Primarily for Undergraduates
Mathematics Ma (formerly Mathematics Xa). Introduction to Functions and Calculus I Catalog Number: 1981 Enrollment: Normally limited to 15 students per section. Juliana Belding, Jameel AlAidroos, Robin Gottlieb and members of the Department Half course (fall term). Section meeting times: Section I: M., W., F., at 10; Section II: M., W., F., at 11; Section III: M. W. F., at 12 (with sufficient enrollment); and a twice weekly lab session to be arranged. The study of functions and their rates of change. Fundamental ideas of calculus are introduced early and used to provide a framework for the study of mathematical modeling involving algebraic, exponential, and logarithmic functions. Thorough understanding of differential calculus promoted by year long reinforcement. Applications to biology and economics emphasized according to the interests of our students. Note: Required first meeting: Wednesday, September 2, 8:30 am, Science Center D. Participation in a one and a half hour workshop is required each week. Participation in the weekly problem sessions is also encouraged. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning. This course, when taken for a letter grade together with Mathematics Mb, meets the Core area requirement for Quantitative Reasoning.
Mathematics Mb (formerly Mathematics Xb). Introduction to Functions and Calculus II
Mathematics 1a. Introduction to Calculus
Mathematics 1b. Calculus, Series, and Differential Equations
Mathematics 19a. Modeling and Differential Equations for the Life Sciences
Mathematics 19b. Linear Algebra, Probability, and Statistics for the Life Sciences
Mathematics 20. Algebra and Multivariable Mathematics for Social Sciences
Mathematics 21a. Multivariable Calculus
Mathematics 21b. Linear Algebra and Differential Equations
Mathematics 23a. Linear Algebra and Real Analysis I
Mathematics 23b. Linear Algebra and Real Analysis II
Mathematics 25a. Honors Linear Algebra and Real Analysis I
Mathematics 25b. Honors Linear Algebra and Real Analysis II
*Mathematics 55a. Honors Abstract Algebra
Mathematics 55b. Honors Real and Complex Analysis
*Mathematics 60r. Reading Course for Senior Honors Candidates
*Mathematics 91r. Supervised Reading and Research
*Mathematics 99r. Tutorial

For Undergraduates and Graduates
See also Applied Mathematics and Statistics. Mathematics 101. Sets, Groups and Topology Catalog Number: 8066 Benedict H. Gross Half course (spring term). M., W., F., at 11. EXAM GROUP: 4 An introduction to rigorous mathematics, axioms, and proofs, via topics such as set theory, symmetry groups, and lowdimensional topology. Note: Familiarity with algebra, geometry and/or calculus is desirable. Students who have already taken Mathematics 23a,b, 25a,b or 55a,b should not take this course for credit. This course, when taken for a letter grade, meets the General Education requirement for Empirical and Mathematical Reasoning or the Core area requirement for Quantitative Reasoning. Prerequisite: An interest in mathematical reasoning.
Mathematics 112. Introductory Real Analysis
Mathematics 113. Analysis I: Complex Function Theory
Mathematics 114. Analysis II: Measure, Integration and Banach Spaces
Mathematics 115. Methods of Analysis
Mathematics 116. Convexity and Optimization with Applications
Mathematics 118r. Dynamical Systems
Mathematics 121. Linear Algebra and Applications
Mathematics 122. Algebra I: Theory of Groups and Vector Spaces
Mathematics 123. Algebra II: Theory of Rings and Fields
Mathematics 124. Number Theory
Mathematics 129. Number Fields
Mathematics 130 (formerly Mathematics 138). Classical Geometry
Mathematics 131. Topology I: Topological Spaces and the Fundamental Group
Mathematics 132. Topology II: Smooth Manifolds
Mathematics 136. Differential Geometry
Mathematics 137. Algebraic Geometry
Mathematics 141. Introduction to Mathematical Logic
[Mathematics 143. Set Theory]
[Mathematics 144. Model Theory and Algebra]
Mathematics 152. Discrete Mathematics
Mathematics 153. Mathematical BiologyEvolutionary Dynamics
Mathematics 154 (formerly Mathematics 191). Probability Theory
Mathematics 155r (formerly Mathematics 192r). Combinatorics
Mathematics 160. John Wallis and Transcendence: Measuring the Circle  (New Course)

New Course
Topics in Additive CombinatoricsBen Green
Entire Year 200910 M W F at 10 beginning Friday September 18 In Science Center 232
A large part of Additive Combinatorics is concerned with approximate algebraic objects and their relation to genuine ones. What is an approximate group? An approximate ring? An approximate homomorphism? An approximate polynomial? We will study these questions and more. To give an example, one notion of an approximate subgroup of Z is that it is a finite set A with the property that A+A = {x + y: x, y in A} is covered by a few translates of A. A remarkable theorem of Freiman and Ruzsa, which we shall prove in the course, describes the structure of sets like this in quite explicit terms. Additive Combinatorics is perhaps better known for its applications: Szemeredi's theorem about arithmetic progressions in dense sets of integers, or the theorem that the primes contain arbitrarily long progressions. We will discuss these and other applications according to the tastes of the audience (for example, we could look at the work of Tao and Vu on random matrices, or some of the work of BourgainGamburdSarnak on nonlinear sieve problems). A special emphasis will be given to the role of nilpotent Lie groups, although this is not yet fully understood. These seem to form part of a "higher Fourier analysis", the proper development of which is something I am keen to get people interested in.
Primarily for Graduates
Mathematics 212a (formerly Mathematics 212ar). Real Analysis Catalog Number: 5446 HorngTzer Yau Half course (fall term). Tu., Th., 10–11:30. EXAM GROUP: 12, 13 Banach spaces, Hilbert spaces and functional analysis. Distributions, spectral theory and the Fourier transform. Prerequisite: Mathematics 114 or equivalent.
Mathematics 212br. Advanced Real Analysis
Mathematics 213a. Complex Analysis
Mathematics 213br. Advanced Complex Analysis
Mathematics 221. Commutative Algebra
Mathematics 222. Lie Groups and Lie Algebras
[Mathematics 223a (formerly Mathematics 251a). Algebraic Number Theory]
[Mathematics 223b (formerly Mathematics 251b). Algebraic Number Theory]
Mathematics 224. Representations of Reductive Lie Groups  (New Course)
Mathematics 229x. Introduction to Analytic Number Theory  (New Course)
Mathematics 230a. Differential Geometry
Mathematics 230br. Advanced Differential Geometry
Mathematics 231a (formerly Mathematics 272a). Algebraic Topology
Mathematics 231br (formerly Mathematics 272b). Advanced Algebraic Topology
Mathematics 232a (formerly Mathematics 260a). Introduction to Algebraic Geometry I
Mathematics 232br (formerly Mathematics 260b). Introduction to Algebraic Geometry II
Mathematics 233a (formerly Mathematics 261a). Theory of Schemes I
Mathematics 233br (formerly Mathematics 261b). Theory of Schemes II
Mathematics 243 (formerly Mathematics 234). Evolutionary Dynamics
Mathematics 252x. Chromatic Homotopy Theory  (New Course)
Mathematics 264. Introduction to the Dynamics of Large Quantum Systems  (New Course)
Mathematics 269x. Integrable Systems and Algebraic Geometry  (New Course)
Mathematics 272x. padic Modular Forms and Analytic Continuation  (New Course)
Mathematics 275z. Riemann Surfaces and Hyperbolic Geometry  (New Course)
Mathematics 278y. Algebraic Topology Literature  (New Course)
Mathematics 279. Aspects of the padic Langlands Correspondence  (New Course)
Mathematics 284. Loop Groups and their Flag Varieties  (New Course)
Mathematics 285. Topics in LowDimensional Topology  (New Course)
Mathematics 291. Symplectic and Contact Topology  (New Course)
Mathematics 292. Galois Representations  (New Course)
Mathematics 293. Topics in the Moduli Theory of Sheaves  (New Course)
Mathematics 299r. Graduate Tutorial in Number Theory

Crosslisted Courses
Applied Mathematics 105a. Complex and Fourier Analysis Applied Mathematics 105b. Ordinary and Partial Differential Equations Applied Mathematics 107. Graph Theory and Combinatorics [Empirical and Mathematical Reasoning 14. Fat Chance]  (New Course) *Freshman Seminar 24i. Mathematical Problem Solving *Freshman Seminar 26s. Mathematical Structures and Gödel’s Completeness Theorem *History of Science 206r. The Continuum of Motion, Space and Change in Aristotle and the Aristotelian Tradition: Seminar  (New Course) Philosophy 144. Logic and Philosophy 
Reading and Research
*Mathematics 300. Teaching Undergraduate Mathematics Catalog Number: 3996 Robin Gottlieb and Jameel AlAidroos Half course (fall term). Tu., 1–2:30. This course is for all firstyear graduate students in Mathematics.
[*Mathematics 301. Theory and Practice of Teaching in the Mathematical Sciences]
*Mathematics 304. Topics in Algebraic Topology
*Mathematics 306. Topics in Representation Theory
*Mathematics 308. Topics in Number Theory and Modular Forms
*Mathematics 313. Topics in Geometrical Representation Theory  (New Course)
*Mathematics 314. Topics in Differential Geometry and Mathematical Physics
*Mathematics 316. Topics in Algebraic Topology / Arithmetic Geometry  (New Course)
*Mathematics 317. Topics in Number Theory and Algebraic Geometry  (New Course)
*Mathematics 318. Topics in Number Theory
*Mathematics 320. Topics in Deformation Theory  (New Course)
*Mathematics 321. Topics in Mathematical Physics
*Mathematics 327. Topics in Several Complex Variables
*Mathematics 333. Topics in Complex Analysis, Dynamics and Geometry
*Mathematics 335. Topics in Differential Geometry and Analysis
*Mathematics 338. Topics in Complex Dynamics  (New Course)
*Mathematics 345. Topics in Geometry and Topology
*Mathematics 346y. Topics in Analysis: Quantum Dynamics
*Mathematics 349. Topics in Algebraic Number Theory
*Mathematics 350. Topics in Mathematical Logic
*Mathematics 351. Topics in Algebraic Number Theory
*Mathematics 352. Topics in Algebraic Number Theory  (New Course)
*Mathematics 356. Topics in Harmonic Analysis
*Mathematics 365. Topics in Differential Geometry
*Mathematics 369. Topics in Derived Algebraic Geometry
*Mathematics 372. Topics in Mathematical Relativity
*Mathematics 373. Topics in Algebraic Topology  (New Course)
*Mathematics 379. Topics in Combinatorics
*Mathematics 381. Introduction to Geometric Representation Theory
*Mathematics 382. Topics in Algebraic Geometry
*Mathematics 388. Topics in Mathematics and Biology
*Mathematics 389. Topics in Number Theory
*Mathematics 394. Topics in ManyBody Quantum System
*Mathematics 395. Topics in Symplectic, Contact, and Low  Dimensional Topology  (New Course)

General education course
Source
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