MATHEMATICS

110.105 (Q)

INTRODUCTION TO CALCULUS (4) Staff   Limit 30 per section   This course starts from scratch and provides students with all the background necessary for the study of calculus. It includes a review of algebra, trigonometry, exponential and logarithmic functions, coordinates and graphs. Each of these tools will be introduced in its cultural and historical context. The concept of the rate of change of a function will be introduced. Not open to students who have studied calculus in high school.

Lec.

Sec. 01

02

MWF 9-9:50

T 1:30-2:20

T 3-3:50

110.106 (Q)

CALCULUS I (FOR BIOLOGICAL AND SOCIAL SCIENCE) (4) Salch
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, applications for systems of linear differential equations, probability distributions. Many applications to the biological and social sciences will be discussed.  

Lec.I

Sec. 01

02

03

04

05

Lec. II

06

07

08

09

MWF 10-10:50

T 4:30-5:20

Th 3-3:50

Th 4:30-5:20

T 1:30-2:20

Th 1:30-2:20

MWF 11-11:50

Th 1:30-2:20

T 3-3:50

T 1:30-2:20

Th 3-3:50

110.107 (Q)   

CALCULUS II  (FOR BIOLOGICAL AND SOCIAL SCIENCE) (4) Porod Wang Limit 30 per section  Prereq: C- or better in Calculus I 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

04

MWF 10-10:50

T 4:30-5:20

T 3-3:50

Th 3-3:50

 Th 1:30-2:20

110.108 (Q)

CALCULUS I (FOR PHYSICAL SCIENCES AND ENGINEERING) (4) Chu   Limit 28 per section  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. I

Sec. 01

02

Lec. II

03

04

MWF 10-10:50

T 1:30-2:20

T 3-3:50

MWF 11-11:50

Th 1:30-2:20

Th 3-3:50

110.109 (Q)

CALCULUS II (FOR PHYSICAL SCIENCES AND ENGINEERING)  (4) Ambro   Limit 28 per section  Prereq: C- or better in 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.  Some applications to the physical sciences and engineering will be discussed, and the courses are designed to meet the needs of students in these disciplines.

Lec. I

Sec. 01

02

03

04

Lec. II

Sec. 05

06

07

08

MWF 10-10:50

T 1:30-2:20

T 4:30-5:20

Th 1:30-2:20

Th 3-3:50

MWF 11-11:50

Th 4:30-5:20

T 3-3:50

T 1:30-2:20

Th 3-3:50

110.113 (Q)

HONORS ONE VARIABLE CALCULUS (4) Staff   Limit 35   This is an honors alternative to the calculus sequences 110.106-107 or 110.108-109 and meets the general requirements for both Calculus I and II (although the credit hours count for only one course). It is a more theoretical treatment of one variable differential and integral calculus and is based on our modern understanding of the real number system as explained by Cantor, Dedekind, and Weierstrass. Students who want to know the “why’s and how’s” of calculus will find this course rewarding. Previous background in calculus is not assumed. Students will learn differential calculus (derivatives, differentiation, chain rule, optimization, related rates, etc.), the theory of integration, the fundamental theorem(s) of calculus, applications of integration, and Taylor series. Prerequisite: A strong ability to learn mathematics quickly and on a higher level than that of the regular calculus sequences.

Lec.

Sec. 01

MW 1:30-2:45

F 1:30-2:20

110.201 (Q)

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

Lec.

Sec. 01

02

03

04

05

06

MWF 10-10:50

T 1:30-2:20

T 3-3:50

T 4:30-5:20

Th 1:30-2:20

Th 3-3:50

Th 4:30-5:20

110.202 (Q)

CALCULUS III (4) Wilson Limit 28 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.

Lec. I

Sec. 01

02

03

04

Lec. II

05

06

07

08

 09

MWF 11-11:50

T 1:30-2:20

T 3-3:50

 Th 4:30-5:20

Th 3-3:50

MWF 12-12:50

 T 4:30-5:20

Th 1:30-2:20

Th 3-3:50

T 1:30-2:20

T 3-3:50

110.211 (Q)

HONORS MULTIVARIABLE CALCULUS (4) Wilkin Limit 40 35 Prereq: B+ or better in Calculus II, or 5 on the Calculus BC AP Exam, or 110.113. This course includes the material in Calculus III (110.202) with some additional applications and theory. Recommended for mathematically able students majoring in physical science, engineering, or especially mathematics.

Lec.

Sec. 01

MW 12-12:50

F 12-12:50

110.212 (Q)

HONORS LINEAR ALGEBRA (4) Zucker    Limit 30   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. 211-212 used to be an integrated yearlong course, but now the two are independent courses and can be taken in either order.  This course satisfies a requirement for the math major that its non-honors sibling does not.

Lec.

Sec. 01

MW 1:30-2:45

F 1:30-2:20

110.302 (E,Q)

DIFFERENTIAL EQUATIONS WITH APPLICATIONS (4) Brown Limit 35 per section.  Prereq: Calculus II 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. The specific subjects to be covered include first order differential equations, second order linear differential equations, applications to electric circuits, oscillation of solutions, power series solutions, systems of linear differential equations, autonomous systems, Laplace transforms and linear differential equations, mathematical models (e.g., in the sciences or economics).

Lec. I

Sec. 01

02

03

04

Lec. II

Sec. 04

05

06

07

MWF 12-12:50

T 1:30-2:20

T 3-3:50

Th 3-3:50

Th 4:30-5:20

MWF 1:30-2:20

Th 4:30-5:20

T 4:30-5:20

Th 1:30-2:20

Th 3-3:50

110.304 (Q)

ELEMENTARY NUMBER  THEORY (4) Shalika  Limit 25   Prereq: Calculus II and Linear Algebra. The student is provided with many historical examples of topics, each of which serves as an illustration of and provides a background for many years of current research in number theory. 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.

Sec. 01

TTh 9-10:15

110.311 (Q)

METHODS OF COMPLEX ANALYSIS (4) Ha  Prereq: Calculus III    Limit 35  
This course is an introduction to the theory of functions of one complex variable. Its emphasis is on techniques and applications, and it serves as a basis for more advanced courses. Functions of a complex variable and their derivatives; power series and Laurent expansions; Cauchy integral theorem and formula; calculus of residues and contour integrals; harmonic functions.

Sec. 01

TTh 12-1:15

110.401 (Q)

ADVANCED ALGEBRA I (4) Consani  Limit 40   Prereq: Linear Algebra An introduction to the basic notions of modern algebra. Elements of group theory: groups, subgroups, normal subgroups, quotients, homomorphisms. Generators and relations, free groups, products, commutative (Abelian) groups, finite groups. Groups acting on sets, the Sylow theorems. Definition and examples of rings and ideals. Introduction to field theory. Linear algebra over a field. Field extensions, constructible polygons, non-trisectability.

Lec.

Sec. 01

MW 12-1:15

F 12-12:50

110.405 (Q)

INTRODUCTION TO REAL ANALYSIS (4) Wang Khosravi  Limit 55  Prereq: Calculus III and Linear Algebra  This course is designed to give a firm grounding in the basic tools of analysis. It is recommended as preparation (but may not be a prerequisite) for other advanced analysis courses. Real and complex number systems, topology of metric spaces, limits, continuity, infinite sequences and series, differentiation, Riemann-Stieltjes integration.

Lec.

Sec. 01

MW 1:30-2:45

F 1:30-2:20

110.415 (Q)

HONORS ANALYSIS I (4) Goldberg Limit 25   Prereq: B+ or higher in Calculus III and Linear Algebra. This highly theoretical sequence in analysis is reserved for the most able students. The sequence covers the real number system, metric spaces, basic functional analysis, the Lebesgue integral, and other topics.

Lec.

Sec. 01

MW 1:30-2:45

F 1:30-2:20

110.427 (Q)

INTRODUCTION TO THE CALCULUS OF VARIATIONS (4) Kim   Limit  25 Prereq: Calculus I, II and III The calculus of variations is concerned with finding optimal solutions (shapes, functions, etc.) where optimality is measured by minimizing a functional (usually an integral involving the unknown functions) possibly with constraints.  This introductory (self-contained) course will cover one dimensional problems (often geometric):  brachistochrone, geodesics, minimum surface area of revolution, isoperimetric problem, curvature flows.  Additional material as required (some differential geometry of curves and surfaces) holding prerequisites to a minimum.

Sec. 01

TTh 10:30-11:45

110.439 (Q)

INTRODUCTION TO DIFFERENTIAL GEOMETRY (4) Spruck   Limit 35  
Prereq: Calculus III, Linear Algebra Theory of curves and surfaces in Euclidean space: Frenet equations, fundamental forms, curvatures of a surface, theorems of Gauss and Mainardi-Codazzi, curves on a surface; introduction to tensor analysis and Riemannian geometry; theorema egregium; elementary global theorems.

Sec. 01

TTh 1:30-2:45

110.443 (E,Q)

FOURIER ANALYSIS (4) Wang, C.   Limit 25   Prereq: Calculus III, Linear Algebra. Recommend: 110.405. An introduction to the Fourier transform and the construction of fundamental solutions of linear partial differential equations. Homogeneous distributions on the real line: the Dirac delta function, the Heaviside step function. Operations with distributions: convolution, differentiation, Fourier transforms. Construction of fundamental solutions of the wave, heat, Laplace and Schrödinger equations. Singularities of fundamental solutions and their physical interpretations (e.g., wave fronts). Fourier analysis of singularities, oscillatory integrals, method of stationary phase.

Sec. 01

TTh 1:30-2:45

110.601

ALGEBRA Kong   Limit 25  Prereq: 110.401-402 or equivalent. 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, rudiments of category theory, homological algebra, field theory, Galois theory, and non-commutative algebras.

Sec. 01

TTh 10:30-11:45 12-1:15

110.605

REAL VARIABLES Sogge Limit 25    Prereq: 110.405, 110.415, 110.413 or equivalent. Measure and integration on abstract and locally compact spaces (extension of measures, decompositions of measures, product measures, the Lebesgue integral, differentiation, Lp-spaces); introduction to functional analysis; integration on groups; Fourier transforms.

Sec. 01

MW 1:30-2:45 TTh 12-1:15

110.612 611

COMPLEX GEOMETRY Shiffman   Limit 25    

Sec. 01

TTh 10:30-11:45

110.615

ALGEBRAIC TOPOLOGY Boardman    Limit 25   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

TTh 12-1:15 1:30-2:45

110.617

NUMBER THEORY Consani   Prereq: 110.601-602   Limit 25  Topics in advanced algebra and number theory, including local fields and adeles, Iwasawa-Tate theory of zeta functions and connections with Hecke’s treatment, semisimple algebras over local and number fields, adeles geometry.

Sec. 01

MW 1:30-2:45

110.631

PARTIAL DIFFERENTIAL EQUATIONS Zelditch Limit 25    Prereq: 110.605
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

MW 12-1:15

110.643

ALGEBRAIC GEOMETRY Shokurov   Prereq: 110.601-602. Limit 25  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, connections with complex analytic geometry and topology, schemes.

Sec. 01

TTh 1:30-2:45

110.645

RIEMANNIAN GEOMETRY Rubinstein   Limit 25 Prereq: 110.405 or 110.415, 110.413 or equivalent. Recommended: 110.406. Differential manifolds, vector fields, flows, 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 theorem, Jacobi equation, variation of arc length and area, Chern-Gauss-Bonnet theorems.

Sec. 01

MW 1:30-2:45

110.665

REPRESENTATION THEORY Boardman   Limit 25

Sec. 01

TTh 9-10:15

110.727

TOPICS IN ALGEBRAIC TOPOLOGY Wilson  Limit 20 Course added 4/28/08

Sec. 01

TBA

110.729

TOPICS IN SEVERAL COMPLEX VARIABLES Staff   Limit 25

Sec. 01

W 3-5:30

110.733

TOPICS IN ALGEBRAIC NUMBER THEORY Ono Limit 25    

Sec. 01

TTh 10:30-11:45

110.737

TOPICS IN ALGEBRAIC GEOMETRY Shokurov Limit 25  Course added 4/15/08

Sec. 01

TTh 10:30-11:45

110.741

TOPICS IN PARTIAL DIFFERENTIAL EQUATIONS Minicozzi Limit 25

Sec. 01

MW 1:30-2:45

110.745

INTRODUCTION TO CURVATURE FLOWS Spruck  Limit 25

Sec. 01

MW TTh 10:30-11:45

110.800

INDEPENDENT STUDY -GRADUATES  Staff

Sec. 01

TBA

110.801

THESIS RESEARCH  Staff

Sec. 01

TBA