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Course 18: Mathematics
IAP/Spring 2024


Logic

18.504 Seminar in Logic
______

Not offered academic year 2023-2024Undergrad (Fall)
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
______

Not offered academic year 2024-2025Undergrad (Fall)
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.
H. Cohn

18.515 Mathematical Logic
______

Graduate (Spring)
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
______

Undergrad (Fall, Spring) Rest Elec in Sci & Tech
Prereq: Calculus II (GIR)
Units: 4-0-8
Credit cannot also be received for 6.3700, 6.3702
Lecture: MWF10 (26-100) Recitation: R10 (2-190) or R12 (1-390) or R3 (1-390) or R4 (1-390) +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.
Fall: S. Sheffield
Spring: J. Kelner
No required or recommended textbooks

18.615 Introduction to Stochastic Processes
______

Graduate (Fall)
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.
J. He

18.619[J] Discrete Probability and Stochastic Processes
(New)
______

Graduate (Spring)
(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
Lecture: MW2.30-4 (E25-111)
______
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
No textbook information available

18.642 Topics in Mathematics with Applications in Finance
______

Undergrad (Fall)
Prereq: 18.03, 18.06, and (18.05 or 18.600)
Units: 3-0-9
______
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

18.650[J] Fundamentals of Statistics
______

Undergrad (Fall, Spring)
(Same subject as IDS.014[J])
(Subject meets with 18.6501)
Prereq: 6.3700 or 18.600
Units: 4-0-8
Lecture: MWF1 (2-190) Recitation: R10 (4-270) or R3 (4-153) or R4 (4-153) +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.
Fall: P. Rigollet
Spring: A. Katsevich
Textbooks (Spring 2024)

18.6501 Fundamentals of Statistics
______

Graduate (Fall, Spring)
(Subject meets with 18.650[J], IDS.014[J])
Prereq: 6.3700 or 18.600
Units: 4-0-8
Lecture: MWF1 (2-190) Recitation: R10 (4-270) or R3 (4-153) or R4 (4-153) +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
Textbooks (Spring 2024)

18.655 Mathematical Statistics
______

Graduate (Spring)
Prereq: (18.650 and (18.100A, 18.100A, 18.100P, or 18.100Q)) or permission of instructor
Units: 3-0-9
Lecture: TR11-12.30 (2-143)
______
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
Textbooks (Spring 2024)

18.656[J] Mathematical Statistics: a Non-Asymptotic Approach
______

Graduate (Spring)
(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
Lecture: TR1-2.30 (46-3002)
______
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
No required or recommended textbooks

18.657 Topics in Statistics
______

Not offered academic year 2024-2025Graduate (Spring) Can be repeated for credit
Prereq: Permission of instructor
Units: 3-0-9
Lecture: TR9.30-11 (2-146)
______
Topics vary from term to term.
P. Rigollet
No required or recommended textbooks

18.675 Theory of Probability
______

Graduate (Fall)
Prereq: 18.100A, 18.100B, 18.100P, or 18.100Q
Units: 3-0-9
______
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

18.676 Stochastic Calculus
______

Graduate (Spring)
Prereq: 18.675
Units: 3-0-9
Lecture: MW11-12.30 (E25-117)
______
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
Textbooks (Spring 2024)

18.677 Topics in Stochastic Processes
______

Graduate (Spring) Can be repeated for credit
Prereq: 18.675
Units: 3-0-9
Lecture: TR1-2.30 (2-131)
______
Topics vary from year to year.
S. Sheffield
No required or recommended textbooks

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
______

Undergrad (Fall) Rest Elec in Sci & Tech
Prereq: Calculus II (GIR)
Units: 3-0-9
Credit cannot also be received for 6.C06, 18.06, 18.C06
______
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.
K. Vashaw

18.701 Algebra I
______

Undergrad (Fall)
Prereq: 18.100A, 18.100B, 18.100P, 18.100Q, 18.090, or permission of instructor
Units: 3-0-9
______
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.
Z. Yun

18.702 Algebra II
______

Undergrad (Spring)
Prereq: 18.701
Units: 3-0-9
Lecture: MWF11 (54-100) +final
______
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.
Z. Yun
Textbooks (Spring 2024)

18.703 Modern Algebra
______

Undergrad (Spring)
Prereq: Calculus II (GIR)
Units: 3-0-9
Lecture: TR2.30-4 (2-135)
______
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
Textbooks (Spring 2024)

18.704 Seminar in Algebra
______

Undergrad (Fall, Spring)
Prereq: 18.701, (18.06 and 18.703), or (18.700 and 18.703)
Units: 3-0-9
Lecture: TR2.30-4 (2-142)
______
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.
Fall: E. Bodish
Spring: E. Bodish
Textbooks (Spring 2024)

18.705 Commutative Algebra
______

Graduate (Fall)
Prereq: 18.702
Units: 3-0-9
______
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.
D. Maulik

18.706 Noncommutative Algebra
______

Not offered academic year 2023-2024Graduate (Spring)
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.
R. Bezrukavnikov

18.708 Topics in Algebra
______

Not offered academic year 2023-2024Graduate (Fall) Can be repeated for credit
Prereq: 18.705
Units: 3-0-9
______
Topics vary from year to year.
Staff

18.715 Introduction to Representation Theory
______

Not offered academic year 2023-2024Graduate (Spring)
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.
G. Lusztig

18.721 Introduction to Algebraic Geometry
______

Undergrad (Spring)
Prereq: 18.702 and 18.901
Units: 3-0-9
Lecture: TR1-2.30 (2-143)
______
Presents basic examples of complex algebraic varieties, affine and projective algebraic geometry, sheaves, cohomology.
D. Yang
No required or recommended textbooks

18.725 Algebraic Geometry I
______

Graduate (Fall)
Prereq: None. Coreq: 18.705
Units: 3-0-9
______
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.
R. Bezrukavnikov

18.726 Algebraic Geometry II
______

Graduate (Spring)
Prereq: 18.725
Units: 3-0-9
Lecture: MW9.30-11 (2-142)
______
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.
A. Landesman
No required or recommended textbooks

18.727 Topics in Algebraic Geometry
______

Not offered academic year 2023-2024Graduate (Spring) Can be repeated for credit
Prereq: 18.725
Units: 3-0-9
______
Topics vary from year to year.
Staff

18.737 Algebraic Groups
______

Not offered academic year 2023-2024Graduate (Spring)
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.
J.-L. Kim

18.745 Lie Groups and Lie Algebras I
______

Graduate (Fall)
Prereq: (18.701 or 18.703) and (18.100A, 18.100B, 18.100P, or 18.100Q)
Units: 3-0-9
______
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.
J. Kim

18.747 Infinite-dimensional Lie Algebras
______

Not offered academic year 2024-2025Graduate (Fall)
Prereq: 18.745
Units: 3-0-9
______
Topics vary from year to year.
L. Rybnikov

18.748 Topics in Lie Theory
______

Not offered academic year 2024-2025Graduate (Spring) Can be repeated for credit
Prereq: Permission of instructor
Units: 3-0-9
Lecture: MW1-2.30 (2-143)
______
Topics vary from year to year.
L. Rybnikov
No textbook information available

18.755 Lie Groups and Lie Algebras II
______

Graduate (Spring)
Prereq: 18.745 or permission of instructor
Units: 3-0-9
Lecture: TR11-12.30 (2-147)
______
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.
P. Etingof
No required or recommended textbooks

18.757 Representations of Lie Groups
______

Not offered academic year 2024-2025Graduate (Fall)
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.
P. Etingof

18.781 Theory of Numbers
______

Undergrad (Spring)
Prereq: None
Units: 3-0-9
Lecture: TR9.30-11 (4-159) +final
______
An elementary introduction to number theory with no algebraic prerequisites. Primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, partitions.
T. Rud
Textbooks (Spring 2024)

18.782 Introduction to Arithmetic Geometry
______

Not offered academic year 2023-2024Undergrad (Spring)
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.
S. Chidambaram

18.783 Elliptic Curves
______

Not offered academic year 2024-2025Undergrad (Fall)
(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
______

Not offered academic year 2024-2025Graduate (Fall)
(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
______

Undergrad (Spring)
Prereq: 18.701 or (18.703 and (18.06 or 18.700))
Units: 3-0-9
Lecture: MW11-12.30 (2-146)
______
Topics vary from year to year. Students present and discuss the subject matter. Instruction and practice in written and oral communication provided. Enrollment limited.
J.Kim
Textbooks (Spring 2024)

18.785 Number Theory I
______

Graduate (Fall)
Prereq: None. Coreq: 18.705
Units: 3-0-9
______
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.
B. Poonen

18.786 Number Theory II
______

Graduate (Spring)
Prereq: 18.785
Units: 3-0-9
Lecture: MW11-12.30 (2-139)
______
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.
A. Sutherland
No required or recommended textbooks

18.787 Topics in Number Theory
______

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

Mathematics Laboratory

18.821 Project Laboratory in Mathematics
______

Undergrad (Fall, Spring) Institute Lab
Prereq: Two mathematics subjects numbered 18.100 or above
Units: 3-6-3
URL: http://math.mit.edu/classes/18.821/
Lecture: MW2-4 (LIMITED 27) (2-147) 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: W. Zhang
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
______

Graduate (Spring); 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]
Begins Apr 1. Lecture: TR9.30-11 (32-082)
______
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
No required or recommended textbooks

18.897[J] Leadership and Professional Strategies & Skills Training (LEAPS), Part II: Developing Your Leadership Competencies
______

Graduate (Spring); 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]
Ends Mar 22. Lecture: TR9.30-11 (32-082)
______
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
No required or recommended textbooks

Topology and Geometry

18.900 Geometry and Topology in the Plane
______

Undergrad (Spring)
Prereq: 18.03 or 18.06
Units: 3-0-9
Lecture: TR9.30-11 (2-139) +final
______
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.
J. Zung
No required or recommended textbooks

18.901 Introduction to Topology
______

Undergrad (Fall, Spring)
(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 (66-168) +final
______
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.
Fall: A. Pieloch
Spring: R. Jiang
Textbooks (Spring 2024)

18.9011 Introduction to Topology
______

Graduate (Fall, Spring)
(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 (66-168) +final
______
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
Textbooks (Spring 2024)

18.904 Seminar in Topology
______

Undergrad (Fall)
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.
J. Zung

18.905 Algebraic Topology I
______

Graduate (Fall)
Prereq: 18.901 and (18.701 or 18.703)
Units: 3-0-9
______
Singular homology, CW complexes, universal coefficient and Künneth theorems, cohomology, cup products, Poincaré duality.
P. Seidel

18.906 Algebraic Topology II
______

Graduate (Spring)
Prereq: 18.905 and (18.101 or 18.965)
Units: 3-0-9
Lecture: MW9.30-11 (2-131)
______
Continues the introduction to Algebraic Topology from 18.905. Topics include basic homotopy theory, spectral sequences, characteristic classes, and cohomology operations.
T. S. Mrowka
No textbook information available

18.917 Topics in Algebraic Topology
______

Not offered academic year 2024-2025Graduate (Spring) Can be repeated for credit
Prereq: 18.906
Units: 3-0-9
Lecture: TR1-2.30 (2-139)
______
Content varies from year to year. Introduces new and significant developments in algebraic topology with the focus on homotopy theory and related areas.
D. Alvarez-Gavela
Textbooks (Spring 2024)

18.919 Graduate Topology Seminar
______

Graduate (Spring)
Prereq: 18.906
Units: 3-0-9
Lecture: TR11-12.30 (2-132)
______
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.
J. Hahn
No required or recommended textbooks

18.937 Topics in Geometric Topology
______

Not offered academic year 2023-2024Graduate (Spring) Can be repeated for credit
Prereq: Permission of instructor
Units: 3-0-9
______
Content varies from year to year. Introduces new and significant developments in geometric topology.
Staff

18.950 Differential Geometry
______

Undergrad (Fall)
(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
______
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.
G. Franz

18.9501 Differential Geometry
______

Graduate (Fall)
(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
______
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

18.952 Theory of Differential Forms
______

Undergrad (Spring)
Not offered regularly; consult department
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.
V. W. Guillemin

18.965 Geometry of Manifolds I
______

Graduate (Fall)
Prereq: 18.101, 18.950, or 18.952
Units: 3-0-9
______
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.
K. Naff

18.966 Geometry of Manifolds II
______

Graduate (Spring)
Prereq: 18.965
Units: 3-0-9
URL: http://math.mit.edu/classes/18.966
Lecture: TR9.30-11 (2-361)
______
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
No required or recommended textbooks

18.968 Topics in Geometry
______

Not offered academic year 2023-2024Graduate (Spring) Can be repeated for credit
Prereq: 18.965
Units: 3-0-9
______
Content varies from year to year.
Staff

18.979 Graduate Geometry Seminar
______

Graduate (Spring) Can be repeated for credit
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
______

Not offered academic year 2023-2024Undergrad (Spring)
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
______

Graduate (Fall, IAP, Spring, Summer) Can be repeated for credit
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.
Fall: T. Cummings
IAP: T. Cummings
Spring: T. Cummings
Summer: T. Cummings
No required or recommended textbooks

18.C20[J] Introduction to Computational Science and Engineering
______

Undergrad (Fall, Spring); 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: 3-0-3
Credit cannot also be received for 6.100B
Begins Apr 1. Lecture: MW3-4.30 (1-390)
______
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 textbook information available

18.C25[J] Real World Computation with Julia
(New)
______

Undergrad (Fall)
(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
______
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

18.UR Undergraduate Research
______

Undergrad (Fall, IAP, Spring, Summer) Can be repeated for credit
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.
Fall: H. Lloyd
IAP: K. Myatt
Spring: K. Myatt
Summer: K. Myatt
No required or recommended textbooks

18.THG Graduate Thesis
______

Graduate (Fall, IAP, Spring, Summer) Can be repeated for credit
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.
Fall: T. Cummings
IAP: T. Cummings
Spring: T. Cummings
Summer: T. Cummings
No required or recommended textbooks

18.S096 Special Subject in Mathematics
______

Undergrad (IAP, Spring) Can be repeated for credit
Prereq: Permission of instructor
Units arranged
Lecture: TR2.30-4 (2-151)
______
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.
IAP: S. Johnson
Spring: H. Cohn
No required or recommended textbooks

18.S097 Special Subject in Mathematics
______

Undergrad (IAP) Can be repeated for credit
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.
A. Edelman
No textbook information available

18.S190 Special Subject in Mathematics
______

Undergrad (IAP) Can be repeated for credit
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.
M. Sherman-Bennett
No required or recommended textbooks

18.S191 Special Subject in Mathematics
______

Not offered academic year 2024-2025Undergrad (IAP) Can be repeated for credit
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.
C. Rackauckas
No required or recommended textbooks

18.S995 Special Subject in Mathematics
______

Not offered academic year 2023-2024Graduate (Spring) Can be repeated for credit
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
______

Graduate (Spring) Can be repeated for credit
Prereq: Permission of instructor
Units arranged
Lecture: MW2.30-4 (2-143)
______
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.
J. Bush
No required or recommended textbooks

18.S997 Special Subject in Mathematics
______

Graduate (Spring) Can be repeated for credit
Prereq: Permission of instructor
Units arranged
Lecture: TR1-2.30 (66-168)
______
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.
B. Berger
No required or recommended textbooks

18.S998 Special Subject in Mathematics
______

Not offered academic year 2023-2024Graduate (Fall, IAP, Spring) Can be repeated for credit
Prereq: Permission of instructor
Units arranged
Subject Cancelled Subject Cancelled
______
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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Produced: 18-APR-2024 05:10 PM