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  • MATH 305: Applied Real Analysis

    3.00 Credits
    Williams College

    Real analysis or the theory of calculus--derivatives, integrals, continuity, convergence--starts with a deeper understanding of real numbers and limits. Applications in the calculus of variations or "infinite-dimensional calculus" include geodesics, harmonic functions, minimal surfaces, Hamilton's action and Lagrange's equations, optimal economic strategies, nonEuclidean geometry, and general relativity. Prerequisite:    Mathematics 105 and 211, or permission of instructor
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  • MATH 312: Abstract Algebra

    3.00 Credits
    Williams College

    Algebra gives us the tools to solve equations. Sets such as the integers or real numbers have special properties which make algebra work or not work according to the circumstances. In this course, we generalize algebraic processes and the sets upon which they operate in order to better understand, theoretically, when equations can and cannot be solved. We define and study the abstract algebraic structures called groups, rings and fields, as well as the concepts of factor group, quotient ring, homomorphism, isomorphism, and various types of field extensions. Prerequisite:    Mathematics 211 and one or more of the following: Mathematics 209, 251 or Statistics 201, or permission of instructor
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  • MATH 313: Introduction to Number Theory

    3.00 Credits
    Williams College

    The study of numbers dates back thousands of years, and is fundamental in mathematics. In this course, we will investigate both classical and modern questions about numbers. In particular, we will explore the integers, and examine issues involving primes., divisibility, and congruences. We will also look at the ideas of number and prime in more general settings, and consider fascinating questions that are simple to understand, but can be quite difficult to answer. Prerequisite:    Math 211 or permission of instructor
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  • MATH 316: Protecting Information: Applications of Abstract Algebra and Quantum Physics

    3.00 Credits
    Williams College

    Living in the early decades of the information age, we find ourselves depending more and more on codes that protect messages against either noise or eavesdropping. This course examines some of the most important codes currently being used to protect information, including linear codes, which in addition to being mathematically elegant are the most practical codes for error correction, and the RSA public key cryptographic scheme, popular nowadays for internet applications. We also study the standard AES system as well as an increasingly popular cryptographic strategy based on elliptic curves. Looking ahead by a decade or more, we show how a "quantum computer" could crack any RSA code scheme in short order, and how quantum cryptographic devices will achieve security through the Heisenberg uncertainty principleinherent unpredictability of quantum events. Prerequisite:    Physics 210 or Mathematics 211 (possibly concurrent) or permission of the instructors; students not satisfying the course prerequisites but who have completed Mathematics 209 or Mathematics 251 are particularly encouraged to ask to be admitted
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  • MATH 317: Applied Abstract Algebra

    3.00 Credits
    Williams College

    The abstract algebraic structures called groups, rings and fields have proven to have surprisingly many applications. For example, groups have been used to build secure cryptosystems and to study the symmetry of molecules. We will study the abstract properties of groups, rings and fields and then study several applications of this theory. Possible topics include cryptography, puzzles, error correcting codes, computer software applications, symmetry, tiling, networks, and grobner bases. Prerequisite:    Mathematics 211 and one or more of the following: Mathematics 209, 251 or Statistics 201 or permission of the instructor
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  • MATH 324: Topology

    3.00 Credits
    Williams College

    Topology is the study of when one geometric object can be continuously deformed and twisted into another object. Determining when two objects are topologically the same is incredibly difficult and is still the subject of a tremendous amount of research, including current work on the Poincare Conjecture, one of the million-dollar millennium-prize problems. The first part of the course on "Point-set Topology" establishes a framework based on "open sets" for studying continuity and compactness in very general spaces. The second part on "Homotopy Theory" develops refined methods for determining when objects are the same. We will prove for example that you cannot twist a basketball into a doughnut. Prerequisite:    Mathematics 301, or permission of instructor and Mathematics 305 or 312. Not open to students who have taken Mathematics 323
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  • MATH 327: Computational Geometry

    3.00 Credits
    Williams College

    The subject of computational geometry started just 25 years ago, and this course is designed to introduce its fundamental ideas. Our goal is to explore "visualization" and "shape" in real world problems. We focus on both theoretic ideas (such as visibility, polyhedra, Voronoi diagrams, triangulations, motion) as well as applications (such as cartography, origami, robotics, surface meshing, rigidity). This is a beautiful subject with a tremendous amount of active research and numerous unsolved problems, relating powerful ideas from mathematics and computer science. Prerequisite:    Mathematics 211 or Mathematics 251 or Computer Science 256
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  • MATH 341: Probability

    3.00 Credits
    Williams College

    While probability began with a study of games, it has grown to become a discipline with numerous applications throughout mathematics and the sciences. Drawing on gaming examples for motivation, this course will present axiomatic and mathematical aspects of probability. Included will be discussions of random variables, expectation, independence, laws of large numbers, and the Central Limit Theorem. Many interesting and important applications will also be presented, including some from coding theory, number theory and nuclear physics. Prerequisite:    Mathematics 211 or 251 or permission of instructor
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  • MATH 354:

    3.00 Credits
    Williams College

    No course description available.
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  • MATH 361:

    3.00 Credits
    Williams College

    No course description available.
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