MATHEMATICS

110.106 (Q)

CALCULUS I (4) Budur For Biological and Social Sciences Majors   Limit 25 per section    Differential and integral calculus. Includes analytic geometry, functions, limits, integrals and derivatives, introduction to differential equations, functions of several variables, linear systems, and applications for systems of linear differential equations, probability distributions.

Lec.

Sec. 01

02

03

MTW 10

F 9

F 12

Th 10:30-11:20

110.107 (Q)

CALCULUS II (4) Ching For Biological and Social Sciences Majors    Limit 28 per section Prereq: Calculus I    Differential and integral calculus. Includes analytic geometry, functions, limits, integrals and derivatives, introduction to differential equations, functions of several variables, linear systems, applications for systems of linear differential equations, probability distributions.

Lec.

Sec. 01 & 02

03

04

 05

06

MTW 10

F 9

F 12

F 12

Th 10:30-11:20

Th 10:30-11:20

110.109 (Q)

CALCULUS II (4) DeSilva For Physical Sciences and Engineering Majors   Limit 28 per section Prereq: Calculus I    Differential and integral calculus. Includes analytic geometry, functions, limits, integrals and derivatives, polar coordinates, parametric equations, Taylor's theorem and applications, infinite sequences and series.

Lec.

Sec. 01

02

03

04 & 05

06

MTW 10

F 9

F 12

F 12

Th 10:30-11:20

F 9

110.201 (Q)

LINEAR ALGEBRA (4) Goldberg Limit 25 per section   Prereq: Calculus I Vector spaces, matrices, and linear transformations. Solutions of systems of linear equations. Eigenvalues, eigenvectors, and diagonalization of matrices. Applications to differential equations.

Lec. I

Sec. 01

02

03

04

05

06

MTW 10

F 9

Th 10:30-11:20

F 12

F 12

F 9

Th 10:30-11:20

110.202 (Q)

CALCULUS III (4) Mese - Secs. 01,02,04 / Blair - Secs. 05,06,07,08 Limit 25 per section  Prereq: 110.107, 110.109 or 110.112. Calculus of functions of more than one variable: partial derivatives, and applications; multiple integrals, line and surface integrals; Green's Theorem, Stokes' Theorem, and Gauss' Divergence Theorem.

Sec. 03 canceled 01/06/06

Lec. I

Sec. 01

02

03

04

Lec. II

05

06

07

08

MTW 10

Th 9

Th 12

F 9

F 12

MTW 10

Th 9

Th 12

F 9

F 12

110.204 (Q)

ELEMENTARY NUMBER THEORY (4) Zhang   Limit 30   Prereq: a good high school background including a year of Calculus. Primes and prime factorization, congruences, Euler's function, quadratic reciprocity, primitive roots, solutions to polynomial congruences (Chevalley's theorem), Diophantine equations including the Pythagorean and Pell equations, Gaussian integers, Dirichlet's theorem on primes.

Lec.

Sec. 01

MTW 11

F 12

110.211 (Q)

HONORS CALCULUS III (4) Ching Limit 35    Prereq: B+ or better in Calculus II or 5 in the BC AP exam   This course includes the material in Calculus III (202) with some additional applications and theory. Recommended for mathematically able students majoring in physical science, engineering, or especially mathematics. 110.211-212 used to be an integrated yearlong course, but now the two are independent courses and can be taken in either order.

Lec.

Sec. 01

MTW 12

F 12

110.212 (Q)

HONORS LINEAR ALGEBRA  (4) Kong   Limit 45    Prereq: Calculus II or III or equivalent, preferably honors. This course includes the material in Linear Algebra (201) with some additional applications and theory. Recommended for mathematically able students majoring in physical science, engineering, or mathematics.

Lec.

Sec. 01

MTW 12

F 12

110.302 (E,Q)

DIFFERENTIAL EQUATIONS WITH APPLICATIONS (4) Goldberg/Brown Limit 35 per section   Prereq: Calculus II III This is an applied course in ordinary differential equations, which is primarily for students in the biological, physical and social sciences, and engineering. The purpose of the course is to familiarize the student with the techniques of solving ordinary differential equations.

Secs. 05,06,07 added 11/21/05

Lec. 1

Sec. 01

02

05

06

Lec. 2

Sec. 03

04

07

MTW 12 1

F 9 Th 10:30-11:20

Th 12

Th 12

F 12

MTW 1

Th 10:30F  9

F 12

F 12 10:30

110.402 (Q)

ADVANCED ALGEBRA II (4.5) Consani  Limit 30   Prereq: 110.401     Splitting field of a polynomial, algebraic closure of a field. Galois theory: correspondence between subgroups and subfields. Solvability of polynomial equations by radicals.

Lec.

Sec. 01

MTW 11

Th 9

110.405 (Q)

ANALYSIS I (4.5) DeSilva     Limit 35    Prereq: Calculus III, Linear Algebra Real and complex number systems, topology of metric spaces, limits, continuity, infinite sequences and series, differentiation, Riemann-Stieltjes integration.

Lec.

Sec. 01

MTW 1

F 9

110.406 (Q)

ANALYSIS II (4.5) Song   Limit 35 Prereq: 110.405     This course continues 110.405 notions of modern analysis. Sequences and series of functions, Fourier series, equicontinuity and the Arzela-Ascoli theorem, the Stone-Weierstrass theorem. Functions of several variables, the inverse and implicit function theorems, introduction to the Lebesgue integral.

Lec.

Sec. 01

MTW 1

F 10:30 9

110.409 (Q)

INTRODUCTION TO ALGEBRAIC NUMBER THEORY (4) Ono    Prereq: Algebra 110.401-402 or equivalent    The unique factorization theorem for ideals in rings of algebraic integers, integral bases, the discriminant, the different, ramification, the finiteness theorem for ideal-class groups, Dirichlet's theorem on groups of units of rings of algebraic integers etc.

Sec. 01

MTW 10

110.413 (Q)

INTRODUCTION TO TOPOLOGY (4.5) Morava Topological spaces, connectedness, compactness, quotient spaces, metric spaces, function spaces. An introduction to algebraic topology: covering spaces, the fundamental group, and other topics as time permits.

Sec. 01

MTW 2

110.417 (E,Q)

PARTIAL DIFFERENTIAL EQUATIONS FOR APPLICATIONS (4.5) Song   Limit 35 Prereq: Calculus III, Linear Algebra    Recommended: 110.405 Classification of second order equations, well-posed problems. separation of variables and expansions of solutions. The wave equation: Cauchy problem, Poisson's solution, energy inequalities, domains of influence and dependence.

Sec. 01

MTW 12

110.423 (Q)

LIE GROUPS FOR UNDERGRADUATES (4) Spinu   Prereq: Calculus III , Prior knowledge of group theory would be helpful.   This course is an introduction to Lie Groups and their representations at the upper undergraduate level. It will cover basic Lie Groups such as SU (2), U (n), the Euclidean Motion Group and Lorentz Group. This course is useful for students who want a working knowledge of group representations. We will also discuss some aspects of the role of symmetry groups in particle physics such as some of the formal aspects of the electroweak and the strong interactions. A good reference is the book Lie Algebras in Particle Physics by Howard Georgi.

Sec. 01

MTW 1

110.602

ALGEBRA Shokurov Prereq: 110.401-402    An introductory graduate course on fundamental topics in algebra to provide the student with the foundations for Number Theory, Algebraic Geometry, and other advanced courses. Topics include group theory, commutative algebra, Noetherian rings, local rings, modules, and rudiments of category theory, homological algebra, field theory, Galois theory, and non-commutative algebras.

Sec. 01

MTW 2 MT 12-1:15

110.606

LIE GROUPS AND LIE ALGEBRA Shalika Prereq: 110.404 Lie groups and Lie algebras, classification of complex semi-simple Lie algebras, compact forms, representations and Weyl formulas, symmetric Riemannian spaces. Measure and integration on abstract and locally compact spaces (extension of measures, decompositions of measures, product measures, the Lebesgue integral, differentiation, L^p-spaces); introduction to functional analysis; integration on groups; Fourier transforms. Course canceled 10/31/05

Sec. 01

MTW 11

110.607

COMPLEX VARIABLES Shiffman Prereq: 110.311, 110.405    Analytic functions of one complex variable.  Topics include Mittag-Leffler Theorem, Weierstrass factorization theorem, elliptic functions, Riemann-Roch theorem, Picard theorem, and Nevanlinna theory.

Sec. 01

MTW 1 10 MT 10-11:15

110.612

COMPLEX GEOMETRY Shiffman

Sec. 01

MT 2-3:15 1-2:15

110.616

ALGEBRAIC TOPOLOGY Boardman Prereq: 110.401, 110.413   Polyhedra, simplicial and singular homology theory, Lefschetz fixed-point theorem, cohomology and products, homological algebra, Künneth and universal coefficient theorems, Poincaré and Alexander duality theorems.

Sec. 01

ThF 2-3:15

110.617

NUMBER THEORY Shalika Limit: 20  Course added 10/31/05

Sec. 01

MTW 11

110.632

PARTIAL DIFFERENTIAL EQUATIONS Minicozzi  Prereq: 110.605-606 An introductory graduate course in partial differential equations. Classical topics include first order equations and characteristics, the Cauchy-Kowalevski theorem, Laplace's equation, heat equation, wave equation, fundamental solutions, weak solutions, Sobolev spaces, maximum principles. The second term focuses on special topics such as second order elliptic theory.

Sec. 01

MTW 10 2

110.644

ALGEBRAIC GEOMETRY Shokurov Affine varieties and commutative algebra. Hilbert's theorems about polynomials in several variables with their connections to geometry. General varieties and projective geometry. Dimension theory and smooth varieties. Sheaf theory and cohomology. Applications of sheaves to geometry; e.g., the Riemann-Roch Theorem. Other topics may include Jacobian varieties, resolution of singularities, geometry on surfaces, schemes, connections with complex analytic geometry and topology.

Sec. 01

MTW 12 MT 2-3:15

110.645

RIEMANNIAN GEOMETRY Mese   Prereq: 110.405, 110.413 Differential manifolds, vector fields, Frobenius’ theorem. Differential forms, deRham’s theorem, vector bundles, connections, curvature, Chern classes, Cartan structure equations. Riemannian manifolds, Bianchi identities, geodesics, exponential maps. Geometry of submanifolds, hypersurfaces in Euclidean space. Other topics as time permits, e.g. harmonic forms and Hodge’s theorem, Jacobi equation, variation of arc length and area, Chern-Gauss-Bonnet theorems.

Sec. 01

MTW 11 12

110.660

QUALIFYING EXAM PROBLEMS Staff

Sec. 01

TBA

110.726

TOPICS IN ANALYSIS  Zelditch 
 

Sec. 01

MTW 1

110.727

TOPICS IN ALGEBRAIC TOPOLOGY Boardman

Sec. 01

ThF 3:30-4:45

110.729

TOPICS IN COMPLEX GEOMETRY Wentworth Limit: 20 Course added 10/31/05 Course canceled 11/22/05 Course added 12/05/05 Course canceled 01/30/06

Sec. 01

TBA W 3-4:30

110.732

TOPICS IN MATHEMATICAL PHYSICS Wentworth Course added 11/22/05 Course canceled 10/31/05

Sec. 01

W 3-4:30

110.734

TOPICS IN ALGEBRAIC NUMBER THEORY Ono

Sec. 01

TBA

110.799

THESIS RESEARCH Staff

110.800

INDEPENDENT STUDY