# Course Descriptions: AY 2013-14

Algebra and Number
Theory

Geometry and Topology

Analysis

Numerical Analysis

Applied Mathematics

Probability and
Statistics

All course descriptions are subject to change

**MATH-GA 2110.001,
2120.001 LINEAR ALGEBRA I, II**

3 points per
term. Fall and spring terms.

Tuesday, 5:10-7:00 (fall); Monday, 5:10-7:00 (spring), F.
Greenleaf.

Fall Term

Prerequisites: Undergraduate linear algebra or permission of the instructor.

Linear spaces, subspaces. Linear dependence, linear independence; span, basis, dimension, isomorphism. Quotient spaces. Linear functionals, Dual spaces. Linear mappings, null space, range, fundamental theorem of linear algebra. Underdetermined systems of linear equations. Composition, inverse, transpose of linear maps, algebra of linear maps. Similarity transformations. Matrices, matrix Multiplication, Matrix Inverse, Matrix Representation of Linear Maps, Determinant, Laplace expansion, Cramer's rule. Eigenvalue problem, eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem. Diagonalization.

Text: Friedberg, S.H., Insel, A.J., & Spence, L.E. (2003).Linear algebra(4^{th}ed.). Upper Saddle River, NJ: Prentice-Hall/ Pearson Education.

Recommended Text: Lipschutz, S., & Lipson, M. (2012).Schaum's Outlines[Series].Schaum's outline of linear algebra(5^{th}ed.). New York, NY: McGraw-Hill.

Spring Term

Prerequisites: Linear Algebra I or permission of the instructor.

Trace, determinant, characteristic and minimal polynomial. Eigenvalues, diagonalization, spectral theorem and spectral mapping theorem. Rayleigh quotient and minimax theorem. When diagonalization fails: nilpotent operators and their structure, generalized eigenspace decomposition, Jordan canonical form. Complexification and diagonalization over R. Matrix norms, series and the matrix exponential map, applications to ODE. Bilinear and quadratic forms and their normal forms. The classical matrix groups: unitary, orthogonal, symplectic. Implicit function theorem, smooth surfaces in R^n and their tangent spaces. Intro to matrix Lie algebras and Lie groups.

Text: Friedberg, S.H., Insel, A.J., & Spence, L.E. (2003).Linear algebra(4^{th}ed.). Upper Saddle River, NJ: Prentice-Hall/ Pearson Education.

Cross-listing: MATH-UA 0141.001, 0142.001.

**MATH-GA 2110.001 LINEAR
ALGEBRA I**

3 points. Spring term.

Tuesday, 5:10-7:00, S. Marques.

Prerequisites: Undergraduate Linear Algebra or permission of the instructor.

Linear spaces, subspaces. Linear dependence, linear independence; span, basis, dimension, isomorphism. Quotient spaces. Linear functionals, dual spaces. Linear mappings, null space, range, fundamental theorem of linear algebra. Underdetermined systems of linear equations. Composition, inverse, transpose of linear maps, algebra of linear maps. Similarity transformations. Matrices, matrix Multiplication, Matrix Inverse, Matrix Representation of Linear Maps determinant, Laplace expansion, Cramer's rule. Eigenvalue problem, eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem. Diagonalization.

Text: Lax, P.D. (2007).Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts[Series, Bk. 78].Linear algebra and its applications(2^{nd}ed.). Hoboken, NJ: John Wiley & Sons/ Wiley-Interscience.

**MATH-GA 2111.001 LINEAR ALGEBRA
(one-term format)**

3 points. Fall term.

Thursday, 9:00-10:50, A. Rangan.

Prerequisites: Undergraduate linear algebra.

Linear algebra is two things in one: a general methodology for solving linear systems, and a beautiful abstract structure underlying much of mathematics and the sciences. This course will try to strike a balance between both. We will follow the book of our own Peter Lax, which does a superb job in describing the mathematical structure of linear algebra, and complement it with applications and computing. The most advanced topics include spectral theory, convexity, duality, and various matrix decompositions.

Text: Lax, P.D. (2007).Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts[Series, Bk. 78].Linear algebra and its applications(2^{nd}ed.). Hoboken, NJ: John Wiley & Sons/ Wiley-Interscience.

Recommended Text: Strang, G. (2005).Linear algebra and its applications(4^{th}ed.). Stamford, CT: Cengage Learning.

**MATH-GA 2130.001, 2140.001 ALGEBRA I, II**

3 points per term. Fall and spring terms.

Tuesday, 7:10-9:00, Y.
Tschinkel (fall); Monday, 7:10-9:00, F.
Bogomolov (spring).

__Prerequisites__: Elements of linear algebra and the theory of rings and fields.

Fall TermFall Term

Basic concepts of groups, rings and fields. Symmetry groups, linear groups, Sylow theorems; quotient rings, polynomial rings, ideals, unique factorization, Nullstellensatz; field extensions, finite fields.

Spring Term

Representations of finite groups. Characters, orthogonality of characters of irreducible representations, a ring of representations. Induced representations, Artin's theorem, Brauer's theorem. Representations of compact groups and the Peter-Weyl theorem. Lie groups, examples of Lie groups, representations and characters of Lie group. Lie algebras associated with Lie groups. Applications of the group representations in algebra and physics. Elements of algebraic geometry.

Text: Fulton, W., Harris, J. (2008).Graduate Texts in Mathematics/ Readings in Mathematics[Series, Bk.129].Representation theory: A first course(Corrected ed.). New York, NY: Springer-Verlag.

Recommended Texts: Lang, S. (2005).Graduate Texts in Mathematics[Series, Bk. 211].Algebra(3^{rd}ed.). New York, NY: Springer-Verlag.

Serre, J.P. (1977).Graduate Texts in Mathematics[Series, Bk. 42].Linear representations of finite groups. New York, NY: Springer-Verlag.

Reid, M. (1989).London Mathematical Society Student Texts[Series].Undergraduate algebraic geometry. New York, NY: Cambridge University Press.

James, G., & Liebeck, M. (1993).Cambrdige Mathematical Textbooks[Series].Representations and characters of groups. New York, NY: Cambrdige University Press.

Artin, M. (2010).Algebra(2^{nd}ed.). Upper Saddle River, NJ: Prentice-Hall/ Pearson Education.

Sagan, B.E. (1991).Wadsworth Series in Computer Information Systems[Series].The symmetric group: Representations, combinatorial algorithms, and symmetric functions. Pacific Grove, CA: Wadsworth & Brooks/Cole.

Brocker, T., & Dieck, T. (2003).Graduate Texts in Mathematics[Series, Bk. 98].Representations of compact lie groups. New York, NY: Springer-Verlag.