provides a correspondence between two different concepts in abstract algebra: fields and groups. Our study of fields will use linear algebra in interesting ways. For example, we will see how to show certain polynomials are irreducible using the concept of dimension. Only towards the end of the course will group theory be needed in a serious way, at which point what we need from group theory will be reviewed.
Prerequisites: Math 3230(216).
Taught by: Keith Conrad. Meets: 12:30-1:45 TuTh
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Description: This course serves as an introduction to mathematical logic
and three of its subfields: model theory, computability theory, and set
theory. The emphasis will be on first-order logic, though propositional
logic will receive significant attention. The course will cover the
celebrated Completeness and Incompleteness Theorems, the Compactness
Theorem, and other topics traditionally found within an introductory logic
course.
Taught by: $Unique empty. Meets: 11:00-12:15 TuTh
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Description:
Based on Math 2410 (211), Math 3410 (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 2410 (211).
Taught By: $Unique empty. Meets: 9:00-9:50 MWF
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Description:
This is a second introductory course to modern numerical techniques, i.e., a sequel to Math 3510. It starts with a survey of modern approximation techniques and explains how, why, and when the techniques can be expected to work. Using this background the course covers difference equations, numerical methods for the solution of ordinary and partial differential equations, eigenvalue computations. The course demonstrates the relevance of numerical analysis to a variety of disciplines and provides ample practice for students. The applications chosen demonstrate concisely how numerical methods can be, and often must be, applied in real-life situations. The exercise sets include many applied problems from diverse areas of engineering, as well as from the physical, computer, biological, and social sciences.
Taught by: Vadim Olshevsky. Meets: 4:00-5:15 MW
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Description:
What makes expert problem solvers so effective? In this course we will examine some of the techniques and insights of expert problem solvers like George Polya, whose book is a classic in the subject. For the most part, we will illustrate the different strategies by solving problems, with occasional peeks at some of the research on the subject. The problems will be drawn from a broad range of areas, giving students the chance to see how useful mathematical ways of thinking can be. Prerequisites include two semesters of calculus and an inquisitive mind.
Taught by: Charles Vinsonhaler and Thomas C. DeFranco. Meets: 5:15-6:30 MW
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University of Connecticut | 
