General Mathematics
18.01 Calculus
 ( ,  ) 
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: P. Seidel
Spring: T. Rud
No textbook information available
18.01A Calculus
() ; 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.
D. Jerison
18.02 Calculus
(, ) 
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 (2-146) or MW11 (2-146) or MW12 (2-136) or MW1 (2-146) or MW2 (2-146) +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: D. Maulik
Spring: M. King
Textbooks (Spring 2025)
18.02A Calculus
(,  , ) ; 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
IAP:  Lecture: MTWRF1 (54-100) Recitation: TR10-11.30 (2-135, 2-139, 2-143) or TR2-3.30 (2-135, 2-139, 2-143) +final
Spring: Attend any 18.02 recitation. Lecture: TR11,F2 (BEGINS MAR 31) (54-100) Recitation: TBA +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: J. Bush
IAP: J. Bush
Spring: M. King
Textbooks (IAP 2025); Textbooks (Spring 2025)
18.022 Calculus
()
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.
J. Zung
18.03 Differential Equations
(, ) 
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, 2-131) 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: H. Cohn
Spring: B. Poonen
No textbook information available
18.031 System Functions and the Laplace Transform
()
Prereq: None. Coreq: 18.03
Units: 1-0-2 [P/D/F]
Begins Jan 13. Lecture: MWF11-1 (2-146)
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.
S. Nadimpalli
No required or recommended textbooks
18.032 Differential Equations
()
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 textbook information available
18.04 Complex Variables with Applications
()
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.
J. Dunkel
18.05 Introduction to Probability and Statistics
()
Prereq: Calculus II (GIR)
Units: 4-0-8
Lecture: TR2.30-4,F4 (32-082) or TR1-2.30,F2 (32-082) +final
A unified introduction to probability, Bayesian inference, and frequentist statistics. Topics include: combinatorics, random variables, (joint) distributions, covariance, central limit theorem; Bayesian updating, odds, posterior prediction; significance tests, confidence intervals, bootstrapping, regression. Students also develop computational skills and statistical thinking by using R to simulate, analyze, and visualize data; and by exploring privacy, fairness, and causality in contemporary media and research. Flipped subject taught in a Technology Enabled Active Learning (TEAL) classroom to facilitate discussion, group problem solving, and coding studios with ample mentorship.
J. Bloom
No textbook information available
18.06 Linear Algebra
(, )
Prereq: Calculus II (GIR)
Units: 4-0-8
Credit cannot also be received for 6.C06, 18.700, 18.C06, ES.1806
Lecture: MWF10 (26-100) Recitation: T9 (2-131) or T11 (2-136, 4-159) or T12 (4-159, 2-105) or T1 (2-132, 2-135) or T2 (2-132) or T3 (2-361) or T10 (2-131, 2-132) +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: W. Minicozzi
Spring: N. Sun
Textbooks (Spring 2025)
18.C06[J] Linear Algebra and Optimization
()
(Same subject as 6.C06[J])
Prereq: Calculus II (GIR)
Units: 5-0-7
Credit cannot also be received for 18.06, 18.700, ES.1806
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.
S. Johnson
18.062[J] Mathematics for Computer Science
(, )
(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 textbook information available
18.063 Matrix Calculus for Machine Learning and Beyond
(New)
(); partial term
Prereq: Calculus II (GIR) and 18.06
Units: 1-0-2 [P/D/F]
Begins Jan 13. Lecture: MWF11-1 (2-131)
Covers a coherent approach to matrix calculus, showing techniques that allow the student to think of a matrix holistically, rather than as an array of scalars; generalize and compute derivatives of important matrix factorizations and other complicated-looking operations; and understand how differentiation formulas must be reimagined in large-scale computing. Discusses "adjoint" or "reverse-mode" differentiation, custom vector-Jacobian products, and how modern automatic differentiation is more computer science than calculus.
S. Johnson
No textbook information available
18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning
()
(Subject meets with 18.0651)
Prereq: 18.06
Units: 3-0-9
Lecture: TR9.30-11 (32-141)
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 2025)
18.0651 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning
 ()
(Subject meets with 18.065)
Prereq: 18.06
Units: 3-0-9
Lecture: TR9.30-11 (32-141)
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
No textbook information available
18.075 Methods for Scientists and Engineers
 ()
(Subject meets with 18.0751)
Prereq: Calculus II (GIR) and 18.03
Units: 3-0-9
Credit cannot also be received for 18.04
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
18.0751 Methods for Scientists and Engineers
()
(Subject meets with 18.075)
Prereq: Calculus II (GIR) and 18.03
Units: 3-0-9
Credit cannot also be received for 18.04
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
18.085 Computational Science and Engineering I
(, ,  )
(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: P. Chao
Spring: P. Chao
Summer: Staff
No textbook information available
18.0851 Computational Science and Engineering I
(, , )
(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
No textbook information available
18.086 Computational Science and Engineering II
()
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
()
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
()
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.
Staff
18.090 Introduction to Mathematical Reasoning
()
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.
S. Dyatlov
No textbook information available
18.091 Introduction to Metric Spaces
(New)
(); partial term
Prereq: 18.100A
Units: 1-0-2 [P/D/F]
Begins Jan 13. Lecture: MWF10 (2-146)
Covers metrics, open and closed sets, continuous functions (from a topological perspective), function spaces, completeness, and compactness. Aims to provide more complex concepts and proofs for students who have taken 18.100A as their real analysis subject.
S. Nadimpalli
No required or recommended textbooks
18.094[J] Teaching College-Level Science and Engineering
()
(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
() 
Prereq: Calculus I (GIR)
Units: 2-0-4 [P/D/F]
Lecture: MWF1-2.30 (2-190) Recitation: R10.30-12 (2-147) or R1-2.30 (2-147)
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 textbook information available
18.098 Internship in Mathematics
(, , , )
Prereq: Permission of instructor
Units arranged [P/D/F]
IAP: TBA.
Spring: 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
No required or recommended textbooks
18.099 Independent Study
(, , , )
Prereq: Permission of instructor
Units arranged
IAP: TBA.
Spring: 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
No required or recommended textbooks
Analysis
18.1001 Real Analysis
(, )
(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: TR1-2.30 (4-270) +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 textbook information available
18.1002 Real Analysis
(, )
(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: TR9.30-11 (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
No textbook information available
18.100A Real Analysis
(, )
(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: TR1-2.30 (4-270) +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: T. Chow
Spring: J. Zhu
No required or recommended textbooks
18.100B Real Analysis
(, )
(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: TR9.30-11 (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: L. Guth
Spring: T. Colding
Textbooks (Spring 2025)
18.100P Real Analysis
()
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: MW9.30-11 (LIMITED 30) (2-131) Recitation: F11 (2-151) or F1 (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.
R. Maguire
No required or recommended textbooks
18.100Q Real Analysis
()
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.
M. Lipton
18.101 Analysis and Manifolds
()
(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.
R. Jiang
18.1011 Analysis and Manifolds
()
(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.
Staff
18.102 Introduction to Functional Analysis
()
(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.
C. Kehle
No textbook information available
18.1021 Introduction to Functional Analysis
()
(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.
Staff
No textbook information available
18.103 Fourier Analysis: Theory and Applications
()
(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.
Staff
18.1031 Fourier Analysis: Theory and Applications
()
(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.
Staff
18.104 Seminar in Analysis
(, )
Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q
Units: 3-0-9
Lecture: MW11-12.30 (2-151)
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: L. Alon
Spring: T. Ozuch-Meersseman
No required or recommended textbooks
18.112 Functions of a Complex Variable
()
(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.
J. Shi
18.1121 Functions of a Complex Variable
()
(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.
Staff
18.116 Riemann Surfaces
()
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
()
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.
Staff
18.118 Topics in Analysis
()
Prereq: Permission of instructor
Units: 3-0-9
Topics vary from year to year.
Staff
18.125 Measure Theory and Analysis
()
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
 ()
Prereq: Permission of instructor
Units: 3-0-9
Topics vary from year to year.
T. Colding
18.152 Introduction to Partial Differential Equations
()
(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
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.
F. Hernandez
18.1521 Introduction to Partial Differential Equations
()
(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
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
18.155 Differential Analysis I
()
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.
A. Logunov
18.156 Differential Analysis II
()
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 textbook information available
18.157 Introduction to Microlocal Analysis
()
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.
Staff
18.158 Topics in Differential Equations
()
Prereq: 18.157
Units: 3-0-9
Lecture: MW1-2.30 (2-147)
Topics vary from year to year.
A. Logunov
No textbook information available
18.199 Graduate Analysis Seminar
()
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.
Staff
Discrete Applied Mathematics
18.200 Principles of Discrete Applied Mathematics
()
Prereq: None. Coreq: 18.06
Units: 4-0-11
Credit cannot also be received for 18.200A
Limited 45. Lecture: MWF2 (2-190) Recitation: R10 (2-146) or R1 (2-146) or R2 (2-146)
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. Shor
No textbook information available
18.200A Principles of Discrete Applied Mathematics
()
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
(, )
Prereq: ((6.1200 or 18.200) and (18.06, 18.700, or 18.701)) or permission of instructor
Units: 3-0-9
Lecture: TR2.30-4 (LIMITED 15 EACH S .. (2-151) or TR2.30-4 (4-237, 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: M. Dhar
Spring: M. Dhar
No textbook information available
18.211 Combinatorial Analysis
()
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
()
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 textbook information available
18.217 Combinatorial Theory
()
Prereq: Permission of instructor
Units: 3-0-9
Content varies from year to year.
A. Postnikov
18.218 Topics in Combinatorics
()
Prereq: Permission of instructor
Units: 3-0-9
Topics vary from year to year.
Staff
18.219 Seminar in Combinatorics
()
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
()
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
()
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.
S. Luo
Continuous Applied Mathematics
18.300 Principles of Continuum Applied Mathematics
()
Prereq: Calculus II (GIR) and (18.03 or 18.032)
Units: 3-0-9
Lecture: TR2.30-4 (4-145)
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
No textbook information available
18.303 Linear Partial Differential Equations: Analysis and Numerics
()
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.
Staff
18.305 Advanced Analytic Methods in Science and Engineering
()
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
()
Prereq: (18.03 or 18.032) and (18.04, 18.075, or 18.112)
Units: 3-0-9
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.
Staff
18.327 Topics in Applied Mathematics
()
Prereq: Permission of instructor
Units: 3-0-9
Topics vary from year to year.
Staff
18.330 Introduction to Numerical Analysis
()
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.
Staff
18.335[J] Introduction to Numerical Methods
()
(Same subject as 6.7310[J])
Prereq: 18.06, 18.700, or 18.701
Units: 3-0-9
Lecture: MW9.30-11 (2-190)
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.
S. Chen
Textbooks (Spring 2025)
18.336[J] Fast Methods for Partial Differential and Integral Equations
(, )
(Same subject as 6.7340[J])
Prereq: 6.7300, 16.920, 18.085, 18.335, or permission of instructor
Units: 3-0-9
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.
Staff
18.337[J] Parallel Computing and Scientific Machine Learning
()
(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.
Staff
18.338 Eigenvalues of Random Matrices
()
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
()
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
()
(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
()
(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 (2-143)
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.
N. Derr
No textbook information available
18.3541 Nonlinear Dynamics: Continuum Systems
()
(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 (2-143)
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 textbook information available
18.355 Fluid Mechanics
()
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
()
Prereq: 2.25, 12.800, 18.354, 18.355, or permission of instructor
Units: 3-0-9
Lecture: MW3-4.30 (2-151)
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. Bush
No textbook information available
18.358[J] Nonlinear Dynamics and Turbulence
()
(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
()
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.
L. Demanet
18.369[J] Mathematical Methods in Nanophotonics
()
(Same subject as 8.315[J])
Prereq: 8.07, 18.303, or permission of instructor
Units: 3-0-9
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
18.376[J] Wave Propagation
()
Not offered regularly; consult department
(Same subject as 1.138[J], 2.062[J])
Prereq: 2.003 and 18.075
Units: 3-0-9
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
()
(Same subject as 1.685[J], 2.034[J])
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.
Staff
18.384 Undergraduate Seminar in Physical Mathematics
()
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
()
(Same subject as 2.036[J])
Prereq: 18.03 or 18.032
Units: 3-0-9
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.
Staff
18.397 Mathematical Methods in Physics
()
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
()
(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.
R. Williams
Textbooks (Spring 2025)
18.404 Theory of Computation
()
(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
()
(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
()
(Same subject as 6.5410[J])
Prereq: 18.404
Units: 3-0-9
Lecture: TR11-12.30 (4-163)
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. Minzer
No textbook information available
18.408 Topics in Theoretical Computer Science
()
Prereq: Permission of instructor
Units: 3-0-9
Study of areas of current interest in theoretical computer science. Topics vary from term to term.
J. Kelner
18.410[J] Design and Analysis of Algorithms
(, )
(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 2025)
18.413 Introduction to Computational Molecular Biology
()
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
()
(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.
D. Karger
18.416[J] Randomized Algorithms
()
(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.
Staff
18.417 Introduction to Computational Molecular Biology
()
(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
()
(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
()
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.
J. Gaitonde
18.425[J] Foundations of Cryptography
()
(Same subject as 6.5620[J])
Prereq: 6.1220, 6.1400, or 18.4041
Units: 3-0-9
A rigorous introduction to modern cryptography. Emphasis on the fundamental cryptographic primitives such as public-key encryption, digital signatures, and pseudo-random number generation, as well as advanced cryptographic primitives such as zero-knowledge proofs, homomorphic encryption, and secure multiparty computation.
Staff
18.434 Seminar in Theoretical Computer Science
()
Prereq: 6.1220
Units: 3-0-9
Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.
D. Minzer
18.435[J] Quantum Computation
()
(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.
P. Shor
18.436[J] Quantum Information Science
()
(Same subject as 6.6420[J], 8.371[J])
Prereq: 18.435
Units: 3-0-9
Lecture: MW9.30-11 (4-163)
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.
I. Chuang
Textbooks (Spring 2025)
18.437[J] Distributed Algorithms
()
(Same subject as 6.5250[J])
Prereq: 6.1220
Units: 3-0-9
Design and analysis of algorithms, emphasizing those suitable for use in distributed networks. Covers various topics including distributed graph algorithms, locality constraints, bandwidth limitations and communication complexity, process synchronization, allocation of computational resources, fault tolerance, and asynchrony. No background in distributed systems required.
M. Ghaffari, N. A. Lynch
18.453 Combinatorial Optimization
()
(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
()
(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
()
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
()
(Same subject as 6.7230[J])
Prereq: 6.7210 or 15.093
Units: 3-0-9
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.
Staff
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