Logic
18.504 Seminar in Logic
  ( )
Prereq: (18.06, 18.510, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
Units: 3-0-9
Students present and discuss the subject matter taken from current journals or books. Topics vary from year to year. Instruction and practice in written and oral communication provided. Enrollment limited.
Staff
18.510 Introduction to Mathematical Logic and Set Theory
()
Prereq: None
Units: 3-0-9
Propositional and predicate logic. Zermelo-Fraenkel set theory. Ordinals and cardinals. Axiom of choice and transfinite induction. Elementary model theory: completeness, compactness, and Lowenheim-Skolem theorems. Godel's incompleteness theorem.
Staff
18.515 Mathematical Logic
 ( )
Not offered regularly; consult department
Prereq: Permission of instructor
Units: 3-0-9
More rigorous treatment of basic mathematical logic, Godel's theorems, and Zermelo-Fraenkel set theory. First-order logic. Models and satisfaction. Deduction and proof. Soundness and completeness. Compactness and its consequences. Quantifier elimination. Recursive sets and functions. Incompleteness and undecidability. Ordinals and cardinals. Set-theoretic formalization of mathematics.
Staff
Probability and Statistics
18.600 Probability and Random Variables
(, ) 
Prereq: Calculus II (GIR)
Units: 4-0-8
Credit cannot also be received for 6.3700, 6.3702
Lecture: MW11-12.30 (34-101) Recitation: F10 (4-163) or F11 (32-141) or F12 (2-190) or F1 (1-190) +final
Probability spaces, random variables, distribution functions. Binomial, geometric, hypergeometric, Poisson distributions. Uniform, exponential, normal, gamma and beta distributions. Conditional probability, Bayes theorem, joint distributions. Chebyshev inequality, law of large numbers, and central limit theorem. Credit cannot also be received for 6.041A or 6.041B.
E. Mossel
Textbooks (Fall 2024)
18.604 Seminar In Probability Theory
(New)
()
Prereq: 18.05 or 18.600
Units: 3-0-9
Students work on group presentations on topics selected by students from a provided list of suggestions. Topics may include Benford's law, random walks and electrical networks, and Brownian motions. Assignments include three group presentations, two individual presentations, and a final individual term paper. Instruction in oral and written communication provided to effectively communicate about probability theory. Limited to 16.
Staff
18.615 Introduction to Stochastic Processes
()
Prereq: 6.3700 or 18.600
Units: 3-0-9
Basics of stochastic processes. Markov chains, Poisson processes, random walks, birth and death processes, Brownian motion.
Staff
18.619[J] Discrete Probability and Stochastic Processes
()
(Same subject as 6.7720[J], 15.070[J])
Prereq: 6.3702, 6.7700, 18.100A, 18.100B, or 18.100Q
Units: 3-0-9
Provides an introduction to tools used for probabilistic reasoning in the context of discrete systems and processes. Tools such as the probabilistic method, first and second moment method, martingales, concentration and correlation inequalities, theory of random graphs, weak convergence, random walks and Brownian motion, branching processes, Markov chains, Markov random fields, correlation decay method, isoperimetry, coupling, influences and other basic tools of modern research in probability will be presented. Algorithmic aspects and connections to statistics and machine learning will be emphasized.
G. Bresler
18.642 Topics in Mathematics with Applications in Finance
()
Prereq: 18.03, 18.06, and (18.05 or 18.600)
Units: 3-0-9
Lecture: TR2.30-4 (32-124)
Introduction to mathematical concepts and techniques used in finance. Lectures focusing on linear algebra, probability, statistics, stochastic processes, and numerical methods are interspersed with lectures by financial sector professionals illustrating the corresponding application in the industry. Prior knowledge of economics or finance helpful but not required.
P. Kempthorne, V. Strela, J. Xia
No required or recommended textbooks
18.650[J] Fundamentals of Statistics
(, )
(Same subject as IDS.014[J])
(Subject meets with 18.6501)
Prereq: 6.3700 or 18.600
Units: 4-0-8
Lecture: MWF10 (2-190) Recitation: R11 (4-149) or R12 (4-149) or R2 (4-149) +final
A rapid introduction to the theoretical foundations of statistical methods that are useful in many applications. Covers a broad range of topics in a short amount of time with the goal of providing a rigorous and cohesive understanding of the modern statistical landscape. Mathematical language is used for intuition and basic derivations but not proofs. Main topics include: parametric estimation, confidence intervals, hypothesis testing, Bayesian inference, and linear and logistic regression. Additional topics may include: causal inference, nonparametric estimation, and classification.
P. Rigollet
Textbooks (Fall 2024)
18.6501 Fundamentals of Statistics
(, )
(Subject meets with 18.650[J], IDS.014[J])
Prereq: 6.3700 or 18.600
Units: 4-0-8
Lecture: MWF10 (2-190) Recitation: R11 (4-149) or R12 (4-149) or R2 (4-149) +final
A rapid introduction to the theoretical foundations of statistical methods that are useful in many applications. Covers a broad range of topics in a short amount of time with the goal of providing a rigorous and cohesive understanding of the modern statistical landscape. Mathematical language is used for intuition and basic derivations but not proofs. Main topics include: parametric estimation, confidence intervals, hypothesis testing, Bayesian inference, and linear and logistic regression. Additional topics may include: causal inference, nonparametric estimation, and classification. Students in Course 18 must register for the undergraduate version, 18.650.
Fall: P. Rigollet
Spring: A. Katsevich
No textbook information available
18.655 Mathematical Statistics
()
Prereq: (18.650 and (18.100A, 18.100A, 18.100P, or 18.100Q)) or permission of instructor
Units: 3-0-9
Decision theory, estimation, confidence intervals, hypothesis testing. Introduces large sample theory. Asymptotic efficiency of estimates. Exponential families. Sequential analysis. Prior exposure to both probability and statistics at the university level is assumed.
P. Kempthorne
18.656[J] Mathematical Statistics: a Non-Asymptotic Approach
()
(Same subject as 9.521[J], IDS.160[J])
Prereq: (6.7700, 18.06, and 18.6501) or permission of instructor
Units: 3-0-9
Introduces students to modern non-asymptotic statistical analysis. Topics include high-dimensional models, nonparametric regression, covariance estimation, principal component analysis, oracle inequalities, prediction and margin analysis for classification. Develops a rigorous probabilistic toolkit, including tail bounds and a basic theory of empirical processes
S. Rakhlin, P. Rigollet
18.657 Topics in Statistics
() 
Prereq: Permission of instructor
Units: 3-0-9
Topics vary from term to term.
P. Rigollet
18.675 Theory of Probability
()
Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q
Units: 3-0-9
Lecture: TR2.30-4 (32-155) +final
Sums of independent random variables, central limit phenomena, infinitely divisible laws, Levy processes, Brownian motion, conditioning, and martingales. Prior exposure to probability (e.g., 18.600) recommended.
K. Kavvadias
No required or recommended textbooks
18.676 Stochastic Calculus
()
Prereq: 18.675
Units: 3-0-9
Introduction to stochastic processes, building on the fundamental example of Brownian motion. Topics include Brownian motion, continuous parameter martingales, Ito's theory of stochastic differential equations, Markov processes and partial differential equations, and may also include local time and excursion theory. Students should have familiarity with Lebesgue integration and its application to probability.
N. Sun
18.677 Topics in Stochastic Processes
()
Prereq: 18.675
Units: 3-0-9
Topics vary from year to year.
Staff
For additional related subjects in Statistics, see:
Civil and Environmental Engineering: 1.202, 1.203J, and 1.205
Electrical Engineering and Computer Science: 6.0002, 6.041, 6.231, 6.245, 6.262, 6.431, 6.434J, 6.435, 6.436J, 6.437, and 6.438
Management: 15.034, 15.060, 15.070J, 15.071, 15.075J, 15.077J, 15.098, and 15.456
IDSS: IDS.012J, IDS.013J, IDS.014J, IDS.136J, and IDS.700J
Economics: 14.30, 14.310, 14.36, 14.381, 14.382, 14.384, and 14.386
Algebra and Number Theory
18.700 Linear Algebra
()
Prereq: Calculus II (GIR)
Units: 3-0-9
Credit cannot also be received for 6.C06, 18.06, 18.C06, ES.1806
Lecture: MW9.30-11 (6-120) +final
Vector spaces, systems of linear equations, bases, linear independence, matrices, determinants, eigenvalues, inner products, quadratic forms, and canonical forms of matrices. More emphasis on theory and proofs than in 18.06.
S. Schiavone
Textbooks (Fall 2024)
18.701 Algebra I
()
Prereq: 18.100A, 18.100B, 18.100P, 18.100Q, 18.090, or permission of instructor
Units: 3-0-9
Lecture: TR1-2.30 (54-100)
18.701-18.702 is more extensive and theoretical than the 18.700-18.703 sequence. Experience with proofs necessary. 18.701 focuses on group theory, geometry, and linear algebra.
P. Etingof
Textbooks (Fall 2024)
18.702 Algebra II
()
Prereq: 18.701
Units: 3-0-9
Continuation of 18.701. Focuses on group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.
Staff
18.703 Modern Algebra
()
Prereq: Calculus II (GIR)
Units: 3-0-9
Focuses on traditional algebra topics that have found greatest application in science and engineering as well as in mathematics: group theory, emphasizing finite groups; ring theory, including ideals and unique factorization in polynomial and Euclidean rings; field theory, including properties and applications of finite fields. 18.700 and 18.703 together form a standard algebra sequence.
V. G. Kac
18.704 Seminar in Algebra
()
Prereq: 18.701, (18.06 and 18.703), or (18.700 and 18.703)
Units: 3-0-9
Lecture: MW1-2.30 (2-146)
Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Some experience with proofs required. Enrollment limited.
J. Kim
Textbooks (Fall 2024)
18.705 Commutative Algebra
()
Prereq: 18.702
Units: 3-0-9
Lecture: MW3-4.30 (2-139)
Exactness, direct limits, tensor products, Cayley-Hamilton theorem, integral dependence, localization, Cohen-Seidenberg theory, Noether normalization, Nullstellensatz, chain conditions, primary decomposition, length, Hilbert functions, dimension theory, completion, Dedekind domains.
M. Moreira
Textbooks (Fall 2024)
18.706 Noncommutative Algebra
()
Prereq: 18.702
Units: 3-0-9
Topics may include Wedderburn theory and structure of Artinian rings, Morita equivalence and elements of category theory, localization and Goldie's theorem, central simple algebras and the Brauer group, representations, polynomial identity rings, invariant theory growth of algebras, Gelfand-Kirillov dimension.
Staff
18.708 Topics in Algebra
()
Prereq: 18.705
Units: 3-0-9
Topics vary from year to year.
Staff
18.715 Introduction to Representation Theory
 ()
Prereq: 18.702 or 18.703
Units: 3-0-9
Algebras, representations, Schur's lemma. Representations of SL(2). Representations of finite groups, Maschke's theorem, characters, applications. Induced representations, Burnside's theorem, Mackey formula, Frobenius reciprocity. Representations of quivers.
Staff
18.721 Introduction to Algebraic Geometry
()
Prereq: 18.702 and 18.901
Units: 3-0-9
Presents basic examples of complex algebraic varieties, affine and projective algebraic geometry, sheaves, cohomology.
Staff
18.725 Algebraic Geometry I
()
Prereq: None. Coreq: 18.705
Units: 3-0-9
Lecture: MWF9 (2-142)
Introduces the basic notions and techniques of modern algebraic geometry. Covers fundamental notions and results about algebraic varieties over an algebraically closed field; relations between complex algebraic varieties and complex analytic varieties; and examples with emphasis on algebraic curves and surfaces. Introduction to the language of schemes and properties of morphisms. Knowledge of elementary algebraic topology, elementary differential geometry recommended, but not required.
B. Poonen
No required or recommended textbooks
18.726 Algebraic Geometry II
()
Prereq: 18.725
Units: 3-0-9
Continuation of the introduction to algebraic geometry given in 18.725. More advanced properties of the varieties and morphisms of schemes, as well as sheaf cohomology.
Staff
18.727 Topics in Algebraic Geometry
()
Prereq: 18.725
Units: 3-0-9
Topics vary from year to year.
Staff
18.737 Algebraic Groups
()
Prereq: 18.705
Units: 3-0-9
Structure of linear algebraic groups over an algebraically closed field, with emphasis on reductive groups. Representations of groups over a finite field using methods from etale cohomology. Some results from algebraic geometry are stated without proof.
Staff
18.745 Lie Groups and Lie Algebras I
()
Prereq: (18.701 or 18.703) and (18.100A, 18.100B, 18.100P, or 18.100Q)
Units: 3-0-9
Lecture: TR2.30-4 (2-146)
Covers fundamentals of the theory of Lie algebras and related groups. Topics may include theorems of Engel and Lie; enveloping algebra, Poincare-Birkhoff-Witt theorem; classification and construction of semisimple Lie algebras; the center of their enveloping algebras; elements of representation theory; compact Lie groups and/or finite Chevalley groups.
V. Kac
No textbook information available
18.747 Infinite-dimensional Lie Algebras
()
Prereq: 18.745
Units: 3-0-9
Topics vary from year to year.
Staff
18.748 Topics in Lie Theory
()
Prereq: Permission of instructor
Units: 3-0-9
Topics vary from year to year.
Staff
18.755 Lie Groups and Lie Algebras II
()
Prereq: 18.745 or permission of instructor
Units: 3-0-9
A more in-depth treatment of Lie groups and Lie algebras. Topics may include homogeneous spaces and groups of automorphisms; representations of compact groups and their geometric realizations, Peter-Weyl theorem; invariant differential forms and cohomology of Lie groups and homogeneous spaces; complex reductive Lie groups, classification of real reductive groups.
Staff
18.757 Representations of Lie Groups
()
Prereq: 18.745 or 18.755
Units: 3-0-9
Covers representations of locally compact groups, with emphasis on compact groups and abelian groups. Includes Peter-Weyl theorem and Cartan-Weyl highest weight theory for compact Lie groups.
Staff
18.781 Theory of Numbers
()
Prereq: None
Units: 3-0-9
An elementary introduction to number theory with no algebraic prerequisites. Primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, partitions.
Staff
18.782 Introduction to Arithmetic Geometry
()
Prereq: 18.702
Units: 3-0-9
Exposes students to arithmetic geometry, motivated by the problem of finding rational points on curves. Includes an introduction to p-adic numbers and some fundamental results from number theory and algebraic geometry, such as the Hasse-Minkowski theorem and the Riemann-Roch theorem for curves. Additional topics may include Mordell's theorem, the Weil conjectures, and Jacobian varieties.
Staff
18.783 Elliptic Curves
()
(Subject meets with 18.7831)
Prereq: 18.702, 18.703, or permission of instructor
Units: 3-0-9
Computationally focused introduction to elliptic curves, with applications to number theory and cryptography. Topics include point-counting, isogenies, pairings, and the theory of complex multiplication, with applications to integer factorization, primality proving, and elliptic curve cryptography. Includes a brief introduction to modular curves and the proof of Fermat's Last Theorem.
A. Sutherland
18.7831 Elliptic Curves
()
(Subject meets with 18.783)
Prereq: 18.702, 18.703, or permission of instructor
Units: 3-0-9
Computationally focused introduction to elliptic curves, with applications to number theory and cryptography. Topics include point-counting, isogenies, pairings, and the theory of complex multiplication, with applications to integer factorization, primality proving, and elliptic curve cryptography. Includes a brief introduction to modular curves and the proof of Fermat's Last Theorem. Students in Course 18 must register for the undergraduate version, 18.783.
A. Sutherland
18.784 Seminar in Number Theory
()
Prereq: 18.701 or (18.703 and (18.06 or 18.700))
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.
Staff
18.785 Number Theory I
()
Prereq: None. Coreq: 18.705
Units: 3-0-9
Lecture: TR11-12.30 (2-139)
Dedekind domains, unique factorization of ideals, splitting of primes. Lattice methods, finiteness of the class group, Dirichlet's unit theorem. Local fields, ramification, discriminants. Zeta and L-functions, analytic class number formula. Adeles and ideles. Statements of class field theory and the Chebotarev density theorem.
W. Zhang
No required or recommended textbooks
18.786 Number Theory II
()
Prereq: 18.785
Units: 3-0-9
Continuation of 18.785. More advanced topics in number theory, such as Galois cohomology, proofs of class field theory, modular forms and automorphic forms, Galois representations, or quadratic forms.
Staff
18.787 Topics in Number Theory
()
Prereq: Permission of instructor
Units: 3-0-9
Topics vary from year to year.
Staff
Mathematics Laboratory
18.821 Project Laboratory in Mathematics
(, ) 
Prereq: Two mathematics subjects numbered 18.100 or above
Units: 3-6-3
URL: http://math.mit.edu/classes/18.821/
Lecture: MW2-4 (2-135) Lab: TBA
Guided research in mathematics, employing the scientific method. Students confront puzzling and complex mathematical situations, through the acquisition of data by computer, pencil and paper, or physical experimentation, and attempt to explain them mathematically. Students choose three projects from a large collection of options. Each project results in a laboratory report subject to revision; oral presentation on one or two projects. Projects drawn from many areas, including dynamical systems, number theory, algebra, fluid mechanics, asymptotic analysis, knot theory, and probability. Enrollment limited.
Fall: R. Bezrukavnikov
Spring: R. Bezrukavnikov
No required or recommended textbooks
18.896[J] Leadership and Professional Strategies & Skills Training (LEAPS), Part I: Advancing Your Professional Strategies and Skills
(); second half of term
(Same subject as 5.961[J], 8.396[J], 9.980[J], 12.396[J])
Prereq: None
Units: 2-0-1 [P/D/F]
Part I (of two parts) of the LEAPS graduate career development and training series. Topics include: navigating and charting an academic career with confidence; convincing an audience with clear writing and arguments; mastering public speaking and communications; networking at conferences and building a brand; identifying transferable skills; preparing for a successful job application package and job interviews; understanding group dynamics and different leadership styles; leading a group or team with purpose and confidence. Postdocs encouraged to attend as non-registered participants. Limited to 80.
A. Frebel
18.897[J] Leadership and Professional Strategies & Skills Training (LEAPS), Part II: Developing Your Leadership Competencies
(); first half of term
(Same subject as 5.962[J], 8.397[J], 9.981[J], 12.397[J])
Prereq: None
Units: 2-0-1 [P/D/F]
Part II (of two parts) of the LEAPS graduate career development and training series. Topics covered include gaining self awareness and awareness of others, and communicating with different personality types; learning about team building practices; strategies for recognizing and resolving conflict and bias; advocating for diversity and inclusion; becoming organizationally savvy; having the courage to be an ethical leader; coaching, mentoring, and developing others; championing, accepting, and implementing change. Postdocs encouraged to attend as non-registered participants. Limited to 80.
D. Rigos
18.899 Internship in Mathematics
(New)
(, ,  )
Prereq: None
Units arranged [P/D/F]
TBA.
Provides academic credit for students pursuing internships to gain practical experience applications of mathematical concepts and methods as related to their field of research.
T. Cummings
No required or recommended textbooks
Topology and Geometry
18.900 Geometry and Topology in the Plane
()
Prereq: 18.03 or 18.06
Units: 3-0-9
Introduction to selected aspects of geometry and topology, using concepts that can be visualized easily. Mixes geometric topics (such as hyperbolic geometry or billiards) and more topological ones (such as loops in the plane). Suitable for students with no prior exposure to differential geometry or topology.
Staff
18.901 Introduction to Topology
(, )
(Subject meets with 18.9011)
Prereq: 18.100A, 18.100B, 18.100P, 18.100Q, or permission of instructor
Units: 3-0-9
Lecture: TR2.30-4 (4-163)
Introduces topology, covering topics fundamental to modern analysis and geometry. Topological spaces and continuous functions, connectedness, compactness, separation axioms, covering spaces, and the fundamental group.
A. Raksit
Textbooks (Fall 2024)
18.9011 Introduction to Topology
(, )
(Subject meets with 18.901)
Prereq: 18.100A, 18.100B, 18.100P, 18.100Q, or permission of instructor
Units: 3-0-9
Lecture: TR2.30-4 (4-163)
Introduces topology, covering topics fundamental to modern analysis and geometry. Topological spaces and continuous functions, connectedness, compactness, separation axioms, covering spaces, and the fundamental group. Students in Course 18 must register for the undergraduate version, 18.901.
Fall: A. Pieloch
Spring: R. Jiang
No textbook information available
18.904 Seminar in Topology
()
Prereq: 18.901
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.
Staff
18.905 Algebraic Topology I
()
Prereq: 18.901 and (18.701 or 18.703)
Units: 3-0-9
Lecture: MWF10 (32-141) +final
Singular homology, CW complexes, universal coefficient and Künneth theorems, cohomology, cup products, Poincaré duality.
J. Hahn
No required or recommended textbooks
18.906 Algebraic Topology II
()
Prereq: 18.905 and (18.101 or 18.965)
Units: 3-0-9
Continues the introduction to Algebraic Topology from 18.905. Topics include basic homotopy theory, spectral sequences, characteristic classes, and cohomology operations.
Staff
18.917 Topics in Algebraic Topology
()
Prereq: 18.906
Units: 3-0-9
Content varies from year to year. Introduces new and significant developments in algebraic topology with the focus on homotopy theory and related areas.
Staff
18.919 Graduate Topology Seminar
()
Prereq: 18.906
Units: 3-0-9
 Subject Cancelled
Study and discussion of important original papers in the various parts of topology. Open to all students who have taken 18.906 or the equivalent, not only prospective topologists.
Staff
18.937 Topics in Geometric Topology
(, )
Prereq: Permission of instructor
Units: 3-0-9
Lecture: TR9.30-11 (66-144)
Content varies from year to year. Introduces new and significant developments in geometric topology.
S. Bai
No required or recommended textbooks
18.950 Differential Geometry
()
(Subject meets with 18.9501)
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
Units: 3-0-9
Lecture: TR1-2.30 (2-131)
Introduction to differential geometry, centered on notions of curvature. Starts with curves in the plane, and proceeds to higher dimensional submanifolds. Computations in coordinate charts: first and second fundamental form, Christoffel symbols. Discusses the distinction between extrinsic and intrinsic aspects, in particular Gauss' theorema egregium. The Gauss-Bonnet theorem. Geodesics. Examples such as hyperbolic space.
J. Zhu
Textbooks (Fall 2024)
18.9501 Differential Geometry
()
(Subject meets with 18.950)
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
Units: 3-0-9
Lecture: TR1-2.30 (2-131)
Introduction to differential geometry, centered on notions of curvature. Starts with curves in the plane, and proceeds to higher dimensional submanifolds. Computations in coordinate charts: first and second fundamental form, Christoffel symbols. Discusses the distinction between extrinsic and intrinsic aspects, in particular Gauss' theorema egregium. The Gauss-Bonnet theorem. Geodesics. Examples such as hyperbolic space. Students in Course 18 must register for the undergraduate version, 18.950.
G. Franz
No textbook information available
18.952 Theory of Differential Forms
()
Prereq: 18.101 and (18.700 or 18.701)
Units: 3-0-9
Multilinear algebra: tensors and exterior forms. Differential forms on Rn: exterior differentiation, the pull-back operation and the Poincaré lemma. Applications to physics: Maxwell's equations from the differential form perspective. Integration of forms on open sets of Rn. The change of variables formula revisited. The degree of a differentiable mapping. Differential forms on manifolds and De Rham theory. Integration of forms on manifolds and Stokes' theorem. The push-forward operation for forms. Thom forms and intersection theory. Applications to differential topology.
Staff
18.965 Geometry of Manifolds I
()
Prereq: 18.101, 18.950, or 18.952
Units: 3-0-9
Lecture: TR11-12.30 (24-121)
Differential forms, introduction to Lie groups, the DeRham theorem, Riemannian manifolds, curvature, the Hodge theory. 18.966 is a continuation of 18.965 and focuses more deeply on various aspects of the geometry of manifolds. Contents vary from year to year, and can range from Riemannian geometry (curvature, holonomy) to symplectic geometry, complex geometry and Hodge-Kahler theory, or smooth manifold topology. Prior exposure to calculus on manifolds, as in 18.952, recommended.
T. Ozuch-Meersseman
No required or recommended textbooks
18.966 Geometry of Manifolds II
()
Prereq: 18.965
Units: 3-0-9
Continuation of 18.965, focusing more deeply on various aspects of the geometry of manifolds. Contents vary from year to year, and can range from Riemannian geometry (curvature, holonomy) to symplectic geometry, complex geometry and Hodge-Kahler theory, or smooth manifold topology.
W. Minicozzi
18.968 Topics in Geometry
()
Prereq: 18.965
Units: 3-0-9
Content varies from year to year.
Staff
18.979 Graduate Geometry Seminar
()
Not offered regularly; consult department
Prereq: Permission of instructor
Units: 3-0-9
Content varies from year to year. Study of classical papers in geometry and in applications of analysis to geometry and topology.
Staff
18.994 Seminar in Geometry
()
Prereq: (18.06, 18.700, or 18.701) and (18.100A, 18.100B, 18.100P, or 18.100Q)
Units: 3-0-9
Students present and discuss subject matter taken from current journals or books. Topics vary from year to year. Instruction and practice in written and oral communication provided. Enrollment limited.
Staff
18.999 Research in Mathematics
(,  , , )
Prereq: Permission of instructor
Units arranged
TBA.
Opportunity for study of graduate-level topics in mathematics under the supervision of a member of the department. For graduate students desiring advanced work not provided in regular subjects.
T. Cummings
No required or recommended textbooks
18.C20[J] Introduction to Computational Science and Engineering
(, ); second half of term
(Same subject as 9.C20[J], 16.C20[J], CSE.C20[J])
Prereq: 6.100A; Coreq: 8.01 and 18.01
Units: 2-0-4
Credit cannot also be received for 6.100B
Begins Oct 28. Lecture: MW3-4.30 (33-419)
Provides an introduction to computational algorithms used throughout engineering and science (natural and social) to simulate time-dependent phenomena; optimize and control systems; and quantify uncertainty in problems involving randomness, including an introduction to probability and statistics. Combination of 6.100A and 16.C20J counts as REST subject.
Fall: D.L. Darmofal, N. Seethapathi
Spring: D.L. Darmofal, N. Seethapathi
No required or recommended textbooks
18.C25[J] Real World Computation with Julia
()
(Same subject as 1.C25[J], 6.C25[J], 12.C25[J], 16.C25[J], 22.C25[J])
Prereq: 6.100A, 18.03, and 18.06
Units: 3-0-9
Lecture: MW1-2.30 (4-149)
Focuses on algorithms and techniques for writing and using modern technical software in a job, lab, or research group environment that may consist of interdisciplinary teams, where performance may be critical, and where the software needs to be flexible and adaptable. Topics include automatic differentiation, matrix calculus, scientific machine learning, parallel and GPU computing, and performance optimization with introductory applications to climate science, economics, agent-based modeling, and other areas. Labs and projects focus on performant, readable, composable algorithms, and software. Programming will be in Julia. Expects students to have some familiarity with Python, Matlab, or R. No Julia experience necessary.
A. Edelman, R. Ferrari, B. Forget, C. Leiseron,Y. Marzouk, J. Williams
No required or recommended textbooks
18.UR Undergraduate Research
(, , , )
Prereq: Permission of instructor
Units arranged [P/D/F]
TBA.
Undergraduate research opportunities in mathematics. Permission required in advance to register for this subject. For further information, consult the departmental coordinator.
K. Myatt
No required or recommended textbooks
18.TAC Classroom Teaching in Mathematics
(New)
(, , )
Prereq: None
Units: 12-0-0 [P/D/F]
TBA.
For classroom training in Mathematics, in cases where teaching assignment is to fulfill academic teaching requirement by the department.
T. Cummings
No required or recommended textbooks
18.THG Graduate Thesis
(, , , )
Prereq: Permission of instructor
Units arranged
TBA.
Program of research leading to the writing of a Ph.D. thesis; to be arranged by the student and an appropriate MIT faculty member.
T. Cummings
No required or recommended textbooks
18.S096 Special Subject in Mathematics
()
Prereq: Permission of instructor
Units arranged
Opportunity for group study of subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to departmental approval.
Staff
18.S097 Special Subject in Mathematics
(, )
Prereq: Permission of instructor
Units arranged [P/D/F]
Opportunity for group study of subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to departmental approval. 18.S097 is graded P/D/F.
Staff
18.S190 Special Subject in Mathematics
()
Prereq: Permission of instructor
Units arranged
Opportunity for group study of subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to departmental approval.
Staff
18.S191 Special Subject in Mathematics
()
Prereq: Permission of instructor
Units arranged
Opportunity for group study of subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to departmental approval.
Staff
18.S995 Special Subject in Mathematics
()
Prereq: Permission of instructor
Units arranged
Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the mathematics faculty on an ad hoc basis, subject to departmental approval.
Staff
18.S996 Special Subject in Mathematics
()
Prereq: Permission of instructor
Units arranged
Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to Departmental approval.
Staff
18.S997 Special Subject in Mathematics
()
Prereq: Permission of instructor
Units arranged
Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to Departmental approval.
Staff
18.S998 Special Subject in Mathematics
()
Prereq: Permission of instructor
Units arranged
Opportunity for group study of advanced subjects in mathematics not otherwise included in the curriculum. Offerings are initiated by members of the Mathematics faculty on an ad hoc basis, subject to departmental approval.
Staff
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