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MATH 141: Introduction to Mathematical Logic
4.00 Credits
Harvard University
An introduction to mathematical logic with applications to computer science and algebra. Formal languages. Completeness and compactness of first order logic. Definability and interpolation. Decidability. Unsolvable problems. Computable functions and Turing machines. Recursively enumerable sets. Transfinite induction.
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MATH 144: Model Theory and Algebra
4.00 Credits
Harvard University
An introduction to model theory with applications to fields and groups. First order languages, structures, and definable sets. Compactness, completeness, and back-and-forth constructions. Quantifier elimination for algebraically closed, differentially closed, and real closed fields. Omitting types, prime extensions, existence and uniqueness of the differential closure, saturation, and homogeneity. Forking, independence, and rank.
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MATH 144 - Model Theory and Algebra
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MATH 152: Discrete Mathematics
4.00 Credits
Harvard University
An introduction to finite groups, finite fields, finite geometry, discrete probability, and graph theory. A unifying theme of the course is the symmetry group of the regular icosahedron, whose elements can be realized as permutations, as linear transformations of vector spaces over finite fields, as collineations of a finite plane, or as vertices of a graph. Taught in a seminar format, and students will gain experience in presenting proofs at the blackboard.
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MATH 152 - Discrete Mathematics
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MATH 154: Probability Theory
4.00 Credits
Harvard University
An introduction to probability theory. Discrete and continuous random variables; distribution and density functions for one and two random variables; conditional probability. Generating functions, weak and strong laws of large numbers, and the central limit theorem. Geometrical probability, random walks, and Markov processes.
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MATH 154 - Probability Theory
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MATH 155r: Combinatorics
4.00 Credits
Harvard University
In combinatorics and elsewhere one often encounters a "design", or a collection of subsets of some finite set S whose elements are evenly distributed in a suitable sense; for instance the collection of edges of a regular graph (each of whose vertices is contained in the same number of edges) or the collection of lines of a finite projective plane (any two of whose points are contained in a unique line). Of particular interest are designs symmetric under a large group of permutations of S. The consideration of specific classical designs and their symmetries will lead us to the general study of designs and permutation groups. We conclude with the construction and detailed analysis of the remarkable designs associated with Mathieu's sporadic groups of permutations of 12- and 24- element sets.
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MATH 162: Introduction to Quantum Computing
4.00 Credits
Harvard University
This course is meant to give an introduction to the fundamental mathematics of quantum computing. Notions from linear algebra, elementary number theory and probability theory are introduced along the way as needed.
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MATH 162 - Introduction to Quantum Computing
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MATH 19a: Modeling and Differential Equations for the Life Sciences
4.00 Credits
Harvard University
Considers the construction and analysis of mathematical models that arise in the life sciences, ecology and environmental life science. Introduces mathematics that include multivariable calculus, differential equations in one or more variables, vectors, matrices, and linear and non-linear dynamical systems. Taught via examples from current literature (both good and bad).
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MATH 19a - Modeling and Differential Equations for the Life Sciences
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MATH 19b: Linear Algebra, Probability, and Statistics for the Life Sciences
4.00 Credits
Harvard University
Probability, statistics and linear algebra with applications to life sciences, chemistry, and environmental life sciences. Linear algebra includes matrices, eigenvalues, eigenvectors, determinants, and applications to probability, statistics, dynamical systems. Basic probability and statistics are introduced, as are standard models, techniques, and their uses including the central limit theorem, Markov chains, curve fitting, regression, and pattern analysis.
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MATH 19b - Linear Algebra, Probability, and Statistics for the Life Sciences
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MATH 1a: Introduction to Calculus
4.00 Credits
Harvard University
The development of calculus by Newton and Leibniz ranks among the greatest achievements of the past millennium. This course will help you see why by introducing: how differential calculus treats rates of change; how integral calculus treats accumulation; and how the fundamental theorem of calculus links the two. These ideas will be applied to problems from many other disciplines.
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MATH 1b: Calculus, Series, and Differential Equations
4.00 Credits
Harvard University
Speaking the language of modern mathematics requires fluency with the topics of this course: infinite series, integration, and differential equations. Model practical situations using integrals and differential equations. Learn how to represent interesting functions using series and find qualitative, numerical, and analytic ways of studying differential equations. Develop both conceptual understanding and the ability to apply it.
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MATH 1b - Calculus, Series, and Differential Equations
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