MATH 4801 SR Senior Seminar 1-0-0 P/F

Instructor: Prof. Enid Steinbart

Description: For undergraduate students only

Prerequisites: Math 2401, Calculus III, Math 2411, Honors Calculus III, or Math 2605, Calculus III for CS

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Time: M 4:35 - 4:55

MATH 4802 BAR Mathematical Problem Solving 2-0-0 P/F

Instructor: Prof. Matthew Baker

Description: This course is intended to teach general mathematical problem solving skills, and to prepare students to take the Putnam Examination.

Prerequisites:

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Time: T 3:05 - 4:55

Math 4803 HP Combinatorial Game Theory 3-0-3

Instructor: Prof. Thomas Morley

Description: Section HP is for Honors Program students.

There has been an explosion in the interest in combinatorial games in the last 10 years. Combinatorial games are (two player) games with no chance moves. Combinatorial games have been important tool in discrete mathematics, with applications to computer science (computability of game strategies) and pure mathematics (number systems). In this class we will define and study various classes of combinatorial games. Topics include Hackenbush, playing several games at once, impartial games and nim-values, periodicity in take away games, the surreal number system, very small games and very large games. Although much is known about combinatorial games, much is unknown, and the subject is ripe with conjectures, some of which are supported by computational evidence. We shall examine some of these.

Prerequisites: The corequisite is MATH 1502.

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Time: MWF 9:05 - 9:55

Math 4803 MOR Combinatorial Game Theory 3-0-3

Instructor: Prof. Thomas Morley

Description: Section MOR is for undergraduate students not in the Honors Program.

There has been an explosion in the interest in combinatorial games in the last 10 years. Combinatorial games are (two player) games with no chance moves. Combinatorial games have been important tool in discrete mathematics, with applications to computer science (computability of game strategies) and pure mathematics (number systems). In this class we will define and study various classes of combinatorial games. Topics include Hackenbush, playing several games at once, impartial games and nim-values, periodicity in take away games, the surreal number system, very small games and very large games. Although much is known about combinatorial games, much is unknown, and the subject is ripe with conjectures, some of which are supported by computational evidence. We shall examine some of these.

Prerequisites: The corequisite is MATH 1502.

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Time: MWF 9:05 - 9:55

Math 8803 LIF Gaussian Stochastic Processes 3-0-3

Instructor: Prof. Mikhail Lifshits

Description: The course provides an introduction to the most advanced part of the theory of random processes. We show how the classical notion of normal distribution looks like when the random variables and vectors are replaced by random processes and which new features appear in this extended setting. The exposition includes detailed consideration of key examples of Gaussian processes and random fields and goes along the following plan:

• Multivariate Gaussian distributions
• Examples of Gaussian random functions
• Infinite-dimensional distributions
• Kernel of Gaussian distribution (RKHS)
• Isoperimetry
• Fundamental inequalities related to Gaussian measures
• Studies based on metric entropy
• Large deviation principle
• Series expansions for Gaussian processes
• Functional law of the iterated logarithm
• Small deviation probabilities and their applications

Prerequisites: Knowledge of basic Probability, Random Processes and Functional Analysis.

Text: The following text will be frequently used: M. A. Lifshits Gaussian Random Functions, 1995, Kluwer. Obviously, more recent updates to the theory not included in existing monographs will be also considered.

Time: TR 4:35 - 5:55

Math 8813 LUB Potential Theory 3-0-3

Instructor: Prof. Doron Lubinsky

Description: Potential theory is a tool that is useful in many areas, including complex analysis, partial differential equations, approximation theory, numerical analysis, and integral equations. We'll study potential theory mainly in the complex plane, but occassionally in higher dimensions. We'll examine energy integrals, logarithmic and other potentials, solutions to certain p.d.e.'s, and some applications to polynomials and ocmplex analysis.

This course was successfully given in Fall 2003. The background required is something like 6327 (Graduate Real Analysis), especially the measure theory there, and some complex analysis. I have a full set of typed notes, which will be available to students.

Prerequisites:

Text: T. Ransford, Potential Theory, Cambridge University Press. (It is a wonderful softcover book, and you'll be able to use it throughout your career as an excellent reference.)

Time: MWF 12:05 - 12:55

Math 8823 ETN Introduction to Geometry and Topology I 3-0-3

Instructor: Prof. John Etnyre

Description:

1. Brief review of basics of point set topology and classification of surfaces (if necessary)
   
2. Fundamental group, van Kampen's theorem, covering spaces
   
3. Definition of differential manifolds
   
4. Vectors bundles
   
5. Tangent vectors, vectors fields and flows
   
6. Smooth functions on manifolds, derivatives
   
7. Regular values, Morse functions, transversality, degree theory
   
8. Tensors and forms

Prerequisites: Math 4431 or Math 4317; or consent of Instructor

Text: Text at the level of Differential Manifolds by Lawrence Conlon and Algebraic Topology by Alan Hatcher

Time: TR 1:35 - 2:55