Registrar Home | Registrar Search:
 
  MIT Course Picker | MIT Course Planner     
Home | Subject Search | Help | Symbols Help | Pre-Reg Help | Final Exam Schedule
 

Course 18: Mathematics
IAP/Spring 2024


General Mathematics

18.01 Calculus
______

Undergrad (Fall, Spring) Calculus I
Prereq: None
Units: 5-0-7
Credit cannot also be received for 18.01A, CC.1801, ES.1801, ES.181A
Lecture: TR11,F2 (2-135) Recitation: MW10 (2-135) +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: L. Guth
Spring: information: W. Minicozzi
No required or recommended textbooks

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, CC.1801, ES.1801, ES.181A
______
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 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.
T. Colding

18.02 Calculus
______

Undergrad (Fall, Spring) Calculus II
Prereq: Calculus I (GIR)
Units: 5-0-7
Credit cannot also be received for 18.022, 18.02A, CC.1802, ES.1802, ES.182A
Lecture: TR11,F2 (54-100) Recitation: MW10 (2-147) or MW11 (2-147, 2-142) or MW12 (2-142, 2-136) or MW1 (2-139) or MW2 (2-136) or MW12 (16-160) +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: Fall: S Dyatlov. Spring: D Jerison
Spring: Fall: S Dyatlov. Spring: D Jerison
No required or recommended textbooks

18.02A Calculus
______

Undergrad (Fall, IAP, Spring) Calculus II; second half of term
Prereq: Calculus I (GIR)
Units: 5-0-7
Credit cannot also be received for 18.02, 18.022, CC.1802, ES.1802, ES.182A
Attend any 18.02 recitation. Lecture: TR11,F2 (BEGINS APR 1) (54-100) Recitation: TBA (TBD) +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 second half of the Spring term; it covers the remaining material in 18.02.
Fall: W. Minicozzi
IAP: W. Minicozzi
Spring: D. Jerison
No required or recommended textbooks

18.022 Calculus
______

Undergrad (Fall) Calculus II
Prereq: Calculus I (GIR)
Units: 5-0-7
Credit cannot also be received for 18.02, 18.02A, CC.1802, ES.1802, ES.182A
______
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.
A. Landesman

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 CC.1803, ES.1803
Lecture: MWF1 (10-250) Recitation: TR9 (2-132) or TR10 (2-147, 26-328) or TR11 (2-139, 2-131, 26-328) or TR12 (2-139, 2-135, 2-131) or TR1 (4-163, 2-105) or TR2 (2-105, 24-121) or TR3 (2-139) +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: J. Dunkel
Spring: Fall: J. Dunkel. Spring: L. Demanet
No required or recommended textbooks

18.031 System Functions and the Laplace Transform
______

Undergrad (IAP)
Prereq: None. Coreq: 18.03
Units: 1-0-2 [P/D/F]
______
Studies basic continuous control theory as well as representation of functions in the complex frequency domain. Covers generalized functions, unit impulse response, and convolution; and Laplace transform, system (or transfer) function, and the pole diagram. Includes examples from mechanical and electrical engineering.
M. Sherman-Bennett
No required or recommended textbooks

18.032 Differential Equations
______

Undergrad (Spring) Rest Elec in Sci & Tech
Prereq: None. Coreq: Calculus II (GIR)
Units: 5-0-7
Lecture: MWF1 (2-142) Recitation: TR11 (2-142) +final
______
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.
A. Lawrie
No required or recommended textbooks

18.04 Complex Variables with Applications
______

Undergrad (Fall)
Prereq: Calculus II (GIR) and (18.03 or 18.032)
Units: 4-0-8
Credit cannot also be received for 18.075, 18.0751
______
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 II (GIR)
Units: 4-0-8
Lecture: TR2.30-4,F3 (32-082) +final
______
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. Bloom
No required or recommended textbooks

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 6.C06, 18.700, 18.C06
Lecture: MWF11 (26-100) Recitation: T9 (2-131) or T10 (2-131, 2-132) or T11 (2-136, 4-159) or T12 (4-159, 2-105) or T1 (2-132, 2-135) or T2 (2-132) or TR12 (24-621) or T3 (2-361) +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 linear algebra software. Compared with 18.700, more emphasis on matrix algorithms and many applications.
Fall: T. Ozuch-Meersseman
Spring: A. Borodin
Textbooks (Spring 2024)

18.C06[J] Linear Algebra and Optimization
______

Undergrad (Fall) Rest Elec in Sci & Tech
(Same subject as 6.C06[J])
Prereq: Calculus II (GIR)
Units: 5-0-7
Credit cannot also be received for 18.06, 18.700
______
Introductory course in linear algebra and optimization, assuming no prior exposure to linear algebra and starting from the basics, including vectors, matrices, eigenvalues, singular values, and least squares. Covers the basics in optimization including convex optimization, linear/quadratic programming, gradient descent, and regularization, building on insights from linear algebra. Explores a variety of applications in science and engineering, where the tools developed give powerful ways to understand complex systems and also extract structure from data.
A. Moitra, P. Parrilo

18.062[J] Mathematics for Computer Science
______

Undergrad (Fall, Spring) Rest Elec in Sci & Tech
(Same subject as 6.1200[J])
Prereq: Calculus I (GIR)
Units: 5-0-7
Lecture: TR2.30-4 (26-100) Recitation: WF10 (38-166, 36-155) or WF11 (38-166, 36-155, 26-168) or WF12 (38-166, 36-156, 26-168) or WF1 (38-166, 36-156, 35-310) or WF2 (38-166, 36-156, 35-308) or WF3 (38-166, 36-156) +final
______
Elementary discrete mathematics for science and engineering, with a focus on mathematical tools and proof techniques useful in computer science. Topics include logical notation, sets, relations, elementary graph theory, state machines and invariants, induction and proofs by contradiction, recurrences, asymptotic notation, elementary analysis of algorithms, elementary number theory and cryptography, permutations and combinations, counting tools, and discrete probability.
Fall: Z. Abel
Spring: Z. Abel
No required or recommended textbooks

18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning
______

Undergrad (Spring)
(Subject meets with 18.0651)
Prereq: 18.06
Units: 3-0-9
Lecture: TR2.30-4 (4-237)
______
Reviews linear algebra with applications to life sciences, finance, engineering, and big data. Covers singular value decomposition, weighted least squares, signal and image processing, principal component analysis, covariance and correlation matrices, directed and undirected graphs, matrix factorizations, neural nets, machine learning, and computations with large matrices.
Z. Chen
Textbooks (Spring 2024)

18.0651 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning
______

Graduate (Spring)
(Subject meets with 18.065)
Prereq: 18.06
Units: 3-0-9
Lecture: TR2.30-4 (4-237)
______
Reviews linear algebra with applications to life sciences, finance, engineering, and big data. Covers singular value decomposition, weighted least squares, signal and image processing, principal component analysis, covariance and correlation matrices, directed and undirected graphs, matrix factorizations, neural nets, machine learning, and computations with large matrices. Students in Course 18 must register for the undergraduate version, 18.065.
Staff
Textbooks (Spring 2024)

18.075 Methods for Scientists and Engineers
______

Undergrad (Spring)
(Subject meets with 18.0751)
Prereq: Calculus II (GIR) and 18.03
Units: 3-0-9
Credit cannot also be received for 18.04
Lecture: MWF2 (2-132)
______
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
No textbook information available

18.0751 Methods for Scientists and Engineers
______

Graduate (Spring)
(Subject meets with 18.075)
Prereq: Calculus II (GIR) and 18.03
Units: 3-0-9
Credit cannot also be received for 18.04
Lecture: MWF2 (2-132)
______
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. Students in Courses 6, 8, 12, 18, and 22 must register for undergraduate version, 18.075.
H. Cheng
No textbook information available

18.085 Computational Science and Engineering I
______

Undergrad (Fall, Spring, Summer)
(Subject meets with 18.0851)
Prereq: Calculus II (GIR) and (18.03 or 18.032)
Units: 3-0-9
Lecture: TR1-2.30 (4-145) +final
______
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.
Fall: D. Kouskoulas
Spring: P. Chao
Summer: Staff
Textbooks (Spring 2024)

18.0851 Computational Science and Engineering I
______

Graduate (Fall, Spring, Summer)
(Subject meets with 18.085)
Prereq: Calculus II (GIR) and (18.03 or 18.032)
Units: 3-0-9
Lecture: TR1-2.30 (4-145) +final
______
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. Students in Course 18 must register for the undergraduate version, 18.085.
Fall: D. Kouskoulas
Summer: Staff
Textbooks (Spring 2024)

18.086 Computational Science and Engineering II
______

Undergrad (Spring)
Not offered regularly; consult department
(Subject meets with 18.0861)
Prereq: Calculus II (GIR) and (18.03 or 18.032)
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.
Staff

18.0861 Computational Science and Engineering II
______

Graduate (Spring)
Not offered regularly; consult department
(Subject meets with 18.086)
Prereq: Calculus II (GIR) and (18.03 or 18.032)
Units: 3-0-9
______
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. Students in Course 18 must register for the undergraduate version, 18.086.
Staff

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.
T. Cummings

18.090 Introduction to Mathematical Reasoning
______

Undergrad (Spring) Rest Elec in Sci & Tech
Prereq: None. Coreq: Calculus II (GIR)
Units: 3-0-9
Lecture: TR9.30-11 (2-142)
______
Focuses on understanding and constructing mathematical arguments. Discusses foundational topics (such as infinite sets, quantifiers, and methods of proof) as well as selected concepts from algebra (permutations, vector spaces, fields) and analysis (sequences of real numbers). Particularly suitable for students desiring additional experience with proofs before going on to more advanced mathematics subjects or subjects in related areas with significant mathematical content.
M. King
No required or recommended textbooks

18.094[J] Teaching College-Level Science and Engineering
______

Graduate (Fall)
(Same subject as 1.95[J], 5.95[J], 7.59[J], 8.395[J])
(Subject meets with 2.978)
Prereq: None
Units: 2-0-2 [P/D/F]
______
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

18.095 Mathematics Lecture Series
______

Undergrad (IAP) Can be repeated for credit
Prereq: Calculus I (GIR)
Units: 2-0-4 [P/D/F]
______
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.
J. Dunkel
No required or recommended textbooks

18.098 Internship in Mathematics
______

Undergrad (Fall, IAP, Spring, Summer) Can be repeated for credit
Prereq: Permission of instructor
Units arranged [P/D/F]
TBA.
______
Provides academic credit for students pursuing internships to gain practical experience in the applications of mathematical concepts and methods.
Fall: T. Cummings
IAP: T. Cummings
Spring: T. Cummings
Summer: T. Cummings
No required or recommended textbooks

18.099 Independent Study
______

Undergrad (Fall, IAP, Spring, Summer) Can be repeated for credit
Prereq: Permission of instructor
Units arranged
TBA.
______
Studies (during IAP) or special individual reading (during regular terms). Arranged in consultation with individual faculty members and subject to departmental approval.  May not be used to satisfy Mathematics major requirements.
Fall: T. Cummings
IAP: T. Cummings
Spring: T. Cummings
Summer: T. Cummings
No required or recommended textbooks

Analysis

18.1001 Real Analysis
______

Graduate (Fall, Spring)
(Subject meets with 18.100A)
Prereq: Calculus II (GIR)
Units: 3-0-9
Credit cannot also be received for 18.1002, 18.100A, 18.100B, 18.100P, 18.100Q
Lecture: MW9.30-11 (4-163) +final
______
Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. Proofs and definitions are less abstract than in 18.100B. Gives applications where possible. Concerned primarily with the real line. Students in Course 18 must register for undergraduate version 18.100A.
Fall: Q. Deng
Spring: J. Zhu
No required or recommended textbooks

18.1002 Real Analysis
______

Graduate (Fall, Spring)
(Subject meets with 18.100B)
Prereq: Calculus II (GIR)
Units: 3-0-9
Credit cannot also be received for 18.1001, 18.100A, 18.100B, 18.100P, 18.100Q
Lecture: TR1-2.30 (2-190) +final
______
Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. More demanding than 18.100A, for students with more mathematical maturity. Places more emphasis on point-set topology and n-space. Students in Course 18 must register for undergraduate version 18.100B.
Fall: R. Melrose
Spring: G. Franz
Textbooks (Spring 2024)

18.100A Real Analysis
______

Undergrad (Fall, Spring)
(Subject meets with 18.1001)
Prereq: Calculus II (GIR)
Units: 3-0-9
Credit cannot also be received for 18.1001, 18.1002, 18.100B, 18.100P, 18.100Q
Lecture: MW9.30-11 (4-163) +final
______
Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. Proofs and definitions are less abstract than in 18.100B. Gives applications where possible. Concerned primarily with the real line.
Fall: Q. Deng
Spring: Fall: Q. Deng. Spring: J. Zhu
No required or recommended textbooks

18.100B Real Analysis
______

Undergrad (Fall, Spring)
(Subject meets with 18.1002)
Prereq: Calculus II (GIR)
Units: 3-0-9
Credit cannot also be received for 18.1001, 18.1002, 18.100A, 18.100P, 18.100Q
Lecture: TR1-2.30 (2-190) +final
______
Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. More demanding than 18.100A, for students with more mathematical maturity. Places more emphasis on point-set topology and n-space.
Fall: R. Melrose
Spring: Fall: R. Melrose. Spring: G. Franz
Textbooks (Spring 2024)

18.100P Real Analysis
______

Undergrad (Spring)
Prereq: Calculus II (GIR)
Units: 4-0-11
Credit cannot also be received for 18.1001, 18.1002, 18.100A, 18.100B, 18.100Q
Lecture: TR9.30-11 (LIMITED 30) (2-143) Recitation: F11 (2-151) or F2 (2-151)
______
Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. Proofs and definitions are less abstract than in 18.100B. Gives applications where possible. Concerned primarily with the real line. Includes instruction and practice in written communication. Enrollment limited.
K. Naff
No required or recommended textbooks

18.100Q Real Analysis
______

Undergrad (Fall)
Prereq: Calculus II (GIR)
Units: 4-0-11
Credit cannot also be received for 18.1001, 18.1002, 18.100A, 18.100B, 18.100P
______
Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs. More demanding than 18.100A, for students with more mathematical maturity. Places more emphasis on point-set topology and n-space. Includes instruction and practice in written communication. Enrollment limited.
C. Oh

18.101 Analysis and Manifolds
______

Undergrad (Fall)
(Subject meets with 18.1011)
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
Units: 3-0-9
______
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.
M. Jezequel

18.1011 Analysis and Manifolds
______

Graduate (Fall)
(Subject meets with 18.101)
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
Units: 3-0-9
______
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. 18.9011 helpful but not required. Students in Course 18 must register for the undergraduate version, 18.101.
M. Jezequel

18.102 Introduction to Functional Analysis
______

Undergrad (Spring)
(Subject meets with 18.1021)
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
Units: 3-0-9
Lecture: MW9.30-11 (4-237) +final
______
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.
M. Jezequel
Textbooks (Spring 2024)

18.1021 Introduction to Functional Analysis
______

Graduate (Spring)
(Subject meets with 18.102)
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
Units: 3-0-9
Lecture: MW9.30-11 (4-237) +final
______
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. Students in Course 18 must register for the undergraduate version, 18.102.
M. Jezequel
Textbooks (Spring 2024)

18.103 Fourier Analysis: Theory and Applications
______

Undergrad (Fall)
(Subject meets with 18.1031)
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
Units: 3-0-9
______
Roughly half the subject devoted to the theory of the Lebesgue integral with applications to probability, and half to Fourier series and Fourier integrals.
J. Shi

18.1031 Fourier Analysis: Theory and Applications
______

Graduate (Fall)
(Subject meets with 18.103)
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
Units: 3-0-9
______
Roughly half the subject devoted to the theory of the Lebesgue integral with applications to probability, and half to Fourier series and Fourier integrals. Students in Course 18 must register for the undergraduate version, 18.103.
J. Shi

18.104 Seminar in Analysis
______

Undergrad (Fall, Spring)
Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q
Units: 3-0-9
Lecture: MW1-2.30 (2-146)
______
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.
Fall: T. Mrowka
Spring: Q. Deng
No required or recommended textbooks

18.112 Functions of a Complex Variable
______

Undergrad (Fall)
(Subject meets with 18.1121)
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
Units: 3-0-9
______
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.
A. Lawrie

18.1121 Functions of a Complex Variable
______

Graduate (Fall)
(Subject meets with 18.112)
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
Units: 3-0-9
______
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. Students in Course 18 must register for the undergraduate version, 18.112.
A. Lawrie

18.116 Riemann Surfaces
______

Not offered academic year 2023-2024Graduate (Fall)
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.
Staff

18.117 Topics in Several Complex Variables
______

Not offered academic year 2023-2024Graduate (Spring) Can be repeated for credit
Prereq: 18.112 and 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.
B. Poonen

18.118 Topics in Analysis
______

Not offered academic year 2024-2025Graduate (Spring) Can be repeated for credit
Prereq: Permission of instructor
Units: 3-0-9
Lecture: TR2.30-4 (2-147)
______
Topics vary from year to year.
D. Stroock
No required or recommended textbooks

18.125 Measure Theory and Analysis
______

Not offered academic year 2023-2024Graduate (Spring)
Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q
Units: 3-0-9
______
Provides a rigorous introduction to Lebesgue's theory of measure and integration. Covers material that is essential in analysis, probability theory, and differential geometry.
Staff

18.137 Topics in Geometric Partial Differential Equations
______

Not offered academic year 2023-2024Graduate (Fall) Can be repeated for credit
Prereq: Permission of instructor
Units: 3-0-9
______
Topics vary from year to year.
T. Colding

18.152 Introduction to Partial Differential Equations
______

Undergrad (Spring)
(Subject meets with 18.1521)
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
Units: 3-0-9
Lecture: MW11-12.30 (2-135) +final
______
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.
T. Chow
Textbooks (Spring 2024)

18.1521 Introduction to Partial Differential Equations
______

Graduate (Spring)
(Subject meets with 18.152)
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
Units: 3-0-9
Lecture: MW11-12.30 (2-135) +final
______
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. Students in Course 18 must register for the undergraduate version, 18.152.
Staff
Textbooks (Spring 2024)

18.155 Differential Analysis I
______

Graduate (Fall)
Prereq: 18.102 or 18.103
Units: 3-0-9
______
First part of a two-subject sequence. 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. Recommended prerequisite: 18.112.
D. Jerison

18.156 Differential Analysis II
______

Graduate (Spring)
Prereq: 18.155
Units: 3-0-9
Lecture: TR1-2.30 (2-142)
______
Second part of a two-subject sequence. Covers variable coefficient elliptic, parabolic and hyperbolic partial differential equations.
L. Guth
No required or recommended textbooks

18.157 Introduction to Microlocal Analysis
______

Not offered academic year 2023-2024Graduate (Spring)
Prereq: 18.155
Units: 3-0-9
______
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 2024-2025Graduate (Spring) Can be repeated for credit
Prereq: 18.157
Units: 3-0-9
Lecture: MW9.30-11 (2-146)
______
Topics vary from year to year.
G. Staffilani
No required or recommended textbooks

18.199 Graduate Analysis Seminar
______

Not offered academic year 2024-2025Graduate (Spring) Can be repeated for credit
Prereq: Permission of instructor
Units: 3-0-9
Lecture: TR11-12.30 (2-146)
______
Studies original papers in differential analysis and differential equations. Intended for first- and second-year graduate students. Permission must be secured in advance.
R. Melrose
No required or recommended textbooks

Discrete Applied Mathematics

18.200 Principles of Discrete Applied Mathematics
______

Undergrad (Spring)
Prereq: None. Coreq: 18.06
Units: 4-0-11
Credit cannot also be received for 18.200A
Lecture: TR11-12.30 (LIMITED 75) (2-190) Recitation: W10 (2-143) or W1 (2-131) or W3 (2-131)
______
Study of illustrative topics in discrete applied mathematics, including probability theory, information theory, coding theory, secret codes, generating functions, and linear programming. Instruction and practice in written communication provided. Enrollment limited.
P. W. Shor, A. Moitra
No required or recommended textbooks

18.200A Principles of Discrete Applied Mathematics
______

Not offered academic year 2023-2024Undergrad (Fall)
Prereq: None. Coreq: 18.06
Units: 3-0-9
Credit cannot also be received for 18.200
______
Study of illustrative topics in discrete applied mathematics, including probability theory, information theory, coding theory, secret codes, generating functions, and linear programming.
Staff

18.204 Undergraduate Seminar in Discrete Mathematics
______

Undergrad (Fall, Spring)
Prereq: ((6.1200 or 18.200) and (18.06, 18.700, or 18.701)) or permission of instructor
Units: 3-0-9
Lecture: TR1-2.30 (LIMITED 15 EACH S .. (2-151) or TR1-2.30 (2-146, 2-147)
______
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: D. Mikulincer, M. Sherman-Bennet, F. Tom
Spring: M. Dhar, J. He, S. Luo
No required or recommended textbooks

18.211 Combinatorial Analysis
______

Undergrad (Fall)
Prereq: Calculus II (GIR) and (18.06, 18.700, or 18.701)
Units: 3-0-9
______
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.
M. Simkin

18.212 Algebraic Combinatorics
______

Undergrad (Spring)
Prereq: 18.701 or 18.703
Units: 3-0-9
Lecture: MWF1 (4-237)
______
Applications of algebra to combinatorics. Topics include walks in graphs, the Radon transform, groups acting on posets, Young tableaux, electrical networks.
A. Postnikov
No required or recommended textbooks

18.217 Combinatorial Theory
______

Graduate (Fall) Can be repeated for credit
Prereq: Permission of instructor
Units: 3-0-9
______
Content varies from year to year.
A. Postnikov

18.218 Topics in Combinatorics
______

Graduate (Spring) Can be repeated for credit
Prereq: Permission of instructor
Units: 3-0-9
Lecture: TR11-12.30 (66-168)
______
Topics vary from year to year.
D. Minzer
No required or recommended textbooks

18.219 Seminar in Combinatorics
______

Graduate (Fall) Can be repeated for credit
Not offered regularly; consult department
Prereq: Permission of instructor
Units: 3-0-9
______
Content varies from year to year. Readings from current research papers in combinatorics. Topics to be chosen and presented by the class.
Staff

18.225 Graph Theory and Additive Combinatorics
______

Not offered academic year 2024-2025Graduate (Fall)
Prereq: ((18.701 or 18.703) and (18.100A, 18.100B, 18.100P, or 18.100Q)) or permission of instructor
Units: 3-0-9
______
Introduction to extremal graph theory and additive combinatorics. Highlights common themes, such as the dichotomy between structure versus pseudorandomness. Topics include Turan-type problems, Szemeredi's regularity lemma and applications, pseudorandom graphs, spectral graph theory, graph limits, arithmetic progressions (Roth, Szemeredi, Green-Tao), discrete Fourier analysis, Freiman's theorem on sumsets and structure. Discusses current research topics and open problems.
Y. Zhao

18.226 Probabilistic Methods in Combinatorics
______

Not offered academic year 2023-2024Graduate (Fall)
Prereq: (18.211, 18.600, and (18.100A, 18.100B, 18.100P, or 18.100Q)) or permission of instructor
Units: 3-0-9
______
Introduction to the probabilistic method, a fundamental and powerful technique in combinatorics and theoretical computer science. Focuses on methodology as well as combinatorial applications. Suitable for students with strong interest and background in mathematical problem solving. Topics include linearity of expectations, alteration, second moment, Lovasz local lemma, correlation inequalities, Janson inequalities, concentration inequalities, entropy method.
Y. Zhao

Continuous Applied Mathematics

18.300 Principles of Continuum Applied Mathematics
______

Undergrad (Fall)
Prereq: Calculus II (GIR) and (18.03 or 18.032)
Units: 3-0-9
______
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.
L. Demanet

18.303 Linear Partial Differential Equations: Analysis and Numerics
______

Undergrad (Fall)
Prereq: 18.06 or 18.700
Units: 3-0-9
______
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. Some programming required for homework and final project.
A. Horning

18.305 Advanced Analytic Methods in Science and Engineering
______

Not offered academic year 2023-2024Graduate (Fall)
Prereq: 18.04, 18.075, or 18.112
Units: 3-0-9
______
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.
Staff

18.306 Advanced Partial Differential Equations with Applications
______

Not offered academic year 2024-2025Graduate (Spring)
Prereq: (18.03 or 18.032) and (18.04, 18.075, or 18.112)
Units: 3-0-9
Lecture: TR11-12.30 (2-151)
______
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.
D. Kouskoulas
Textbooks (Spring 2024)

18.327 Topics in Applied Mathematics
______

Not offered academic year 2023-2024Graduate (Fall) Can be repeated for credit
Prereq: Permission of instructor
Units: 3-0-9
______
Topics vary from year to year.
L. Demanet

18.330 Introduction to Numerical Analysis
______

Undergrad (Fall)
Prereq: Calculus II (GIR) and (18.03 or 18.032)
Units: 3-0-9
______
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 a language such as MATLAB, Python, or Julia is helpful.
J. Urschel

18.335[J] Introduction to Numerical Methods
______

Graduate (Spring)
(Same subject as 6.7310[J])
Prereq: 18.06, 18.700, or 18.701
Units: 3-0-9
Lecture: MW11-12.30 (45-230)
______
Advanced introduction to numerical analysis: accuracy and efficiency of numerical algorithms. In-depth coverage of sparse-matrix/iterative and dense-matrix algorithms in numerical linear algebra (for linear systems and eigenproblems). Floating-point arithmetic, backwards error analysis, conditioning, and stability. Other computational topics (e.g., numerical integration or nonlinear optimization) may also be surveyed. Final project involves some programming.
J. Urschel
Textbooks (Spring 2024)

18.336[J] Fast Methods for Partial Differential and Integral Equations
______

Graduate (Fall, Spring)
(Same subject as 6.7340[J])
Prereq: 6.7300, 16.920, 18.085, 18.335, or permission of instructor
Units: 3-0-9
Lecture: MW9.30-11 (2-136)
______
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.
Fall: A. Horning
Spring: A. Horning
No required or recommended textbooks

18.337[J] Parallel Computing and Scientific Machine Learning
______

Not offered academic year 2023-2024Graduate (Spring)
(Same subject as 6.7320[J])
Prereq: 18.06, 18.700, or 18.701
Units: 3-0-9
______
Introduction to scientific machine learning with an emphasis on developing scalable differentiable programs. Covers scientific computing topics (numerical differential equations, dense and sparse linear algebra, Fourier transformations, parallelization of large-scale scientific simulation) simultaneously with modern data science (machine learning, deep neural networks, automatic differentiation), focusing on the emerging techniques at the connection between these areas, such as neural differential equations and physics-informed deep learning. Provides direct experience with the modern realities of optimizing code performance for supercomputers, GPUs, and multicores in a high-level language.
A. Edelman

18.338 Eigenvalues of Random Matrices
______

Graduate (Fall)
Prereq: 18.701 or permission of instructor
Units: 3-0-9
______
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 Julia helpful, but not required.
A. Edelman

18.352[J] Nonlinear Dynamics: The Natural Environment
______

Undergrad (Fall)
Not offered regularly; consult department
(Same subject as 12.009[J])
Prereq: Calculus II (GIR) and Physics I (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

18.353[J] Nonlinear Dynamics: Chaos
______

Undergrad (Fall)
(Same subject as 2.050[J], 12.006[J])
Prereq: Physics II (GIR) and (18.03 or 18.032)
Units: 3-0-9
______
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. Rosales

18.354[J] Nonlinear Dynamics: Continuum Systems
______

Undergrad (Spring)
(Same subject as 1.062[J], 12.207[J])
(Subject meets with 18.3541)
Prereq: Physics II (GIR) and (18.03 or 18.032)
Units: 3-0-9
Lecture: TR9.30-11 (2-135)
______
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.
B. Primkulov
No required or recommended textbooks

18.3541 Nonlinear Dynamics: Continuum Systems
______

Graduate (Spring)
(Subject meets with 1.062[J], 12.207[J], 18.354[J])
Prereq: Physics II (GIR) and (18.03 or 18.032)
Units: 3-0-9
Lecture: TR9.30-11 (2-135)
______
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. Students in Courses 1, 12, and 18 must register for undergraduate version, 18.354.
B. Primkulov
No required or recommended textbooks

18.355 Fluid Mechanics
______

Not offered academic year 2023-2024Graduate (Spring)
Prereq: 2.25, 12.800, or 18.354
Units: 3-0-9
______
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.
Staff

18.357 Interfacial Phenomena
______

Not offered academic year 2023-2024Graduate (Fall)
Prereq: 2.25, 12.800, 18.354, 18.355, 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

18.358[J] Nonlinear Dynamics and Turbulence
______

Not offered academic year 2023-2024Graduate (Spring)
(Same subject as 1.686[J], 2.033[J])
(Subject meets with 1.068)
Prereq: 1.060A
Units: 3-2-7
______
Reviews theoretical notions of nonlinear dynamics, instabilities, and waves with applications in fluid dynamics. Discusses hydrodynamic instabilities leading to flow destabilization and transition to turbulence. Focuses on physical turbulence and mixing from homogeneous isotropic turbulence. Also covers topics such as rotating and stratified flows as they arise in the environment, wave-turbulence, and point source turbulent flows. Laboratory activities integrate theoretical concepts covered in lectures and problem sets. Students taking graduate version complete additional assignments.
L. Bourouiba

18.367 Waves and Imaging
______

Not offered academic year 2023-2024Graduate (Fall)
Prereq: Permission of instructor
Units: 3-0-9
______
The mathematics of inverse problems involving waves, with examples taken from reflection seismology, synthetic aperture radar, and computerized tomography. Suitable for graduate students from all departments who have affinities with applied mathematics. Topics include acoustic, elastic, electromagnetic wave equations; geometrical optics; scattering series and inversion; migration and backprojection; adjoint-state methods; Radon and curvilinear Radon transforms; microlocal analysis of imaging; optimization, regularization, and sparse regression.
Staff

18.369[J] Mathematical Methods in Nanophotonics
______

Not offered academic year 2024-2025Graduate (Spring)
(Same subject as 8.315[J])
Prereq: 8.07, 18.303, or permission of instructor
Units: 3-0-9
Lecture: MWF2 (2-131)
______
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
Textbooks (Spring 2024)

18.376[J] Wave Propagation
______

Graduate (Spring)
Not offered regularly; consult department
(Same subject as 1.138[J], 2.062[J])
Prereq: 2.003 and 18.075
Units: 3-0-9
Subject Cancelled Subject Cancelled
______
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

18.377[J] Nonlinear Dynamics and Waves
______

Not offered academic year 2024-2025Graduate (Spring)
(Same subject as 1.685[J], 2.034[J])
Prereq: Permission of instructor
Units: 3-0-9
Lecture: TR2.30-4 (3-333)
______
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. Akylas
No required or recommended textbooks

18.384 Undergraduate Seminar in Physical Mathematics
______

Undergrad (Fall)
Prereq: 12.006, 18.300, 18.354, or permission of instructor
Units: 3-0-9
______
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.
N. Derr

18.385[J] Nonlinear Dynamics and Chaos
______

Not offered academic year 2024-2025Graduate (Spring)
(Same subject as 2.036[J])
Prereq: 18.03 or 18.032
Units: 3-0-9
Lecture: MW1-2.30 (2-151)
______
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.
J. Dunkel
Textbooks (Spring 2024)

18.397 Mathematical Methods in Physics
______

Graduate (Fall) Can be repeated for credit
Not offered regularly; consult department
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.
Staff

Theoretical Computer Science

18.400[J] Computability and Complexity Theory
______

Undergrad (Spring)
(Same subject as 6.1400[J])
Prereq: (6.1200 and 6.1210) or permission of instructor
Units: 4-0-8
Lecture: TR2.30-4 (34-304) Recitation: F11 (4-257) or F1 (24-121)
______
Mathematical introduction to the theory of computing. Rigorously explores what kinds of tasks can be efficiently solved with computers by way of finite automata, circuits, Turing machines, and communication complexity, introducing students to some major open problems in mathematics. Builds skills in classifying computational tasks in terms of their difficulty. Discusses other fundamental issues in computing, including the Halting Problem, the Church-Turing Thesis, the P versus NP problem, and the power of randomness.  
D. Minzer
No textbook information available

18.404 Theory of Computation
______

Undergrad (Fall)
(Subject meets with 6.5400[J], 18.4041[J])
Prereq: 6.1200 or 18.200
Units: 4-0-8
______
A more extensive and theoretical treatment of the material in 6.1400J/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

18.4041[J] Theory of Computation
______

Graduate (Fall)
(Same subject as 6.5400[J])
(Subject meets with 18.404)
Prereq: 6.1200 or 18.200
Units: 4-0-8
______
A more extensive and theoretical treatment of the material in 6.1400J/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. Students in Course 18 must register for the undergraduate version, 18.404.
M. Sipser

18.405[J] Advanced Complexity Theory
______

Not offered academic year 2024-2025Graduate (Spring)
(Same subject as 6.5410[J])
Prereq: 18.404
Units: 3-0-9
Lecture: TR2.30-4 (35-225)
______
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.
R. Williams
Textbooks (Spring 2024)

18.408 Topics in Theoretical Computer Science
______

Not offered academic year 2023-2024Graduate (Fall, Spring) Can be repeated for credit
Prereq: Permission of instructor
Units: 3-0-9
______
Study of areas of current interest in theoretical computer science. Topics vary from term to term.
Staff

18.410[J] Design and Analysis of Algorithms
______

Undergrad (Fall, Spring)
(Same subject as 6.1220[J])
Prereq: 6.1200 and 6.1210
Units: 4-0-8
Lecture: TR11-12.30 (32-123) Recitation: F9 (36-112) or F10 (36-112) or F11 (36-112) or F12 (36-112) or F1 (36-112) or F2 (36-112) or F3 (36-112) or F10 (24-121) or F11 (4-265) or F12 (4-265) or F1 (4-265) or F2 (24-121) or F3 (24-121) +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.
Fall: S. Raghuraman
Spring: S. Raghuraman
Textbooks (Spring 2024)

18.413 Introduction to Computational Molecular Biology
______

Undergrad (Spring)
Not offered regularly; consult department
(Subject meets with 18.417)
Prereq: 6.1210 or permission of instructor
Units: 3-0-9
______
Introduction to computational molecular biology with a focus on the basic computational algorithms used to solve problems in practice. Covers classical techniques in the field for solving problems such as genome sequencing, assembly, and search; detecting genome rearrangements; constructing evolutionary trees; analyzing mass spectrometry data; connecting gene expression to cellular function; and machine learning for drug discovery. Prior knowledge of biology is not required. Particular emphasis on problem solving, collaborative learning, theoretical analysis, and practical implementation of algorithms. Students taking graduate version complete additional and more complex assignments.
B. Berger

18.415[J] Advanced Algorithms
______

Not offered academic year 2023-2024Graduate (Fall)
(Same subject as 6.5210[J])
Prereq: 6.1220 and (6.1200, 6.3700, or 18.600)
Units: 5-0-7
______
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.
Staff

18.416[J] Randomized Algorithms
______

Not offered academic year 2023-2024Graduate (Spring)
(Same subject as 6.5220[J])
Prereq: (6.1200 or 6.3700) and (6.1220 or 6.5210)
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 2023-2024Graduate (Spring)
(Subject meets with 18.413)
Prereq: 6.1210 or permission of instructor
Units: 3-0-9
______
Introduction to computational molecular biology with a focus on the basic computational algorithms used to solve problems in practice. Covers classical techniques in the field for solving problems such as genome sequencing, assembly, and search; detecting genome rearrangements; constructing evolutionary trees; analyzing mass spectrometry data; connecting gene expression to cellular function; and machine learning for drug discovery. Prior knowledge of biology is not required. Particular emphasis on problem solving, collaborative learning, theoretical analysis, and practical implementation of algorithms. Students taking graduate version complete additional and more complex assignments.
B. Berger

18.418[J] Topics in Computational Molecular Biology
______

Graduate (Fall) Can be repeated for credit
(Same subject as HST.504[J])
Prereq: 6.8701, 18.417, or permission of instructor
Units: 3-0-9
______
Covers current research topics in computational molecular biology. Recent research papers presented from leading conferences such as the International Conference on Computational Molecular Biology (RECOMB) and the Conference on Intelligent Systems for Molecular Biology (ISMB). Topics include original research (both theoretical and experimental) in comparative genomics, sequence and structure analysis, molecular evolution, proteomics, gene expression, transcriptional regulation, biological networks, drug discovery, and privacy. Recent research by course participants also covered. Participants will be expected to present individual projects to the class.
B. Berger

18.424 Seminar in Information Theory
______

Undergrad (Fall)
Prereq: (6.3700, 18.05, or 18.600) and (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. Shor

18.425[J] Cryptography and Cryptanalysis
______

Graduate (Fall)
(Same subject as 6.5620[J])
Prereq: 6.1220
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.
V. Vaikuntanathan

18.434 Seminar in Theoretical Computer Science
______

Undergrad (Fall, Spring)
Prereq: 6.1220
Units: 3-0-9
Lecture: MW9.30-11 (2-151)
______
Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.
Fall: Y. Sohn
Spring: A. Sridhar
No required or recommended textbooks

18.435[J] Quantum Computation
______

Graduate (Fall)
(Same subject as 2.111[J], 6.6410[J], 8.370[J])
Prereq: 8.05, 18.06, 18.700, 18.701, or 18.C06
Units: 3-0-9
______
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.
A. Harrow

18.436[J] Quantum Information Science
______

Graduate (Spring)
(Same subject as 6.6420[J], 8.371[J])
Prereq: 18.435
Units: 3-0-9
Lecture: MW9.30-11 (37-212)
______
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.
A. Harrow
Textbooks (Spring 2024)

18.437[J] Distributed Algorithms
______

Not offered academic year 2023-2024Graduate (Fall)
(Same subject as 6.5250[J])
Prereq: 6.1220
Units: 3-0-9
______
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.
M. Ghaffari, N. A. Lynch

18.453 Combinatorial Optimization
______

Not offered academic year 2023-2024Undergrad (Spring)
(Subject meets with 18.4531)
Prereq: 18.06, 18.700, or 18.701
Units: 3-0-9
______
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 18.200) helpful.
Staff

18.4531 Combinatorial Optimization
______

Not offered academic year 2023-2024Graduate (Spring)
(Subject meets with 18.453)
Prereq: 18.06, 18.700, or 18.701
Units: 3-0-9
______
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 18.200) helpful. Students in Course 18 must register for the undergraduate version, 18.453.
Staff

18.455 Advanced Combinatorial Optimization
______

Not offered academic year 2023-2024Graduate (Spring)
Prereq: 18.453 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

18.456[J] Algebraic Techniques and Semidefinite Optimization
______

Not offered academic year 2024-2025Graduate (Spring)
(Same subject as 6.7230[J])
Prereq: 6.7210 or 15.093
Units: 3-0-9
Lecture: WF1-2.30 (36-153)
______
Theory and computational techniques for optimization problems involving polynomial equations and inequalities with particular, emphasis on the connections with semidefinite optimization. Develops algebraic and numerical approaches of general applicability, with a view towards methods that simultaneously incorporate both elements, stressing convexity-based ideas, complexity results, and efficient implementations. Examples from several engineering areas, in particular systems and control applications. Topics include semidefinite programming, resultants/discriminants, hyperbolic polynomials, Groebner bases, quantifier elimination, and sum of squares.
P. Parrilo
No textbook information available


left arrow | 18.01-18.499 | 18.50-18.THG | right arrow



Produced: 18-APR-2024 05:10 PM