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Course Criteria
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3.00 Credits
Over the years financial instruments have grown from stocks and bonds to numerous derivatives, such as options to buy and sell at future dates under certain conditions. The 1997 Nobel Prize in Economics was awarded to Robert Merton and Myron Schloles for their Black-Scholes model of the value of financial instruments. This course will study deterministic and random models, futures, options, the Black-Scholes Equation, and additional topics.
Prerequisite:
Mathematics 211 or permission of instructor
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3.00 Credits
Directed 300-level independent study in Mathematics.
Prerequisite:
Permission of department
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3.00 Credits
Directed 300-levelindependent study in Mathematics.
Prerequisite:
Permission of department
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3.00 Credits
The study of measure theory arose from the study of stochastic (probabilistic) systems. Applications of measure theory lie in biology, chemistry, physics as well as in economics. In this course, we develop the abstract concepts of measure theory and ground them in probability spaces. Included will be Lebesgue and Borel measures, measurable functions (random variables). Lebesgue integration, distributions, independence, convergence and limit theorems. This material provides good preparation for graduate studies in mathematics, statistics and economics.
Prerequisite:
Mathematics 301 or 305 or permission of instructor
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3.00 Credits
In the first N math classes of your career, it's possible to get an incomplete picture as to what the real world is truly like. How? You're often given exact problems and told to find exact solutions. The real world is sadly far more complicated. Frequently we cannot exactly solve problems; moreover, the problems we try to solve are themselves merely approximations to the world. We're forced to develop techniques to approximate not just solutions, but even the statement of the problem. In this course we discuss some powerful methods from advanced linear algebra and their applications to the real world, specifically linear programming (and, if time permits, random matrix theory). Linear programming is used to attack a variety of problems, from applied ones such as the traveling salesman problem, determining schedules for major league sports (or a movie theater, or an airline) to designing efficient diets to feed the world, to pure ones such as Hales' proof of the Kepler conjecture.
Prerequisite:
Mathematics 211 and 301 (programming experience is desirable, but not necessary)
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3.00 Credits
Differential geometry studies smooth surfaces in all dimensions, from curves to the universe. Riemannian geometry shows that curvature is the key to understanding shape, from the curvature of a curve in calculus to the curvature of space in general relativity. Sharp corners and black holes are singularities that require extensions of the theory. We will look at some open questions.
Prerequisite:
Mathematics 301 or 305
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3.00 Credits
Mathematical modeling is concerned with translating a natural phenomenon into a mathematical form. In this abstract form the underlying principles of the phenomenon can be carefully examined and real-world behavior can be interpreted in terms of mathematical shapes. The models we investigate include feedback phenomena, phase locked oscillators, multiple population dynamics, reaction-diffusion equations, shock waves, morphogenesis, and the spread of pollution, forest fires, and diseases. Often the natural phenomenon has some aspect we can control--such as how much pollution, electric charge, or chemotherapeutic agent we put into a river, circuit, or cancer patient. We will investigate how to operate such controls in order to achieve a specific goal or optimize some interpretation of performance. We will employ tools from the fields of differential equations and dynamical systems. The course is intended for students in the mathematical, physical, and chemical sciences, as well as for students who are seriously interested in the mathematical aspects of physiology, economics, geology, biology, and environmental studies.
Prerequisite:
Mathematics 209 or Physics 210 and Mathematics 301 or 305 or permission of instructor
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3.00 Credits
This course is an introduction to chaotic dynamical systems. The topics will include bifurcations, the quadratic family, symbolic dynamics, chaos, dynamics of linear systems, and some complex dynamics.
Prerequisite:
Mathematics 211
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3.00 Credits
Mathematics senior honors thesis. Each student carries out an individual research project under the direction of a faculty member that culminates in a thesis. See description under The Degree with Honors in Mathematics.
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3.00 Credits
Mathematics senior honors thesis. Each student carries out an individual research project under the direction of a faculty member that culminates in a thesis. See description under The Degree with Honors in Mathematics.
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