This theorem has an equivalent statement in terms of matrices: Let A be a square matrix with complex entries.

### 2018-12222 Graduate Courses Descriptions

We further present an algorithm to find P and J, assuming that one can factor the characteristic polynomial of A. In developing this algorithm we introduce the eigenstructure picture ESP of a matrix, a pictorial representation that makes JCF clear.

We illustrate this algorithm with copious examples, and provide numerous exercises for the reader. Series: Contemporary Mathematicians , X, p. Published in honor of his 70th birthday, this volume explores and celebrates the work of G. Pete Stewart, a world-renowned expert in computational linear algebra. It is widely accepted that Stewart is the successor to James Wilkinson, the first giant in the field, taking up the perturbation theory research that Wilkinson so ably began and using it as a foundation for algorithmic insights.

His results on perturbation of eigensystems, pseudo-inverses, least-squares problems, and matrix factorizations are fundamental to numerical practice today. His algorithms for the singular value decomposition, updating and downdating matrix factorizations, and the eigenproblem broke new ground and are still widely used in an increasing number of applications.

His six popular textbooks are excellent sources of knowledge and history. Stewart is a member of the National Academy of Engineering and has received numerous additional honors, including the Bauer Prize. Stewart: Selected Works with Commentaries will appeal to graduate students, practitioners, and researchers in computational linear algebra and the history of mathematics. These objects are suitable fodder for standard homotopy theoretic moves, leading to unstable and stable model structures.

The text is largely self-contained. Progress in Mathematical Physics, Vol. Without using the customary Clifford algebras frequently studied in connection with the representations of orthogonal groups, this book gives an elementary introduction to the two component spinor formalism for four-dimensional spaces with any signature. This systematic and self-contained book is suitable as a seminar text, a reference book, and a self-study guide. Reviews from the author's previous book, 3-D Spinors, Spin-Weighted Functions and their Applications: In summary…the book gathers much of what can be done with 3-D spinors in an easy-to-read, self-contained form designed for applications that will supplement many available spinor treatments.

The book…should be appealing to graduate students and researchers in relativity and mathematical physics. Following a nice and descriptive introduction…the final chapter contains some applications of the formalism to general relativity. Dirac Spinors. Mate of a Spinor. Algebraic Classification.

Lie Derivative of Spinors. Ernst Potentials. The Dirac Operator. Numerous examples drawn from the news and current events with an emphasis on health issues illustrate these concepts. Assuming only a pre-calculus background, the author keeps equations to a minimum and demonstrates all computations using SAS. Most of the programs and output are displayed in a self-contained way, with an emphasis on the interpretation of the output in terms of how it relates to the motivating example.

Plenty of exercises conclude every chapter. Introduction; 2. Principles of statistics; 3. Introduction to linear regression; 4.

### Archived seminars in Mathematics

Assessing the regression; 5. Multiple linear regression; 6. Some help with typesetting in TeX may be available. The mentor and the topic are to be selected in consultation with the instructor. Fulfills the Institute scientific writing requirement. Instructor: Ni. The Mathematics of Enzyme Kinetics. Prerequisites: Ma 1a, b. For course description, see Freshman Seminars. Problem Solving in Vector Calculus.

Prerequisites: Concurrent registration in Ph 1b. A two-hour per week, hands-on class for those students enrolled in Ph 1b needing extra practice with problem solving in vector calculus. How to Solve It. 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. Instructor: Katz. Frontiers in Mathematics. Prerequisites: Open for credit to freshman and sophomores. Weekly seminar by a member of the math department or a visitor, to discuss his or her research at an introductory level. The course aims to introduce students to research areas in mathematics and help them gain an understanding of the scope of the field. Ma 92 abc. Senior Thesis. Prerequisites: To register, the student must obtain permission of the mathematics undergraduate representative.

Open only to senior mathematics majors who are qualified to pursue independent reading and research.

- Mathematics.
- Tongue Twisters (Tongue Twisters for Kids).
- Conferences and Meetings on Algebra?
- The Little Pischna: For Intermediate to Early Advanced Piano (Alfred Masterwork Edition).
- Hidden Agendas.
- Homotopy Theory of C*-Algebras : Paul Arne Ostvaer : .

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.

Research in Mathematics. Units to be arranged in accordance with work accomplished. This course is designed to allow students to continue or expand summer research projects and to work on new projects. Students registering for more than 6 units of Ma 97 must submit a brief no more than 3 pages written report outlining the work completed to the undergraduate option rep at the end of the term.

Independent Reading. Occasionally a reading course will be offered after student consultation with a potential supervisor. Topics, hours, and units by arrangement. Ma abc. Classical Analysis. May be taken concurrently with Ma Second term: brief introduction to ordinary differential equations; Lebesgue integration and an introduction to Fourier analysis. Third term: the theory of functions of one complex variable. Instructors: Lazebnik, Durcik, Ivrii. Introduction to Geometry and Topology. Prerequisites: Ma 2 or equivalent, and Ma must be taken previously or concurrently.

First term: aspects of point set topology, and an introduction to geometric and algebraic methods in topology. Second term: the differential geometry of curves and surfaces in two- and three-dimensional Euclidean space. Third term: an introduction to differentiable manifolds.

Transversality, differential forms, and further related topics. Instructors: Markovic, Chen. Prerequisites: Ma or previous exposure to metric space topology, Lebesgue measure. Second term: basic complex analysis: analytic functions, conformal maps and fractional linear transformations, idea of Riemann surfaces, elementary and some special functions, infinite sums and products, entire and meromorphic functions, elliptic functions. Third term: harmonic analysis; operator theory. Operator theory: compact operators, trace and determinant on a Hilbert space, orthogonal polynomials, the spectral theorem for bounded operators.

If time allows, the theory of commutative Banach algebras. Instructors: Makarov, Durcik. Ma ab. Topics in Analysis. This course will discuss advanced topics in analysis, which vary from year to year. Prerequisite: Ma 2 a probability and statistics or equivalent. The first term covers general methods of testing hypotheses and constructing confidence sets, including regression analysis, analysis of variance, and nonparametric methods.

## Frontiers In Mathematics Series

The second term covers permutation methods and the bootstrap, point estimation, Bayes methods, and multistage sampling. Not offered — Mathematical Logic and Axiomatic Set Theory. First term: Introduction to first-order logic and model theory. Definability, elementary equivalence, complete theories, categoricity.

The Skolem-Lowenheim Theorems. The back and forth method and Ehrenfeucht-Fraisse games. Farisse theory. Elimination of quantifiers, applications to algebra and further related topics if time permits. Second and third terms: Axiomatic set theory, ordinals and cardinals, the Axiom of Choice and the Continuum Hypothesis. Models of set theory, independence and consistency results.

Topics in descriptive set theory, combinatorial set theory and large cardinals. Computability Theory. Various approaches to computability theory, e. Theory of computable functions and effectively enumerable sets. Decision problems. 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. Instructors: Kechris, Panagiotopolous. Topics in Mathematical Logic: Geometrical Paradoxes. This course will provide an introduction to the striking paradoxes that challenge our geometrical intuition. Abstract Algebra. This course will discuss advanced topics in algebra. Among them: an introduction to commutative algebra and homological algebra, infinite Galois theory, Kummer theory, Brauer groups, semisimiple algebras, Weddburn theorems, Jacobson radicals, representation theory of finite groups. Instructors: Ramakrishnan, Zhu, Yom Din. Combinatorial Analysis.

## homotopical structure on C*-algebras in nLab

Prerequisite: Ma 5. 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. Partially ordered sets. Latin squares, finite geometries, combinatorial designs, and codes. Algebraic graph theory, graph embedding, and coloring. Instructors: Krause, Katz. Classification of Simple Lie Algebras. Prerequisite: Ma 5 or equivalent. This course is an introduction to Lie algebras and the classification of the simple Lie algebras over the complex numbers.

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. Elliptic Curves. Prerequisites: Ma 5 or equivalent. The ubiquitous elliptic curves will be analyzed from elementary, geometric, and arithmetic points of view.

Other topics may include diophantine approximation and complex multiplication. Algebraic Curves. Prerequisites: Ma 5. An elementary introduction to the theory of algebraic curves. Topics to be covered will include affine and projective curves, smoothness and singularities, function fields, linear series, and the Riemann-Roch theorem.

Possible additional topics would include Riemann surfaces, branched coverings and monodromy, arithmetic questions, introduction to moduli of curves. Information Theory. Prerequisites: Ma 3 For course description, see Electrical Engineering. Error-Correcting Codes.

For course description, see Electrical Engineering. Homological Algebra. This course introduces standard concepts and techniques in homological algebra. Topics will include Abelian and additive categories; Chain complexes, homotopies and the homotopy category; Derived functors; Yoneda extension and its ring structure; Homological dimension and Koszul complexe; Spectral sequences; Triangulated categories, and the derived category.

Instructor: Qi.

## Conferences and Meetings on Algebra

Algebraic Geometry. Prerequisite: Ma or Ma 5 plus additional reading. 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. Instructors: Campbell, Xu. Topics in Algebraic Geometry. This course will cover advanced topics in algebraic geometry that will vary from year to year. Topics will be listed on the math option website prior to the start of classes.

Instructors: Graber, Zhu. Arithmetic Geometry. Prerequisite: Ma