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Course 18: Mathematics
Fall 2014


General Mathematics

18.01 Calculus
______

Undergrad (Fall, Spring) Calculus I
Prereq: None
Units: 5-0-7
Credit cannot also be received for 18.014, 18.01A, CC.181A, ES.1801, ES.181A
URL: http://www-math.mit.edu/18.01/
Add to schedule Lecture: TR1,F2 (54-100) Recitation: MW10 (36-112) or MW11 (36-112) or MW12 (26-322) or MW1 (26-322) or MW2 (26-322) +final
______
Differentiation and integration of functions of one variable, with applications. Informal treatment of limits and continuity. Differentiation: definition, rules, application to graphing, rates, approximations, and extremum problems. Indefinite integration; separable first-order differential equations. Definite integral; fundamental theorem of calculus. Applications of integration to geometry and science. Elementary functions. Techniques of integration. Polar coordinates. L'Hopital's rule. Improper integrals. Infinite series: geometric, p-harmonic, simple comparison tests, power series for some elementary functions.
Fall: J. Speck
Spring: Information: G. Staffilani
Textbooks (Fall 2014)

18.01A Calculus
______

Undergrad (Fall) Calculus I; first half of term
Prereq: Knowledge of differentiation and elementary integration
Units: 5-0-7
Credit cannot also be received for 18.01, 18.014, CC.181A, ES.1801, ES.181A
Add to schedule Ends Oct 17. Lecture: TR1,F2 (10-250) Recitation: MW10 (13-3101) or MW11 (E17-133) or MW12 (E17-133, 36-112) or MW1 (36-112, 36-155) or MW2 (36-155)
______
Six-week review of one-variable calculus, emphasizing material not on the high-school AB syllabus: integration techniques and applications, improper integrals, infinite series, applications to other topics, such as probability and statistics, as time permits. Prerequisites: one year of high-school calculus or the equivalent, with a score of 4 or 5 on the AB Calculus test (or the AB portion of the BC test, or an equivalent score on a standard international exam), or equivalent college transfer credit, or a passing grade on the first half of the 18.01 advanced standing exam.
J. W. Bush
Textbooks (Fall 2014)

18.014 Calculus with Theory
______

Undergrad (Fall) Calculus I
Prereq: None
Units: 5-0-7
Credit cannot also be received for 18.01, 18.01A, CC.181A, ES.1801, ES.181A
URL: http://math.mit.edu/classes/18.014
Add to schedule Lecture: TR1,F2 (E17-133) Recitation: MW2 (E17-133) +final
______
Covers the same material as 18.01, but at a deeper and more rigorous level. Emphasizes careful reasoning and understanding of proofs. Assumes knowledge of elementary calculus. Topics: axioms for the real numbers; the Riemann integral; limits, theorems on continuous functions; derivatives of functions of one variable; the fundamental theorems of calculus; Taylor's theorem; infinite series, power series, rigorous treatment of the elementary functions.
J. Geiger
Textbooks (Fall 2014)

18.02 Calculus
______

Undergrad (Fall, Spring) Calculus II
Prereq: Calculus I (GIR)
Units: 5-0-7
Credit cannot also be received for 18.022, 18.023, 18.024, 18.02A, CC.1802, CC.182A, ES.1802, ES.182A
URL: http://math.mit.edu/classes/18.02
Add to schedule Lecture: TR1,F2 (26-100) Recitation: MW9 (E17-139) or MW10 (E17-139, 13-4101, 8-119) or MW11 (E17-139, 8-119, 13-4101, 26-302) or MW12 (E17-139, 26-302, 13-4101, 13-3101) or MW1 (13-3101, 13-4101, 13-1143, 66-154) or MW2 (66-154, 13-1143, 36-153, 4-145) or MW3 (4-145, 36-153, 13-1143) +final
______
Calculus of several variables. Vector algebra in 3-space, determinants, matrices. Vector-valued functions of one variable, space motion. Scalar functions of several variables: partial differentiation, gradient, optimization techniques. Double integrals and line integrals in the plane; exact differentials and conservative fields; Green's theorem and applications, triple integrals, line and surface integrals in space, Divergence theorem, Stokes' theorem; applications.
Fall: W. Minicozzi
Spring: P. Etingof
Textbooks (Fall 2014)

18.02A Calculus
______

Undergrad (Fall, IAP, Spring) Calculus II
Prereq: Calculus I (GIR)
Units: 5-0-7
Credit cannot also be received for 18.02, 18.022, 18.023, 18.024, CC.1802, CC.182A, ES.1802, ES.182A
URL: http://math.mit.edu/classes/18.02A
Add to schedule Begins Oct 20. Lecture: TR1,F2 (10-250) Recitation: MW10 (13-3101) or MW11 (E17-133) or MW12 (E17-133, 36-112) or MW1 (36-112, 36-155) or MW2 (36-155) +final
______
First half is taught during the last six weeks of the Fall term; covers material in the first half of 18.02 (through double integrals). Second half of 18.02A can be taken either during IAP (daily lectures) or during the first half of the Spring term; it covers the remaining material in 18.02.
Fall: J. W. Bush
Spring: Information: G. Staffilani
Textbooks (Fall 2014)

18.022 Calculus
______

Undergrad (Fall) Calculus II
Prereq: Calculus I (GIR)
Units: 5-0-7
Credit cannot also be received for 18.02, 18.023, 18.024, 18.02A, CC.1802, CC.182A, ES.1802, ES.182A
Add to schedule Lecture: TR1,F2 (E25-111) Recitation: MW11 (13-1143) or MW12 (E17-128, 13-1143) or MW1 (E17-128) or MW2 (4-159) +final
______
Calculus of several variables. Topics as in 18.02 but with more focus on mathematical concepts. Vector algebra, dot product, matrices, determinant. Functions of several variables, continuity, differentiability, derivative. Parametrized curves, arc length, curvature, torsion. Vector fields, gradient, curl, divergence. Multiple integrals, change of variables, line integrals, surface integrals. Stokes' theorem in one, two, and three dimensions.
O. Tamuz
Textbooks (Fall 2014)

18.024 Calculus with Theory
______

Undergrad (Spring) Calculus II
Prereq: Calculus I (GIR), permission of Instructor
Units: 5-0-7
Credit cannot also be received for 18.02, 18.022, 18.023, 18.02A, CC.1802, CC.182A, ES.1802, ES.182A
______
Continues 18.014. Parallel to 18.02, but at a deeper level, emphasizing careful reasoning and understanding of proofs. Considerable emphasis on linear algebra and vector integral calculus.
J. Geiger

18.03 Differential Equations
______

Undergrad (Fall, Spring) Rest Elec in Sci & Tech
Prereq: None. Coreq: Calculus II (GIR)
Units: 5-0-7
Credit cannot also be received for 18.034, 18.036, CC.1803, ES.1803
URL: http://math.mit.edu/classes/18.03
Add to schedule Lecture: MWF1 (54-100) Recitation: TR10 (36-112, 35-308) or TR11 (E17-139, 35-308) or TR12 (E17-139, 35-308) or TR1 (35-308, 36-112, 4-163) or TR2 (4-163, 36-112) or TR3 (36-112) +final
______
Study of differential equations, including modeling physical systems. Solution of first-order ODEs by analytical, graphical, and numerical methods. Linear ODEs with constant coefficients. Complex numbers and exponentials. Inhomogeneous equations: polynomial, sinusoidal, and exponential inputs. Oscillations, damping, resonance. Fourier series. Matrices, eigenvalues, eigenvectors, diagonalization. First order linear systems: normal modes, matrix exponentials, variation of parameters. Heat equation, wave equation. Nonlinear autonomous systems: critical point analysis, phase plane diagrams.
Fall: L. Demanet
Spring: G. Staffilani, D. Jerison
No required or recommended textbooks

18.034 Differential Equations
______

Undergrad (Spring) Rest Elec in Sci & Tech
Prereq: None. Coreq: Calculus II (GIR)
Units: 5-0-7
Credit cannot also be received for 18.03, 18.036, CC.1803, ES.1803
URL: http://math.mit.edu/classes/18.034
______
Covers much of the same material as 18.03 with more emphasis on theory. The point of view is rigorous and results are proven. Local existence and uniqueness of solutions.
J. Lauer

18.04 Complex Variables with Applications
______

Undergrad (Spring)
Prereq: Calculus II (GIR); 18.03 or 18.034
Units: 4-0-8
Credit cannot also be received for 18.075
URL: http://math.mit.edu/18.04/
______
Complex algebra and functions; analyticity; contour integration, Cauchy's theorem; singularities, Taylor and Laurent series; residues, evaluation of integrals; multivalued functions, potential theory in two dimensions; Fourier analysis, Laplace transforms, and partial differential equations.
H. Cheng

18.05 Introduction to Probability and Statistics
______

Undergrad (Spring) Rest Elec in Sci & Tech
Prereq: Calculus I (GIR)
Units: 4-0-8
URL: http://math.mit.edu/classes/18.05
______
Elementary introduction with applications. Basic probability models. Combinatorics. Random variables. Discrete and continuous probability distributions. Statistical estimation and testing. Confidence intervals. Introduction to linear regression.
J. Orloff

18.06 Linear Algebra
______

Undergrad (Fall, Spring) Rest Elec in Sci & Tech
Prereq: Calculus II (GIR)
Units: 4-0-8
Credit cannot also be received for 18.700
URL: http://web.mit.edu/18.06/www/
Add to schedule Lecture: MWF11 (26-100) Recitation: T9 (E17-136) or T10 (E17-136, 56-180) or T11 (E17-136) or T12 (E17-136) or T1 (E17-139) or T2 (E17-139) +final
______
Basic subject on matrix theory and linear algebra, emphasizing topics useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, singular value decomposition, and positive definite matrices. Applications to least-squares approximations, stability of differential equations, networks, Fourier transforms, and Markov processes. Uses MATLAB. Compared with 18.700, more emphasis on matrix algorithms and many applications.
Fall: A. Postnikov
Spring: G. Strang
Textbooks (Fall 2014)

18.062J Mathematics for Computer Science
______

Undergrad (Fall, Spring) Rest Elec in Sci & Tech
(Same subject as 6.042J)
Prereq: Calculus I (GIR)
Units: 5-0-7
URL: http://theory.csail.mit.edu/classes/6.042
Add to schedule Lecture: TR2.30-4 (32-123) Recitation: WF10 (36-144) or WF1 (34-302) or WF2 (34-302) or WF3 (34-302) or WF11 (36-144) or WF12 (36-144) or WF1 (36-144) or WF2 (36-144) or WF3 (36-144) or WF10 (36-155) or WF11 (36-155) or WF12 (34-302) or WF4 (34-302, 36-144) +final
______
Elementary discrete mathematics for computer science and engineering. Emphasis on mathematical definitions and proofs as well as on applicable methods. Topics: formal logic notation, proof methods; induction, well-ordering; sets, relations; elementary graph theory; integer congruences; asymptotic notation and growth of functions; permutations and combinations, counting principles; discrete probability. Further selected topics such as: recursive definition and structural induction; state machines and invariants; recurrences; generating functions.
F. T. Leighton, A. R. Meyer, A. Moitra
No textbook information available

18.075 Methods for Scientists and Engineers
______

Graduate (Spring) H-Level Grad Credit (H except 2, 6, 8, 12, 16, 18, 22)
Prereq: Calculus II (GIR); 18.03
Units: 3-0-9
Credit cannot also be received for 18.04
URL: http://math.mit.edu/classes/18.075
______
Covers functions of a complex variable; calculus of residues. Includes ordinary differential equations; Bessel and Legendre functions; Sturm-Liouville theory; partial differential equations; heat equation; and wave equations.
H. Cheng

18.085 Computational Science and Engineering I
______

Graduate (Fall, Spring, Summer) H-Level Grad Credit (H except 18)
Prereq: Calculus II (GIR); 18.03 or 18.034
Units: 3-0-9
URL: http://math.mit.edu/classes/18.085
Add to schedule T3 meets in 54-100. Lecture: TR10,T3 (1-190)
______
Review of linear algebra, applications to networks, structures, and estimation, finite difference and finite element solution of differential equations, Laplace's equation and potential flow, boundary-value problems, Fourier series, discrete Fourier transform, convolution. Frequent use of MATLAB in a wide range of scientific and engineering applications.
G. Strang
Textbooks (Summer 2014); Textbooks (Fall 2014)

18.086 Computational Science and Engineering II
______

Graduate (Spring) H-Level Grad Credit (H except 18)
Prereq: Calculus II (GIR); 18.03 or 18.034
Units: 3-0-9
URL: http://math.mit.edu/18086/
______
Initial value problems: finite difference methods, accuracy and stability, heat equation, wave equations, conservation laws and shocks, level sets, Navier-Stokes. Solving large systems: elimination with reordering, iterative methods, preconditioning, multigrid, Krylov subspaces, conjugate gradients. Optimization and minimum principles: weighted least squares, constraints, inverse problems, calculus of variations, saddle point problems, linear programming, duality, adjoint methods.
Information: G. Strang

18.089 Review of Mathematics
______

Graduate (Summer)
Prereq: Permission of instructor
Units: 5-0-7
______
One-week review of one-variable calculus (18.01), followed by concentrated study covering multivariable calculus (18.02), two hours per day for five weeks. Primarily for graduate students in Course 2N. Degree credit allowed only in special circumstances.
Information: G. Staffilani
Textbooks (Summer 2014)

18.094J Teaching College-Level Science and Engineering
______

Graduate (Fall)
(Same subject as 1.95J, 5.95J, 6.982J, 7.59J, 8.395J)
(Subject meets with 2.978)
Prereq: None
Units: 2-0-2 [P/D/F]
URL: http://web.mit.edu/physics/subjects/index.html
Add to schedule Lecture: R9-11 (4-149)
______
Participatory seminar focuses on the knowledge and skills necessary for teaching science and engineering in higher education. Topics include theories of adult learning; course development; promoting active learning, problemsolving, and critical thinking in students; communicating with a diverse student body; using educational technology to further learning; lecturing; creating effective tests and assignments; and assessment and evaluation. Students research and present a relevant topic of particular interest. Appropriate for both novices and those with teaching experience.
J. Rankin
No required or recommended textbooks

18.095 Mathematics Lecture Series
______

Undergrad (IAP) Can be repeated for credit
Prereq: Calculus I (GIR)
Units: 2-0-4 [P/D/F]
URL: http://math.mit.edu/classes/18.095/
______
Ten lectures by mathematics faculty members on interesting topics from both classical and modern mathematics. All lectures accessible to students with calculus background and an interest in mathematics. At each lecture, reading and exercises are assigned. Students prepare these for discussion in a weekly problem session.
Information: G. Staffilani

18.098 Independent Study
______

Undergrad (IAP) Can be repeated for credit
Prereq: Permission of instructor
Units arranged [P/D/F]
______
Studies or special individual reading arranged in consultation with individual faculty members and subject to departmental approval.
Information: G. Staffilani

18.099 Independent Study
______

Undergrad (Fall, IAP, Spring, Summer) Can be repeated for credit
Prereq: Permission of instructor
Units arranged
Add to schedule TBA.
______
Studies (during IAP) or special individual reading (during regular terms). Arranged in consultation with individual faculty members and subject to departmental approval.
Information: G. Staffilani
No textbook information available (Summer 2014); No required or recommended textbooks (Fall 2014)

Analysis

18.100A Real Analysis
______

Graduate (Fall, Spring) H-Level Grad Credit (H except 18)
Prereq: Calculus II (GIR); or 18.014 and Coreq: Calculus II (GIR)
Units: 3-0-9
Credit cannot also be received for 18.100B, 18.100C
URL: http://math.mit.edu/classes/18.100a
Add to schedule Lecture: MWF1 (4-163) +final
______
Textbooks (Fall 2014)

18.100B Real Analysis
______

Graduate (Fall, Spring) H-Level Grad Credit (H except 18)
Prereq: Calculus II (GIR); or 18.014 and Coreq: Calculus II (GIR)
Units: 3-0-9
Credit cannot also be received for 18.100A, 18.100C
URL: http://math.mit.edu/~datchev/18.100B/18.100B.html
Add to schedule Lecture: TR9.30-11 (4-237) +final
______
Textbooks (Fall 2014)

18.100C Real Analysis
______

Undergrad (Fall, Spring)
Prereq: Calculus II (GIR); or 18.014 and Coreq: Calculus II (GIR)
Units: 4-0-11
Credit cannot also be received for 18.100A, 18.100B
Add to schedule Lecture: MWF10 (4-153) Recitation: F12 (4-257) or F3 (4-257) +final
______
Three options offered, each covering fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Each option shows the utility of abstract concepts and teaches understanding and construction of proofs. Option A: Proofs and definitions are less abstract. Gives applications where possible. Concerned primarily with the real line. Option B: More demanding; for students with more mathematical maturity. Places more emphasis on point-set topology and n-space. Option C: 15-unit (4-0-11) variant of Option B, with further instruction and practice in written communication. Enrollment limited in Option C.
Fall: 18.100A: A. P. Mattuck
18.100B: P. Isett
18.100C: E. Baer

Spring: 18.100A: S. Dyatlov
18.100B: J.-L. Kim
18.100C: R. Bezrukavnikov

Textbooks (Fall 2014)

18.101 Analysis and Manifolds
______

Graduate (Fall) H-Level Grad Credit (H except 18)
Prereq: 18.100; 18.06, 18.700, or 18.701
Units: 3-0-9
URL: http://math.mit.edu/classes/18.101/
Add to schedule Lecture: MWF11 (E17-129) +final
______
Introduction to the theory of manifolds: vector fields and densities on manifolds, integral calculus in the manifold setting and the manifold version of the divergence theorem. 18.901 helpful but not required.
V. W. Guillemin
No required or recommended textbooks

18.102 Introduction to Functional Analysis
______

Graduate (Spring) H-Level Grad Credit (H except 18)
Prereq: 18.100; 18.06, 18.700, or 18.701
Units: 3-0-9
URL: http://math.mit.edu/classes/18.102
______
Normed spaces, completeness, functionals, Hahn-Banach theorem, duality, operators. Lebesgue measure, measurable functions, integrability, completeness of L-p spaces. Hilbert space. Compact, Hilbert-Schmidt and trace class operators. Spectral theorem.
R. B. Melrose

18.103 Fourier Analysis: Theory and Applications
______

Graduate (Fall) H-Level Grad Credit (H except 18)
Prereq: 18.100; 18.06, 18.700, or 18.701
Units: 3-0-9
URL: http://math.mit.edu/classes/18.103
Add to schedule Lecture: MWF2 (E17-129) +final
______
Roughly half the subject devoted to the theory of the Lebesgue integral with applications to probability, and half to Fourier series and Fourier integrals.
L. Guth
Textbooks (Fall 2014)

18.104 Seminar in Analysis
______

Undergrad (Fall)
Prereq: 18.100
Units: 3-0-9
URL: http://math.mit.edu/~datchev/18.104/18.104.html
Add to schedule Lecture: MWF1 (E17-129)
______
Students present and discuss material from books or journals. Topics vary from year to year. Instruction and practice in written and oral communication provided. Enrollment limited.
V. W. Guillemin
Textbooks (Fall 2014)

18.112 Functions of a Complex Variable
______

Graduate (Fall) H-Level Grad Credit (H except 18)
Prereq: 18.100; 18.06, 18.700, or 18.701
Units: 3-0-9
URL: http://math.mit.edu/classes/18.112
Add to schedule Lecture: MWF12 (4-149)
______
Studies the basic properties of analytic functions of one complex variable. Conformal mappings and the Poincare model of non-Euclidean geometry. Cauchy-Goursat theorem and Cauchy integral formula. Taylor and Laurent decompositions. Singularities, residues and computation of integrals. Harmonic functions and Dirichlet's problem for the Laplace equation. The partial fractions decomposition. Infinite series and infinite product expansions. The Gamma function. The Riemann mapping theorem. Elliptic functions.
T. Mrowka
Textbooks (Fall 2014)

18.116 Riemann Surfaces
______

Not offered academic year 2014-2015Graduate (Spring) H-Level Grad Credit
Prereq: 18.112
Units: 3-0-9
______
Riemann surfaces, uniformization, Riemann-Roch Theorem. Theory of elliptic functions and modular forms. Some applications, such as to number theory.
T. S. Mrowka

18.117 Topics in Several Complex Variables
______

Not offered academic year 2014-2015Graduate (Spring) H-Level Grad Credit Can be repeated for credit
Prereq: 18.112, 18.965
Units: 3-0-9
______
Harmonic theory on complex manifolds, Hodge decomposition theorem, Hard Lefschetz theorem. Vanishing theorems. Theory of Stein manifolds. As time permits students also study holomorphic vector bundles on Kahler manifolds.
V. W. Guillemin

18.125 Real and Functional Analysis
______

Graduate (Spring) H-Level Grad Credit
Prereq: 18.100
Units: 3-0-9
URL: http://math.mit.edu/classes/18.125/
______
Provides a rigorous introduction to Lebesgue's theory of measure and integration. Covers material that is essential in analysis, probability theory, and differential geometry.
D. W. Stroock

18.135 Geometric Analysis
______

Not offered academic year 2014-2015Graduate (Fall) H-Level Grad Credit
Prereq: 18.745 or 18.755
Units: 3-0-9
______
A quick description of Riemannian symmetric spaces. Spherical functions and Harish-Chandra's c-function. Fourier transforms and Radon transforms on Riemannian symmetric spaces X. Applications to invariant differential equations, in particular the multitemporal wave equation on X. Eigenspace representations.
S. Helgason

18.137 Topics in Geometric Partial Differential Equations
______

Not offered academic year 2015-2016Graduate (Fall) H-Level Grad Credit Can be repeated for credit
Prereq: Permission of Instructor
Units: 3-0-9
Add to schedule Lecture: TR9.30-11 (E17-139)
______
Topics vary from year to year.
T. Colding
Textbooks (Fall 2014)

18.152 Introduction to Partial Differential Equations
______

Not offered academic year 2014-2015Graduate (Spring) H-Level Grad Credit (H except 18)
Prereq: 18.100; 18.06, 18.700, or 18.701
Units: 3-0-9
URL: http://math.mit.edu/classes/18.152
Subject Cancelled Subject Cancelled
______
Introduces three main types of partial differential equations: diffusion, elliptic, and hyperbolic. Includes mathematical tools, real-world examples and applications, such as the Black-Scholes equation, the European options problem, water waves, scalar conservation laws, first order equations and traffic problems.
W. Minicozzi

18.155 Differential Analysis
______

Graduate (Fall) H-Level Grad Credit
Prereq: 18.102 or 18.103
Units: 3-0-9
URL: http://math.mit.edu/classes/18.155
Add to schedule Lecture: TR11-12.30 (4-159)
______
Textbooks (Fall 2014)

18.156 Differential Analysis
______

Graduate (Spring) H-Level Grad Credit
Prereq: 18.155
Units: 3-0-9
URL: http://math.mit.edu/classes/18.156
______
Fall: Review of Lebesgue integration. Lp spaces. Distributions. Fourier transform. Sobolev spaces. Spectral theorem, discrete and continuous spectrum. Homogeneous distributions. Fundamental solutions for elliptic, hyperbolic and parabolic differential operators. Spring: Variable coefficient elliptic, parabolic and hyperbolic partial differential equations. 18.112 recommended for 18.155.
Fall: R. B. Melrose
Spring: L. Guth

18.157 Introduction to Microlocal Analysis
______

Not offered academic year 2014-2015Graduate (Spring) H-Level Grad Credit
Prereq: 18.155
Units: 3-0-9
URL: http://math.mit.edu/classes/18.157
______
The semi-classical theory of partial differential equations. Discussion of Pseudodifferential operators, Fourier integral operators, asymptotic solutions of partial differential equations, and the spectral theory of Schroedinger operators from the semi-classical perspective. Heavy emphasis placed on the symplectic geometric underpinnings of this subject.
R. B. Melrose

18.158 Topics in Differential Equations
______

Not offered academic year 2015-2016Graduate (Spring) H-Level Grad Credit Can be repeated for credit
Prereq: 18.157
Units: 3-0-9
URL: http://math.mit.edu/classes/18.158/
______
Topics vary from year to year.
L. Saint-Raymond

18.175 Theory of Probability
______

Graduate (Spring) H-Level Grad Credit
Prereq: 18.100
Units: 3-0-9
URL: http://math.mit.edu/classes/18.175
______
Sums of independent random variables, central limit phenomena, infinitely divisible laws, Levy processes, Brownian motion, conditioning, and martingales. Prior exposure to probability (e.g., 18.440) recommended.
V. Gorin

18.176 Stochastic Calculus
______

Not offered academic year 2014-2015Graduate (Fall) H-Level Grad Credit
Prereq: 18.175
Units: 3-0-9
______
A rigorous introduction to stochastic calculus. Topics include Brownian motion and continuous martingales, diffusions and Levy processes, Ito calculus, martingale representation and quadratic variation, Girsanov's theorem, Bessel processes, general existence and uniqueness theory for stochastic differential equations, applications to partial differential equations, and a brief overview of applications to finance and statistical physics.
A. Guionnet

18.177 Topics in Stochastic Processes
______

Graduate (Fall, Spring) H-Level Grad Credit Can be repeated for credit
Prereq: 18.175
Units: 3-0-9
URL: http://math.mit.edu/classes/18.177
Add to schedule Lecture: TR1-2.30 (E17-136)
______
Topics vary from year to year.
Fall: J. Miller
Spring: A. Guionnet
No required or recommended textbooks

18.199 Graduate Analysis Seminar
______

Not offered academic year 2014-2015Graduate (Fall) H-Level Grad Credit Can be repeated for credit
Prereq: Permission of instructor
Units: 3-0-9
______
Studies original papers in differential analysis and differential equations. Intended for first- and second-year graduate students. Permission must be secured in advance.
V. W. Guillemin

18.238 Geometry and Quantum Field Theory
______

Graduate (Spring) H-Level Grad Credit Can be repeated for credit
Not offered regularly; consult department
Prereq: Permission of instructor
Units: 3-0-9
______
A rigorous introduction designed for mathematicians into perturbative quantum field theory, using the language of functional integrals. Basics of classical field theory. Free quantum theories. Feynman diagrams. Renormalization theory. Local operators. Operator product expansion. Renormalization group equation. The goal is to discuss, using mathematical language, a number of basic notions and results of QFT that are necessary to understand talks and papers in QFT and string theory.
Information: P. I. Etingof

18.276 Mathematical Methods in Physics
______

Not offered academic year 2015-2016Graduate (Spring) H-Level Grad Credit Can be repeated for credit
Prereq: 18.745 or some familiarity with Lie theory
Units: 3-0-9
______
Content varies from year to year. Recent developments in quantum field theory require mathematical techniques not usually covered in standard graduate subjects.
V. G. Kac

Applied Mathematics

18.303 Linear Partial Differential Equations: Analysis and Numerics
______

Undergrad (Fall)
Prereq: 18.06 or 18.700
Units: 3-0-9
URL: http://math.mit.edu/classes/18.303
Add to schedule Lecture: MWF1 (4-159)
______
Provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science and engineering, including heat/diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Studies operator adjoints and eigenproblems, series solutions, Green's functions, and separation of variables. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems, including stability and convergence analysis and implicit/explicit timestepping. MATLAB is introduced and used in homework for simple examples.
S. G. Johnson
Textbooks (Fall 2014)

18.304 Undergraduate Seminar in Discrete Mathematics
______

Undergrad (Fall, Spring)
Prereq: 18.310 or 18.062; 18.06, 18.700, or 18.701; or permission of instructor
Units: 3-0-9
Credit cannot also be received for 18.316
URL: http://math.mit.edu/classes/18.304
Add to schedule Lecture: MWF2 (E17-128)
______
Seminar in combinatorics, graph theory, and discrete mathematics in general. Participants read and present papers from recent mathematics literature. Instruction and practice in written and oral communication provided. Enrollment limited.
Fall: P. Csikvari
Spring: J. Novak
No textbook information available

18.305 Advanced Analytic Methods in Science and Engineering
______

Graduate (Fall) H-Level Grad Credit
Prereq: 18.04, 18.075, or 18.112
Units: 3-0-9
URL: http://math.mit.edu/18.305/
Add to schedule Lecture: MWF11 (E17-128)
______
Covers expansion around singular points: the WKB method on ordinary and partial differential equations; the method of stationary phase and the saddle point method; the two-scale method and the method of renormalized perturbation; singular perturbation and boundary-layer techniques; WKB method on partial differential equations.
H. Cheng
Textbooks (Fall 2014)

18.306 Advanced Partial Differential Equations with Applications
______

Graduate (Spring) H-Level Grad Credit
Prereq: 18.03 or 18.034; 18.04, 18.075, or 18.112
Units: 3-0-9
URL: http://math.mit.edu/classes/18.306
______
Concepts and techniques for partial differential equations, especially nonlinear. Diffusion, dispersion and other phenomena. Initial and boundary value problems. Normal mode analysis, Green's functions, and transforms. Conservation laws, kinematic waves, hyperbolic equations, characteristics shocks, simple waves. Geometrical optics, caustics. Free-boundary problems. Dimensional analysis. Singular perturbation, boundary layers, homogenization. Variational methods. Solitons. Applications from fluid dynamics, materials science, optics, traffic flow, etc.
R. R. Rosales

18.310 Principles of Discrete Applied Mathematics
______

Undergrad (Fall)
Prereq: Calculus II (GIR)
Units: 4-0-11
Credit cannot also be received for 18.310A
URL: http://math.mit.edu/classes/18.310
Add to schedule Lecture: MWF12 (4-163) Recitation: R10 (E17-136) or R12 (E17-136) or R1 (E17-139) or R3 (E17-139)
______
Study of illustrative topics in discrete applied mathematics, including sorting algorithms, probability theory, information theory, coding theory, secret codes, generating functions, and linear programming. Instruction and practice in written communication provided. Enrollment limited.
J. Fox, P. W. Shor
No required or recommended textbooks

18.310A Principles of Discrete Applied Mathematics
______

Undergrad (Spring)
Prereq: Calculus II (GIR)
Units: 3-0-9
Credit cannot also be received for 18.310
______
Study of illustrative topics in discrete applied mathematics, including sorting algorithms, probability theory, information theory, coding theory, secret codes, generating functions, and linear programming.
M. X. Goemans

18.311 Principles of Continuum Applied Mathematics
______

Undergrad (Spring)
Prereq: Calculus II (GIR); 18.03 or 18.034
Units: 3-0-9
URL: http://math.mit.edu/classes/18.311
______
Covers fundamental concepts in continuous applied mathematics. Applications from traffic flow, fluids, elasticity, granular flows, etc. Also covers continuum limit; conservation laws, quasi-equilibrium; kinematic waves; characteristics, simple waves, shocks; diffusion (linear and nonlinear); numerical solution of wave equations; finite differences, consistency, stability; discrete and fast Fourier transforms; spectral methods; transforms and series (Fourier, Laplace). Additional topics may include sonic booms, Mach cone, caustics, lattices, dispersion and group velocity. Uses MATLAB computing environment.
R. R. Rosales

18.312 Algebraic Combinatorics
______

Undergrad (Spring)
Prereq: 18.701 or 18.703
Units: 3-0-9
URL: http://math.mit.edu/classes/18.312
______
Applications of algebra to combinatorics. Topics include walks in graphs, the Radon transform, groups acting on posets, Young tableaux, electrical networks.
P. Csikvari

18.314 Combinatorial Analysis
______

Undergrad (Fall)
Prereq: Calculus II (GIR); 18.06, 18.700, or 18.701
Units: 3-0-9
URL: http://math.mit.edu/classes/18.314
Add to schedule Lecture: MWF11 (E17-122) +final
______
Combinatorial problems and methods for their solution. Enumeration, generating functions, recurrence relations, construction of bijections. Introduction to graph theory. Prior experience with abstraction and proofs is helpful.
R. P. Stanley
Textbooks (Fall 2014)

18.315 Combinatorial Theory
______

Graduate (Fall) H-Level Grad Credit Can be repeated for credit
Prereq: Permission of instructor
Units: 3-0-9
URL: http://math.mit.edu/classes/18.315
Add to schedule Lecture: WF1-2.30 (E17-139)
______
Content varies from year to year.
A. Postnikov
No required or recommended textbooks

18.316 Seminar in Combinatorics
______

Not offered academic year 2014-2015Graduate (Fall) H-Level Grad Credit Can be repeated for credit
Prereq: Permission of instructor
Units: 3-0-9
Credit cannot also be received for 18.304
______
Content varies from year to year. Readings from current research papers in combinatorics. Topics to be chosen and presented by the class.
J. Fox

18.318 Topics in Combinatorics
______

Graduate (Spring) H-Level Grad Credit Can be repeated for credit
Prereq: Permission of instructor
Units: 3-0-9
URL: http://math.mit.edu/~apost/courses/18.318/
______
Topics vary from year to year.
C. Lee

18.325 Topics in Applied Mathematics
______

Not offered academic year 2014-2015Graduate (Fall) H-Level Grad Credit Can be repeated for credit
Prereq: Permission of instructor
Units: 3-0-9
URL: http://math.mit.edu/classes/18.325
______
Topics vary from year to year.
L. Demanet

18.330 Introduction to Numerical Analysis
______

Undergrad (Spring)
Prereq: Calculus II (GIR); 18.03 or 18.034
Units: 3-0-9
URL: http://math.mit.edu/classes/18.330/
______
Basic techniques for the efficient numerical solution of problems in science and engineering. Root finding, interpolation, approximation of functions, integration, differential equations, direct and iterative methods in linear algebra. Knowledge of programming in Fortran, C, or MATLAB helpful.
H. Reid

18.335J Introduction to Numerical Methods
______

Graduate (Spring) H-Level Grad Credit
(Same subject as 6.337J)
Prereq: 18.03 or 18.034; 18.06, 18.700, or 18.701
Units: 3-0-9
URL: http://math.mit.edu/classes/18.335
______
Advanced introduction to numerical linear algebra and related numerical methods. Topics include direct and iterative methods for linear systems, eigenvalue and QR/SVD factorizations, stability and accuracy, floating-point arithmetic, sparse matrices, preconditioning, and the memory considerations underlying modern linear algebra software. Starting from iterative methods for linear systems, explores more general techniques for local and global nonlinear optimization, including quasi-Newton methods, trust regions, branch-and-bound, and multistart algorithms. Also addresses Chebyshev approximation and FFTs. MATLAB is introduced for problem sets.
S. G. Johnson

18.336J Fast Methods for Partial Differential and Integral Equations
______

Graduate (Fall) H-Level Grad Credit
(Same subject as 6.335J)
Prereq: 6.336, 16.920, 18.085, 18.335, or permission of instructor
Units: 3-0-9
URL: http://math.mit.edu/classes/18.336
Add to schedule Lecture: TR9.30-11 (E17-128)
______
Unified introduction to the theory and practice of modern, near linear-time, numerical methods for large-scale partial-differential and integral equations. Topics include preconditioned iterative methods; generalized Fast Fourier Transform and other butterfly-based methods; multiresolution approaches, such as multigrid algorithms and hierarchical low-rank matrix decompositions; and low and high frequency Fast Multipole Methods. Example applications include aircraft design, cardiovascular system modeling, electronic structure computation, and tomographic imaging.
A. Townsend
No required or recommended textbooks

18.337J Parallel Computing
______

Not offered academic year 2014-2015Graduate (Fall) H-Level Grad Credit
(Same subject as 6.338J)
Prereq: 18.06, 18.700, or 18.701
Units: 3-0-9
URL: http://beowulf.csail.mit.edu/18.337/index.html
______
Interdisciplinary introduction to parallel computing and modern big data analysis using Julia. Covers scientific computing topics such as dense and sparse linear algebra, N-body problems, and Fourier transforms, and geometric computing topics such as mesh generation and mesh partitioning. Focuses on application of these techniques to machine learning algorithms in big data applications. Provides direct experience with programming traditional-style supercomputing as well as working with modern cloud computing stacks. Designed to separate the realities and myths about the kinds of problems that can be solved on the world's fastest machines.
A. Edelman

18.338 Eigenvalues of Random Matrices
______

Not offered academic year 2015-2016Graduate (Spring) H-Level Grad Credit
Prereq: 18.701 or permission of instructor
Units: 3-0-9
URL: http://www.mit.edu/~18.338/
______
Covers the modern main results of random matrix theory as it is currently applied in engineering and science. Topics include matrix calculus for finite and infinite matrices (e.g., Wigner's semi-circle and Marcenko-Pastur laws), free probability, random graphs, combinatorial methods, matrix statistics, stochastic operators, passage to the continuum limit, moment methods, and compressed sensing. Knowledge of MATLAB hepful, but not required.
A. Edelman

18.352J Theoretical Environmental Analysis
______

Undergrad (Spring)
(Same subject as 12.009J)
Prereq: Physics I (GIR), Calculus II (GIR); Coreq: 18.03
Units: 3-0-9
______
Analyzes cooperative processes that shape the natural environment, now and in the geologic past. Emphasizes the development of theoretical models that relate the physical and biological worlds, the comparison of theory to observational data, and associated mathematical methods. Topics include carbon cycle dynamics; ecosystem structure, stability and complexity; mass extinctions; biosphere-geosphere coevolution; and climate change. Employs techniques such as stability analysis; scaling; null model construction; time series and network analysis.
D. H. Rothman

18.353J Nonlinear Dynamics: Chaos
______

Undergrad (Fall)
(Same subject as 2.050J, 12.006J)
Prereq: 18.03 or 18.034; Physics II (GIR)
Units: 3-0-9
Add to schedule Lecture: TR12.30-2 (2-105)
______
Introduction to nonlinear dynamics and chaos in dissipative systems. Forced and parametric oscillators. Phase space. Periodic, quasiperiodic, and aperiodic flows. Sensitivity to initial conditions and strange attractors. Lorenz attractor. Period doubling, intermittency, and quasiperiodicity. Scaling and universality. Analysis of experimental data: Fourier transforms, Poincare sections, fractal dimension, and Lyapunov exponents. Applications to mechanical systems, fluid dynamics, physics, geophysics, and chemistry. See 12.207J/18.354J for Nonlinear Dynamics: Continuum Systems.
R. Lagrange
Textbooks (Fall 2014)

18.354J Nonlinear Dynamics: Continuum Systems
______

Graduate (Spring) H-Level Grad Credit (H except 1, 18)
(Same subject as 1.062J, 12.207J)
Prereq: 18.03 or 18.034; Physics II (GIR)
Units: 3-0-9
URL: http://math.mit.edu/classes/18.354/
______
General mathematical principles of continuum systems. From microscopic to macroscopic descriptions in the form of linear or nonlinear (partial) differential equations. Exact solutions, dimensional analysis, calculus of variations and singular perturbation methods. Stability, waves and pattern formation in continuum systems. Subject matter illustrated using natural fluid and solid systems found, for example, in geophysics and biology.
J. Dunkel

18.355 Fluid Mechanics
______

Not offered academic year 2015-2016Graduate (Fall) H-Level Grad Credit
Prereq: 18.354, 2.25, or 12.800
Units: 3-0-9
Add to schedule Lecture: MW2-3.30 (E17-136)
______
Topics include the development of Navier-Stokes equations, inviscid flows, boundary layers, lubrication theory, Stokes flows, and surface tension. Fundamental concepts illustrated through problems drawn from a variety of areas, including geophysics, biology, and the dynamics of sport. Particular emphasis on the interplay between dimensional analysis, scaling arguments, and theory. Includes classroom and laboratory demonstrations.
J. W. Bush
No required or recommended textbooks

18.357 Interfacial Phenomena
______

Not offered academic year 2014-2015Graduate (Fall) H-Level Grad Credit
Prereq: 18.354, 18.355, 12.800, 2.25, or permission of instructor
Units: 3-0-9
______
Fluid systems dominated by the influence of interfacial tension. Elucidates the roles of curvature pressure and Marangoni stress in a variety of hydrodynamic settings. Particular attention to drops and bubbles, soap films and minimal surfaces, wetting phenomena, water-repellency, surfactants, Marangoni flows, capillary origami and contact line dynamics. Theoretical developments are accompanied by classroom demonstrations. Highlights the role of surface tension in biology.
J. W. Bush

18.369 Mathematical Methods in Nanophotonics
______

Not offered academic year 2014-2015Graduate (Spring) H-Level Grad Credit
Prereq: 18.305 or permission of instructor
Units: 3-0-9
URL: http://math.mit.edu/classes/18.369
______
High-level approaches to understanding complex optical media, structured on the scale of the wavelength, that are not generally analytically soluable. The basis for understanding optical phenomena such as photonic crystals and band gaps, anomalous diffraction, mechanisms for optical confinement, optical fibers (new and old), nonlinearities, and integrated optical devices. Methods covered include linear algebra and eigensystems for Maxwell's equations, symmetry groups and representation theory, Bloch's theorem, numerical eigensolver methods, time and frequency-domain computation, perturbation theory, and coupled-mode theories.
S. G. Johnson

18.376J Wave Propagation
______

Graduate (Spring) H-Level Grad Credit
(Same subject as 1.138J, 2.062J)
Prereq: 2.003, 18.075
Units: 3-0-9
URL: http://math.mit.edu/classes/18.376/
______
Theoretical concepts and analysis of wave problems in science and engineering with examples chosen from elasticity, acoustics, geophysics, hydrodynamics, blood flow, nondestructive evaluation, and other applications. Progressive waves, group velocity and dispersion, energy density and transport. Reflection, refraction and transmission of plane waves by an interface. Mode conversion in elastic waves. Rayleigh waves. Waves due to a moving load. Scattering by a two-dimensional obstacle. Reciprocity theorems. Parabolic approximation. Waves on the sea surface. Capillary-gravity waves. Wave resistance. Radiation of surface waves. Internal waves in stratified fluids. Waves in rotating media. Waves in random media.
T. R. Akylas, R. R. Rosales

18.377J Nonlinear Dynamics and Waves
______

Not offered academic year 2015-2016Graduate (Spring) H-Level Grad Credit
(Same subject as 1.685J, 2.034J)
Prereq: Permission of instructor
Units: 3-0-9
______
A unified treatment of nonlinear oscillations and wave phenomena with applications to mechanical, optical, geophysical, fluid, electrical and flow-structure interaction problems. Nonlinear free and forced vibrations; nonlinear resonances; self-excited oscillations; lock-in phenomena. Nonlinear dispersive and nondispersive waves; resonant wave interactions; propagation of wave pulses and nonlinear Schrodinger equation. Nonlinear long waves and breaking; theory of characteristics; the Korteweg-de Vries equation; solitons and solitary wave interactions. Stability of shear flows. Some topics and applications may vary from year to year.
T. R. Akylas, R. R. Rosales

18.384 Undergraduate Seminar in Physical Mathematics
______

Undergrad (Fall)
Prereq: 18.311, 18.353, 18.354, or permission of instructor
Units: 3-0-9
URL: http://math.mit.edu/classes/18.384
Add to schedule Lecture: TR9.30-11 (E17-133)
______
Covers the mathematical modeling of physical systems, with emphasis on the reading and presentation of papers. Addresses a broad range of topics, with particular focus on macroscopic physics and continuum systems: fluid dynamics, solid mechanics, and biophysics. Instruction and practice in written and oral communication provided. Enrollment limited.
P.-T. Brun
No required or recommended textbooks

18.385J Nonlinear Dynamics and Chaos
______

Not offered academic year 2015-2016Graduate (Fall) H-Level Grad Credit
(Same subject as 2.036J)
Prereq: 18.03 or 18.034
Units: 3-0-9
URL: http://math.mit.edu/classes/18.385
Add to schedule Lecture: TR11-12.30 (E17-129)
______
Introduction to the theory of nonlinear dynamical systems with applications from science and engineering. Local and global existence of solutions, dependence on initial data and parameters. Elementary bifurcations, normal forms. Phase plane, limit cycles, relaxation oscillations, Poincare-Bendixson theory. Floquet theory. Poincare maps. Averaging. Near-equilibrium dynamics. Synchronization. Introduction to chaos. Universality. Strange attractors. Lorenz and Rossler systems. Hamiltonian dynamics and KAM theory. Uses MATLAB computing environment.
R. R. Rosales
No textbook information available

18.386 Advanced Nonlinear Dynamics and Chaos
______

Not offered academic year 2014-2015Graduate (Spring) H-Level Grad Credit
Prereq: 18.385 or permission of instructor
Units: 3-0-9
URL: http://web.mit.edu/2.037j/www/
______
Advanced subject on the modern theory of nonlinear dynamical systems with an emphasis on applications in science and engineering. Invariant manifolds, homoclinic orbits, global bifurcations. Hamiltonian systems, completely integrable systems, KAM theory. Different mechanisms for chaotic dynamics, Shilnikov-type orbits, attractors, horseshoes, symbolic dynamics. Geometric singular perturbation theory. Physical applications.
Information: R. R. Rosales

18.395 Group Theory with Applications to Physics
______

Not offered academic year 2014-2015Graduate (Fall) H-Level Grad Credit
Prereq: 8.321
Units: 3-0-9
______
Selection of topics from the theory of finite groups, Lie groups, and group representations, motivated by quantum mechanics and particle physics. 8.322 and 8.323 helpful.
D. Z. Freedman

18.396J Supersymmetric Quantum Field Theories
______

Not offered academic year 2015-2016Graduate (Fall) H-Level Grad Credit Can be repeated for credit
(Same subject as 8.831J)
Prereq: Permission of instructor
Units: 3-0-9
Add to schedule Lecture: TR11-12.30 (E17-128)
______
Topics selected from the following: SUSY algebras and their particle representations; Weyl and Majorana spinors; Lagrangians of basic four-dimensional SUSY theories, both rigid SUSY and supergravity; supermultiplets of fields and superspace methods; renormalization properties, and the non-renormalization theorem; spontaneous breakdown of SUSY; and phenomenological SUSY theories. Some prior knowledge of Noether's theorem, derivation and use of Feynman rules, l-loop renormalization, and gauge theories is essential.
D. Z. Freedman
Textbooks (Fall 2014)

18.398 Quantum Field Theories
______

Graduate (Spring) H-Level Grad Credit
Not offered regularly; consult department
Prereq: Permission of instructor
Units: 3-0-9
______
For students who want to have a clear understanding of quantum field theories. Appropriate for students who have not taken such a subject as well as students who have but are not entirely comfortable with the basic concepts and techniques. The topics begin with classical mechanics and end with gauge field theories and the renormalization of the standard model.
Information: H. Cheng

Theoretical Computer Science

18.400J Automata, Computability, and Complexity
______

Undergrad (Spring)
(Same subject as 6.045J)
Prereq: 6.042
Units: 4-0-8
URL: http://math.mit.edu/classes/18.400
______
Provides an introduction to some of the central ideas of theoretical computer science, including circuits, finite automata, Turing machines and computability, efficient algorithms and reducibility, the P versus NP problem, NP-completeness, the power of randomness, cryptography, computational learning theory, and quantum computing. Examines the classes of problems that can and cannot be solved in various computational models.
S. Aaronson

18.404J Theory of Computation
______

Graduate (Fall) H-Level Grad Credit (H except 18)
(Same subject as 6.840J)
Prereq: 18.310 or 18.062J
Units: 4-0-8
URL: http://math.mit.edu/classes/18.404
Add to schedule Lecture: TR2.30-4 (E25-111) Recitation: F11 (E17-136) or F12 (E17-136) or F1 (E17-128) +final
______
A more extensive and theoretical treatment of the material in 6.045J/18.400J, emphasizing computability and computational complexity theory. Regular and context-free languages. Decidable and undecidable problems, reducibility, recursive function theory. Time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems.
M. Sipser
Textbooks (Fall 2014)

18.405J Advanced Complexity Theory
______

Not offered academic year 2015-2016Graduate (Spring) H-Level Grad Credit
(Same subject as 6.841J)
Prereq: 18.404
Units: 3-0-9
______
Current research topics in computational complexity theory. Nondeterministic, alternating, probabilistic, and parallel computation models. Boolean circuits. Complexity classes and complete sets. The polynomial-time hierarchy. Interactive proof systems. Relativization. Definitions of randomness. Pseudo-randomness and derandomizations. Interactive proof systems and probabilistically checkable proofs.
D. Moshkovitz

18.409 Topics in Theoretical Computer Science
______

Graduate (Spring) H-Level Grad Credit Can be repeated for credit
Prereq: Permission of instructor
Units: 3-0-9
URL: http://math.mit.edu/classes/18.409
Subject Cancelled Subject Cancelled
______
Study of areas of current interest in theoretical computer science. Topics vary from term to term.
A. Moitra

18.410J Design and Analysis of Algorithms
______

Undergrad (Fall, Spring)
(Same subject as 6.046J)
Prereq: 6.006
Units: 4-0-8
URL: http://math.mit.edu/classes/18.410
Add to schedule Lecture: TR11-12.30 (26-100) Recitation: F10 (26-302) or F11 (26-302) or F12 (26-168) or F1 (26-168) or F2 (26-302) or F3 (26-302) or F11 (36-156) or F12 (36-156) or F1 (34-301) or F2 (34-301) +final
______
Techniques for the design and analysis of efficient algorithms, emphasizing methods useful in practice. Topics include sorting; search trees, heaps, and hashing; divide-and-conquer; dynamic programming; greedy algorithms; amortized analysis; graph algorithms; and shortest paths. Advanced topics may include network flow; computational geometry; number-theoretic algorithms; polynomial and matrix calculations; caching; and parallel computing.
E. Demaine, M. Goemans
Textbooks (Fall 2014)

18.415J Advanced Algorithms
______

Graduate (Fall) H-Level Grad Credit
(Same subject as 6.854J)
Prereq: 6.041, 6.042, or 18.440; 6.046
Units: 5-0-7
URL: http://theory.lcs.mit.edu/classes/6.854/
Add to schedule Lecture: MWF2.30-4 (32-155)
______
First-year graduate subject in algorithms. Emphasizes fundamental algorithms and advanced methods of algorithmic design, analysis, and implementation. Surveys a variety of computational models and the algorithms for them. Data structures, network flows, linear programming, computational geometry, approximation algorithms, online algorithms, parallel algorithms, external memory, streaming algorithms.
D. R. Karger
No textbook information available

18.416J Randomized Algorithms
______

Not offered academic year 2015-2016Graduate (Spring) H-Level Grad Credit
(Same subject as 6.856J)
Prereq: 6.854J, 6.041 or 6.042J
Units: 5-0-7
______
Studies how randomization can be used to make algorithms simpler and more efficient via random sampling, random selection of witnesses, symmetry breaking, and Markov chains. Models of randomized computation. Data structures: hash tables, and skip lists. Graph algorithms: minimum spanning trees, shortest paths, and minimum cuts. Geometric algorithms: convex hulls, linear programming in fixed or arbitrary dimension. Approximate counting; parallel algorithms; online algorithms; derandomization techniques; and tools for probabilistic analysis of algorithms.
D. R. Karger

18.417 Introduction to Computational Molecular Biology
______

Not offered academic year 2014-2015Graduate (Fall) H-Level Grad Credit
Prereq: 6.01, 6.006, or permission of instructor
Units: 3-0-9
URL: http://www-math.mit.edu/18.417/
______
Introduces the basic computational methods used to model and predict the structure of biomolecules (proteins, DNA, RNA). Covers classical techniques in the field (molecular dynamics, Monte Carlo, dynamic programming) to more recent advances in analyzing and predicting RNA and protein structure, ranging from Hidden Markov Models and 3-D lattice models to attribute Grammars and tree Grammars.
Information: B. Berger

18.418 Topics in Computational Molecular Biology
______

Graduate (Spring) H-Level Grad Credit Can be repeated for credit
Prereq: 18.417, 6.047, or permission of instructor
Units: 3-0-9
URL: http://math.mit.edu/classes/18.418
______
Covers current research topics in computational molecular biology. Recent research papers presented from leading conferences such as the SIGACT International Conference on Computational Molecular Biology (RECOMB). Topics include original research (both theoretical and experimental) in comparative genomics, sequence and structure analysis, molecular evolution, proteomics, gene expression, transcriptional regulation, and biological networks. Recent research by course participants also covered. Participants will be expected to present either group or individual projects to the class.
B. Berger

18.424 Seminar in Information Theory
______

Undergrad (Spring)
Prereq: 18.05, 18.440, or 6.041; 18.06, 18.700, or 18.701
Units: 3-0-9
______
Considers various topics in information theory, including data compression, Shannon's Theorems, and error-correcting codes. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.
P. W. Shor

18.425J Cryptography and Cryptanalysis
______

Graduate (Spring) H-Level Grad Credit
(Same subject as 6.875J)
Prereq: 6.046J
Units: 3-0-9
______
A rigorous introduction to modern cryptography. Emphasis on the fundamental cryptographic primitives of public-key encryption, digital signatures, pseudo-random number generation, and basic protocols and their computational complexity requirements.
S. Goldwasser, S. Micali

18.426J Advanced Topics in Cryptography
______

Not offered academic year 2014-2015Graduate (Fall) H-Level Grad Credit Can be repeated for credit
(Same subject as 6.876J)
Prereq: 6.875
Units: 3-0-9
Subject Cancelled Subject Cancelled
______
Recent results in cryptography, interactive proofs, and cryptographic game theory. Lectures by instructor, invited speakers, and students.
S. Goldwasser, S. Micali

18.433 Combinatorial Optimization
______

Not offered academic year 2015-2016Graduate (Spring) H-Level Grad Credit (H except 18)
Prereq: 18.06, 18.700, or 18.701
Units: 3-0-9
URL: http://math.mit.edu/classes/18.433
______
Thorough treatment of linear programming and combinatorial optimization. Topics include matching theory, network flow, matroid optimization, and how to deal with NP-hard optimization problems. Prior exposure to discrete mathematics (such as 18.310) helpful.
M. X. Goemans

18.434 Seminar in Theoretical Computer Science
______

Undergrad (Spring)
Prereq: 18.410
Units: 3-0-9
URL: http://math.mit.edu/classes/18.434
______
Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.
R. Peng

18.435J Quantum Computation
______

Graduate (Fall) H-Level Grad Credit
(Same subject as 2.111J, 8.370J)
Prereq: Permission of instructor
Units: 3-0-9
Add to schedule Lecture: TR1-2.30 (56-114) +final
______
Provides an introduction to the theory and practice of quantum computation. Topics covered: physics of information processing; quantum algorithms including the factoring algorithm and Grover's search algorithm; quantum error correction; quantum communication and cryptography. Knowledge of quantum mechanics helpful but not required.
I. Chuang, E. Farhi, S. Lloyd, P. Shor
No textbook information available

18.436J Quantum Information Science
______

Graduate (Spring, Summer) H-Level Grad Credit
(Same subject as 6.443J, 8.371J)
Prereq: 18.435
Units: 3-0-9
______
Examines quantum computation and quantum information. Topics include quantum circuits, the quantum Fourier transform and search algorithms, the quantum operations formalism, quantum error correction, Calderbank-Shor-Steane and stabilizer codes, fault tolerant quantum computation, quantum data compression, quantum entanglement, capacity of quantum channels, and quantum cryptography and the proof of its security. Prior knowledge of quantum mechanics required.
Information: P. W. Shor
Textbooks (Summer 2014)

18.437J Distributed Algorithms
______

Graduate (Fall) H-Level Grad Credit
(Same subject as 6.852J)
Prereq: 6.046
Units: 3-0-9
URL: http://theory.csail.mit.edu/classes/6.852/
Add to schedule Lecture: TR11-12.30 (4-237)
______
Design and analysis of concurrent algorithms, emphasizing those suitable for use in distributed networks. Process synchronization, allocation of computational resources, distributed consensus, distributed graph algorithms, election of a leader in a network, distributed termination, deadlock detection, concurrency control, communication, and clock synchronization. Special consideration given to issues of efficiency and fault tolerance. Formal models and proof methods for distributed computation.
N. A. Lynch
Textbooks (Fall 2014)

18.438 Advanced Combinatorial Optimization
______

Not offered academic year 2014-2015Graduate (Spring) H-Level Grad Credit
Prereq: 18.433 or permission of instructor
Units: 3-0-9
______
Advanced treatment of combinatorial optimization with an emphasis on combinatorial aspects. Non-bipartite matchings, submodular functions, matroid intersection/union, matroid matching, submodular flows, multicommodity flows, packing and connectivity problems, and other recent developments.
M. X. Goemans

Probability and Statistics

18.440 Probability and Random Variables
______

Undergrad (Fall, Spring) Rest Elec in Sci & Tech
Prereq: Calculus II (GIR)
Units: 3-0-9
Credit cannot also be received for 6.041, 6.431
URL: http://math.mit.edu/classes/18.440
Add to schedule Lecture: MWF10 (54-100) +final
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Probability spaces, random variables, distribution functions. Binomial, geometric, hypergeometric, Poisson distributions. Uniform, exponential, normal, gamma and beta distributions. Conditional probability, Bayes theorem, joint distributions. Chebyshev inequality, law of large numbers, and central limit theorem.
Fall: A. Guionnet
Spring: J. A. Kelner
Textbooks (Fall 2014)

18.443 Statistics for Applications
______

Graduate (Fall, Spring) H-Level Grad Credit (H except 18)
Prereq: 18.440 or 6.041
Units: 3-0-9
URL: http://math.mit.edu/classes/18.443
Add to schedule Lecture: MWF11 (32-141)
______
A broad treatment of statistics, concentrating on specific statistical techniques used in science and industry. Topics: hypothesis testing and estimation. Confidence intervals, chi-square tests, nonparametric statistics, analysis of variance, regression, correlation, decision theory, and Bayesian statistics.
Fall: R. M. Dudley
Spring: P. Kempthorne
Textbooks (Fall 2014)

18.445 Introduction to Stochastic Processes
______

Graduate (Spring) H-Level Grad Credit
Prereq: 18.440 or 6.041
Units: 3-0-9
______
Basics of stochastic processes. Markov chains, Poisson processes, random walks, birth and death processes, Brownian motion.
H. Wu

18.465 Topics in Statistics
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Not offered academic year 2015-2016Graduate (Spring) H-Level Grad Credit Can be repeated for credit
Prereq: Permission of instructor
Units: 3-0-9
URL: http://math.mit.edu/classes/18.465
Subject Cancelled Subject Cancelled
______
Topics vary from term to term.
R. M. Dudley

18.466 Mathematical Statistics
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Graduate (Fall) H-Level Grad Credit
Prereq: Permission of instructor
Units: 3-0-9
Add to schedule Lecture: TR11.30-1 (56-154)
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Decision theory, estimation, confidence intervals, hypothesis testing. Introduces large sample theory. Asymptotic efficiency of estimates. Exponential families. Sequential analysis.
P. Kempthorne
Textbooks (Fall 2014)

18.472 Topics in Mathematics with Applications in Finance
(New)
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Undergrad (Fall)
Prereq: 18.03; 18.06; 18.05 or 18.440
Units: 3-0-9
Add to schedule Lecture: TR2.30-4 (4-237)
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Introduction to mathematical concepts and techniques used in finance. Lectures focusing on linear algebra, probability, statistics, stochastic processes, and numerical methods are interspersed with lectures by financial sector professionals illustrating the corresponding application in the industry. Prior knowledge of economics or finance helpful but not required.
P. Kempthorne, V. Strela, J. Xia
Textbooks (Fall 2014)

For additional related subjects in Statistics, see:

Civil and Environmental Engineering: 1.151, 1.155,1.202J, 1.203J, 1.205J

Electrical Engineering and Computer Science: 6.041,6.231, 6.245, 6.262, 6.431, 6.432, and 6.435

Management: 15.034, 15.061, 15.065, 15.070, 15.075, 15.076, 15.098, and 15.306

Mathematics: 18.05, 18.175, 18.176, 18.177, 18.440, 18.443, 18.445, 18.465, 18.466, and 18.472

See also: 2.061, 2.830, 5.70, 5.72, 7.02, 8.044, 8.08,10.816, 11.220, 11.221, 16.322, 17.872, 17.874, 22.38, HST.191, and MAS.622J.


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Produced: 30-JUL-2014 05:10 PM