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Honors Calculus

Students who are interested in math or science might also want to consider a more challenging Honors version of Calculus II and III, Math 116 and Math 260 (the analogues of math 114 and math 240, respectively). These courses will cover essentially the same material as 114 and 240, but more in depth and involve discussion of the underlying theory as well as computations.

Math 180 - Analytical Methods in Economics, Law, and Medicine

This course will include units on Decision Analysis, Theory of Games, Finance, Statistical Analysis, and Evidence-Based Medicine. Some very basics calculus and its applications to any of these areas might be discussed. Other units may be added at the discretion of the instructor or if there is demand. Students will be asked to complete projects individually or in groups. (No prior knowledge of calculus required.) Suitable for both pre-law and pre-med students.

Math 202 - Proving Things: Analysis

This course focuses on the creative side of mathematics, with an emphasis on discovery, reasoning, proofs and effective communication, while at the same time studying real and complex numbers, sequences, series, continuity, differentiability and integrability. Small class sizes permit an informal, discussion-type atmosphere, and often the entire class works together on a given problem. Homework is intended to be thought-provoking, rather than skill-sharpening.

Math 203 - Proving Things: Algebra

This course focuses on the creative side of mathematics, with an emphasis on discovery, reasoning, proofs and effective communication, while at the same time studying arithmetic, algebra, linear algebra, groups, rings and fields. Small class sizes permit an informal, discussion-type atmosphere, and often the entire class works together on a given problem. Homework is intended to be thought-provoking, rather than skill-sharpening.

Math 340 - Discrete Mathematics I

Prerequisites: Math 114 or Math 115 or permission of the instructor
Topics will be drawn from some subjects in combinatorial analysis with applications to many other branches of math and science: graphs and networks, generating functions, permutations, posets, asymptotics.

Math 341 - Discrete Mathematics II

Prerequisites: Math 340/Lgic 210 or permission of instructor
Topics will be drawn from some subjects useful in the analysis of information and computation: logic, set theory, theory of computation, number theory, probability, and basic cryptography. See also:
Math 341 web page (Spring 2007).

Math 430 - Introduction to Probability

Random variables, events, special distributions, expectations, independence, law of large numbers, introduction to the central limit theorem, and applications.

Math 432 - Game Theory

We will analyze a wide variety of games by studying several mathematical models, including conbinatorial games, two person (zero-sum and general-sum) games, noncooperative games, and Nash equilibrium. The interdisciplinary approach will be appealing to students in business, economics, mathematics, political science, statistics, etc.

Math 480 -- Topics in Modern Mathematics

Two semester course, 1 cu/semester
Prerequisite: Math240/241 and Math312 or the permission of the instructor
Fall '07 The analytic, geometric and number-theoretic properties of two fractal sets: the so-called Sierpinski gasket of dimension about 1.585 and the Apollonian gasket of dimension about 1.308.
Text: the preliminary draft of Kirillov, A. A. A Tale of two Fractals Spring '08 When a school student first meets mathematics, (s)he is told that it is a science which studies numbers and Þgures. Later, in college, (s)he learns analytic geometry which express geometric notions using numbers. So, it seems that numbers is the only ob ject of study in mathematics. True, if you open a modern mathematical journal and try to read any article, it is very probable that you will see no numbers at all. Instead, authors speak about sets, functions, operators, groups, manifolds, categories, etc. Nevertheless, all these notions in one way or another are based on numbers and the Þnal result of any mathematical theory usually is expressed by a number. So, I think it is useful to discuss with math major students the question posed in the title ("What is number?"). I want to show, what meaning can the term ÒnumberÓ have in modern mathematics, speak of some problems arising in this connection and of their solutuons. Click here for a more complete description by Prof. Kirilov.

Click here for more information on the Spring 2009 Math 480 with Prof. Josh Guffin.

Math 512 -- Advanced Linear Algebra

Prerequisite(s): Math 114 or 115. Math 512 covers Linear Algebra at the advanced level with a theoretical approach. Students can receive credit for at most one of Math 312 and Math 512. Topics will include: Vector spaces, Basis and dimension, quotients; Linear maps and matrices, ; Determinants, Dual spaces and maps; Invariant subspaces, Canonical forms; Scalar products: Euclidean, unitary and symplectic spaces; Orthogonal and unitary operators; Tensor products and polylinear maps; Symmetric and skew-symmetric tensors and exterior algebra.


Stat 530 / Math 546 -- Probability Theory

The required background is (1) enough math background to understand proof techniques in real analysis (closed sets, uniform convergence, fourier series, etc.) and (2) some exposure to probability theory at an intuitive level (a course at the level of Ross's probability text or some exposure to probability in a statistics class).

Homeworks: there will be five homework sets, as well as a take-home final exam in each semester.

After a summary of the necessary results from mesaure theory, we will learn the probabist's lexicon (random variables, independence, etc.). We will then develoop the necessary techniques (Borel Cantelli lemmas, estimates on sums of independent random variables and truncation techniques) to prove the classical laws of large numbers. Next come Fourier techniques and the Central Limit Theorem, followed by combinatorial techniques and the study of random walks.

Texts:
"Probability: theory and examples", 3rd Edition, by R. Durrett.

Stat 531 / Math 547 -- Probability Theory

Required background Stat 530.

Homeworks: there will be four homework sets, as well as a take-home final exam.

The first task will be to understand conditional expectation. This is harder than you think. From there, we will define and study martingales. We will discuss Markov chains, not giving a comprehensive treatment but hitting the basics and some highlights. Lastly we will construct Brownian motion and just barely scratch the surface as to the properties of this strange object.

Texts:
"Probability: theory and examples", 3rd Edition, by R. Durrett.

Math548 -- Topics in Analysis (Operators in Hilbert space)

The main goal: to give a complete and self-contained proof of the Spectral Theorem for self-adjoint operators (not necessary bounded) in a Hilbert space; The main tool: the operational calculus for commuting self-adjoint operators; The main idea: the commuting self-adjoint operators can play the role of real numbers; one can add them, subtract, multiply and use as arguments in functions of one or several real variables.

Prerequisite: Math 360-361 and 312/412 or the permission of the instructor.

Text:
Kirillov A.A. and Gvishiani A.D., “Theorems and Problems in Functional Analysis”, Springer-Verlag, 1982, Chapter 5, pp 116-135 (theory), 219-230 (problems), 325-334 (hints).
ISBN 0-387-90638-X, 3-540-90638-X.
ISBN 0-387-90638-X, 3-540-90638-X.

Math 5xx--Inequalities, Geometry, and the Calculus of Variations

Short description: One of the most famous analytic inequalities of all time is the arithmatic/geometric mean inequality: for any set of numbers (a_1, ..., a_n) (a_1 + ... + a_n)/n >= (a_1 x .... x a_n)^{1/n} with equality only when the numbers are all equal. A similarly famous geometric inequality is the isoperimetric inequality: for any closed curve in the plane, the ratio
Area/Length
is maximized for the round circle. Inequalities such as these are powerful tools in the study of geometry and analysis. This course presents a systematic survey of the basic theory of inequalities, leading up to applications in problems of geometric optimization. We will end with a survey of various forms of the isoperimetric inequality, and some unexpected results of these theorems, including a proof of the Faber-Krahn theorem: Among all domains with unit area in the plane, the round circle has the lowest first eigenvalue of the Laplacian.
Textbook: Hardy, Littlewood and Polya. Inequalities. ($44, paperback)
Recommended Reading: Gelfand and Fomin. Calculus of Variations ($8, paperback)
Click here for a more complete description and syllabus by Prof. Cantarella.