| source Caltech (X) |
level |
department Mathematics (X) |
Review of calculus. Complex numbers, Taylor polynomials, infinite series. Comprehensive presentation of linear algebra. Derivatives of vector functions, multiple integrals, line and path integrals, theorems of Green and Stokes. Ma 1 b, c is divided into two tracks: analytic and practical. Students will be given information helping them to choose a track at the end of the fall term. There will be a special section or sections of Ma 1 a for those students who, because of their background, require more calculus than is provided in the regular Ma 1 a sequence. These students will not learn series in Ma 1 a and will be required to take Ma 1 d. Instructors: Breuer, Graber, Aschbacher, Wilson, Flach, Rains.
Score: 7.759316 Details | Listing | Web page
This is a course intended for those students in the special calculus-intensive sections of Ma 1 a who did not have complex numbers, Taylor polynomials, and infinite series during Ma 1 a. It may not be taken by students who have passed the regular Ma 1 a. Instructor: Staff.
Score: 7.759316 Details | Listing | Web page
Some of the fundamental ideas, techniques, and open problems of basic number theory will be introduced. Examples will be stressed. Topics include Euclidean algorithm, primes, Diophantine equations, including a
Score: 7.759316 Details | Listing | Web page
First term: a survey emphasizing graph theory, algorithms, and applications of algebraic structures. Graphs: paths, trees, circuits, breadth-first and depth-first searches, colorings, matchings. Enumeration techniques; formal power series; combinatorial interpretations. Topics from coding and cryptography, including Hamming codes and RSA. Second term: directed graphs; networks; combinatorial optimization; linear programming. Permutation groups; counting nonisomorphic structures. Topics from extremal graph and set theory, and partially ordered sets. Third term: elements of computability theory and computational complexity. Discussion of the P=NP problem, syntax and semantics of propositional and first-order logic. Introduction to the Gödel completeness and incompleteness theorems. Instructors: Wilson, Balachandran, Epstein.
Score: 7.759316 Details | Listing | Web page
A three-hour per week hands-on class for those students in Ma 1 needing extra practice in problem solving in calculus. Instructor: Lyons.
Score: 7.759316 Details | Listing | Web page
In this course, students will receive training and practice in presenting mathematical material before an audience. In particular, students will present material of their own choosing to other members of the class. There may also be elementary lectures from members of the mathematics faculty on topics of their own research interest. Instructor: Rains.
Score: 7.759316 Details | Listing | Web page
Students will work with the instructor and a mentor to write and revise a self-contained paper dealing with a topic in mathematics. In the first week, an introduction to some matters of style and format will be given in a classroom setting. Some help with typesetting in TeX may be available. Students are encouraged to take advantage of the Hixon Writing Center’s facilities. The mentor and the topic are to be selected in consultation with the instructor. It is expected that in most cases the paper will be in the style of a textbook or journal article, at the level of the student’s peers (mathematics students at Caltech).
Score: 7.759316 Details | Listing | Web page
This course will explore examples of the use and misuse of notions of probability and statistics in popular culture and in scientific research. Basic ideas about random fluctuations will be introduced, along with simple techniques like nonparametric statistics and the bootstrap. Not offered 2008–09.
Score: 7.759316 Details | Listing | Web page
There are many problems in elementary mathematics that require ingenuity for their solution. This is a seminar-type course on problem solving in areas of mathematics where little theoretical knowledge is required. Students will work on problems taken from diverse areas of mathematics; there is no prerequisite and the course is open to freshmen. May be repeated for credit. Graded pass/fail. Instructor: Nelson.
Score: 7.759316 Details | Listing | Web page
Open only to senior mathematics majors who are qualified to pursue independent reading and research. This research must be supervised by a faculty member. The research must begin in the first term of the senior year and will normally follow up on an earlier SURF or independent reading project. Two short presentations to a thesis committee are required: the first at the end of the first term and the second at the midterm week of the third term. A draft of the written thesis must be completed and distributed to the committee one week before the second presentation. Graded pass/fail in the first and second terms; a letter grade will be given in the third term.
Score: 7.759316 Details | Listing | Web page
Occasionally a reading course will be offered after student consultation with a potential supervisor. Topics, hours, and units by arrangement. Graded pass/fail.
Score: 7.759316 Details | Listing | Web page
First term: structure of the real numbers, topology of metric spaces, a rigorous approach to differentiation in R
Score: 7.759316 Details | Listing | Web page
This course will discuss advanced topics in analysis, which vary from year to year. Topics from previous years include potential theory, bounded analytic functions in the unit disk, probabilistic and combinatorial methods in analysis, operator theory, C*-algebras. First term: classical special functions. Third term: Riemann surface theory. Instructors: van de Bult, Zinchenko.
Score: 7.759316 Details | Listing | Web page
The first term covers general methods of testing hypotheses and constructing confidence sets, including regression analysis, analysis of variance, and nonparametric methods. The second term covers permutation methods and the bootstrap, point estimation, Bayes methods, and multistage sampling. Instructor: Lorden.
Score: 7.759316 Details | Listing | Web page
Propositional logic, predicate logic, formal proofs, Gödel completeness theorem, the method of resolution, elements of model theory. Computability, undecidability, Gödel incompleteness theorems. Axiomatic set theory, ordinals, transfinite induction and recursion, iterations and fixed points, cardinals, axiom of choice. Not offered 2008–09.
Score: 7.759316 Details | Listing | Web page
Various approaches to computability theory, e.g., Turing machines, recursive functions, Markov algorithms; proof of their equivalence. Church’s thesis. Theory of computable functions and effectively enumerable sets. Decision problems. Undecidable problems: word problems for groups, solvability of Diophantine equations (Hilbert’s 10th problem). Relations with mathematical logic and the Gödel incompleteness theorems. Decidable problems, from number theory, algebra, combinatorics, and logic. Complexity of decision procedures. Inherently complex problems of exponential and superexponential difficulty. Feasible (polynomial time) computations. Polynomial deterministic vs. nondeterministic algorithms, NP-complete problems and the P = NP question. Instructor: Kechris.
Score: 7.759316 Details | Listing | Web page
This course will provide an introduction to the striking paradoxes that challenge our geometrical intuition. Topics to be discussed include geometrical transformations, especially rigid motions; free groups; amenable groups; group actions; equidecomposability and invariant measures; Tarski’s theorem; the role of the axiom of choice; old and new paradoxes, including the Banach-Tarski paradox, the Laczkovich paradox (solving the Tarski circle-squaring problem), and the Dougherty-Foreman paradox (the solution of the Marczewski problem). Not offered 2008–09.
Score: 7.759316 Details | Listing | Web page
Basic theory of groups, rings, modules, and fields, including free groups; Sylow’s theorem; solvable and nilpotent groups; factorization in commutative rings; integral extensions; Wedderburn theorems; Jacobson radical; semisimple, projective, and injective modules; tensor products; chain conditions; Galois theory; cyclotomic extensions; separability; transcendental extensions. Instructors: Graber, Flach, Mantovan.
Score: 7.759316 Details | Listing | Web page
A survey of modern combinatorial mathematics, starting with an introduction to graph theory and extremal problems. Flows in networks with combinatorial applications. Counting, recursion, and generating functions. Theory of partitions. (0, 1)-matrices. Partially ordered sets. Latin squares, finite geometries, combinatorial designs, and codes. Algebraic graph theory, graph embedding, and coloring. Instructors: Balachandran, van de Bult.
Score: 7.759316 Details | Listing | Web page
Groups of Lie type: classical groups, Coxeter groups, root systems, Chevalley groups, weight theory, linear algebraic groups, buildings. Not offered 2008–09.
Score: 7.759316 Details | Listing | Web page
This course is an introduction to Lie algebras and the classification of the simple Lie algebras over the complex numbers. This will include Lie’s theorem, Engel’s theorem, the solvable radical, and the Cartan Killing trace form. The classification of simple Lie algebras proceeds in terms of the associated reflection groups and a classification of them in terms of their Dynkin diagrams. Not offered 2008–09.
Score: 7.759316 Details | Listing | Web page
For course description, see Electrical Engineering.
Score: 7.759316 Details | Listing | Web page
For course description, see Electrical Engineering.
Score: 7.759316 Details | Listing | Web page
For course description, see Computer Science.
Score: 7.759316 Details | Listing | Web page
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: Morin. Part c not offered 2008–09.
Score: 7.759316 Details | Listing | Web page