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Ma 125: Elliptic Curves
9.00 Credits
California Institute of Technology
The ubiquitous elliptic curves will be analyzed from elementary, geometric, and arithmetic points of view. Possible topics are the group structure via the chord-and-tangent method, the Nagel-Lutz procedure for finding division points, Mordell’s theorem on the finite generation of rational points, points over finite fields through a special case treated by Gauss, Lenstra’s factoring algorithm, integral points. Other topics may include diophantine approximation and complex multiplication. Not offered 2012-13.
Prerequisite:
Ma 5, Ma 3, or equivalents.
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Ma 125 - Elliptic Curves
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Ma 126 ab: Information Theory
9.00 Credits
California Institute of Technology
Shannon’s mathematical theory of communication, 1948–present. Entropy, relative entropy, and mutual information for discrete and continuous random variables. Shannon’s source and channel coding theorems. Mathematical models for information sources and communication channels, including memoryless, first- order Markov, ergodic, and Gaussian. Calculation of capacity and rate-distortion functions. Kolmogorov complexity and universal source codes. Side information in source coding and communications. Network information theory, including multiuser data compression, multiple access channels, broadcast channels, and multiterminal networks. Discussion of philosophical and practical implications of the theory. This course, when combined with EE 112, EE/Ma/CS 127, EE 161, and/or EE 167 should prepare the student for research in information theory, coding theory, wireless communications, and/or data compression. Instructor: Effros.
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Ma 126 ab - Information Theory
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Ma 127: Error-Correcting Codes
9.00 Credits
California Institute of Technology
This course develops from first principles the theory and practical implementation of the most important techniques for combating errors in digital transmission or storage systems. Topics include algebraic block codes, e.g., Hamming, BCH, Reed-Solomon (including a self-contained introduction to the theory of finite fields); and the modern theory of sparse graph codes with iterative decoding, e.g. LDPC codes, turbo codes, fountain coding. Emphasis will be placed on the associated encoding and decoding algorithms, and students will be asked to demonstrate their understanding with a software project. Instructor: Ho.
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Ma 127 - Error-Correcting Codes
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Ma 129 abc: Information and Complexity
9.00 Credits
California Institute of Technology
A basic course in information theory and computational complexity with emphasis on fundamental concepts and tools that equip the student for research and provide a foundation for pattern recognition and learning theory. what information is and what computation is; entropy, source coding, Turing machines, uncomputability. topics in information and complexity; Kolmogorov complexity, channel coding, circuit complexity, NP-completeness. theoretical and experimental projects on current research topics. Not offered 2012–13.
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Ma 129 abc - Information and Complexity
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Ma 130 abc: Algebraic Geometry
9.00 Credits
California Institute of Technology
Plane curves, rational functions, affine and projective varieties, products, local properties, birational maps, divisors, differentials, intersection numbers, schemes, sheaves, general varieties, vector bundles, coherent sheaves, curves and surfaces. Instructor: Graber.
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Ma 130 abc - Algebraic Geometry
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Ma 132 c: Topics in Algebraic Geometry
9.00 Credits
California Institute of Technology
This course will cover advanced topics in algebraic geometry that will vary from year to year. This year, the topic will be deformation theory. Not offered 2012–13.
Prerequisite:
Ma 130 or instructor’s permission.
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Ma 132 c - Topics in Algebraic Geometry
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Ma 135 ab: Arithmetic Geometry
9.00 Credits
California Institute of Technology
The course deals with aspects of algebraic geometry that have been found useful for number theoretic applications. Topics will be chosen from the following: general cohomology theories (étale cohomology, flat cohomology, motivic cohomology, or p-adic Hodge theory), curves and Abelian varieties over arithmetic schemes, moduli spaces, Diophantine geometry, algebraic cycles. Not offered 2012–13.
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Ma 135 ab - Arithmetic Geometry
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Ma 142: Ordinary and Partial Differential Equations
9.00 Credits
California Institute of Technology
The mathematical theory of ordinary and partial differential equations, including a discussion of elliptic regularity, maximal principles, solubility of equations. The method of characteristics. Instructors: Kreuger, Chipeniuk.
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Ma 142 - Ordinary and Partial Differential Equations
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Ma 144 ab: Probability
9.00 Credits
California Institute of Technology
Overview of measure theory. Random walks and the Strong law of large numbers via the theory of martingales and Markov chains. Characteristic functions and the central limit theorem. Poisson process and Brownian motion. Topics in statistics. Not offered 2012–13.
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Ma 144 ab - Probability
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Ma 145 abc: Introduction to Unitary Group Representations
9.00 Credits
California Institute of Technology
The study of representations of a group by unitary operators on a Hilbert space, including finite and compact groups, and, to the extent that time allows, other groups. First term: general representation theory of finite groups. Frobenius’s theory of representations of semidirect products. The Young tableaux and the representations of symmetric groups. Second term: the Peter-Weyl theorem. The classical compact groups and their representation theory. Weyl character formula. Third term: Quantum Groups. Not offered 2012–13.
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Ma 145 abc - Introduction to Unitary Group Representations
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