Adelic analysis on \(\mathrm{GL}(1)\) and \(\mathrm{GL}(2)\), especially Tate thesis and Hecke theory, automorphic representations. Special values of \(L\)-functions and \(p\)-adic \(L\)-functions, arithmetic theory of modular forms, advanced topics in analytic number theory. Arithmetic geometry, especially of modular curves.

Algebraic number theory, including ideal theory, valuations, local fields, cyclotomic fields. Introduction to class-field theory, analytic number theory, \(L\)-functions and class number formulas, and modular forms.

Introduction to theory of cryptography, stressing rigorous definitions and proofs of security. Topics include notions of hardness, one-way functions, hard-core bits, pseudorandom generators, pseudorandom functions and pseudorandom permutations, semantic security, public-key and private-key encryption, secret-sharing, message authentication, digital signatures, interactive proofs, zero-knowledge proofs, collision-resistant hash functions, commitment protocols, key-agreement, contract signing, and two-party secure computation with static security.

Cauchy/Riemann equations. Cauchy theorem. Cauchy integral formula and residue calculus. Power series. Normal families. Harmonic functions. Linear fractional transformations. Conformal mappings. Analytic continuation. Examples of Riemann surfaces. Infinite products. Partial fractions. Classical transcendental functions. Elliptic functions.

Basic measure theory. Measure theory on locally compact spaces. Fubini theorem. Elementary aspects of Banach and Hilbert spaces and linear operators. Function spaces. Radon/Nikodym theorem. Fourier transform and Plancherel on \(\mathbb{R}^n\) and \(\mathbb{T}^n\).

Seminar, one hour; two-day intensive training at beginning of Fall Quarter. Special course for teaching assistants designed to deal with problems and echniques of teaching college mathematics.

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.

Textbook: Jean-Pierre Serre, A Course in Arithmetic, 1st edition.
Textbook: Daniel Bump, Automorphic Forms and Representations, 1st edition.

Dedekind domains, unique factorization of ideals, splitting of primes. Lattice methods, finiteness of the class group, Dirichlet’s unit theorem. Local fields, ramifications, discriminants. Zeta and \(L\)-functions, analytic class number formula. Adeles and ideles. Statements of class field theory and the Chebotarev density theorem.

Textbook: Jürgen Neukirch, Algebraic Number Theory, 1st edition.

Basic arithmetic of \(p\)-adic numbers and the classification of quadratic forms over the field of rational numbers (Hasse-Minkowski theorem). Dirichlet’s proof of the theorem on arithmetic progressions and also basic theory of modular forms. Instruction and practice in written and oral communication provided.

Textbook: Jean-Pierre Serre, A Course in Arithmetic, 1st edition.
Textbook: Fernando Gouvêa, \(p\)-adic Numbers: An Introduction, 1st edition.

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 analytic varieties; and examples with emphasis on algebraic curves and surfaces. Introduction to the language of schemes and properties of morphisms.

Textbook: Ravi Vakil, The Rising Sea: Foundations of Algebraic Geometry, 8 Sept 2024 draft.

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.

Textbook: Michael Atiyah and Ian Macdonald, Introduction to Commutative Algebra, 1st edition.
Textbook: Allen Altman and Steven Kleiman, A Term of Commutative Algebra, 1st edition.

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.

Textbook: Michael Artin, Algebra, 2nd edition.

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.

Textbook: Michael Artin, Algebra, 2nd edition.

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.

Textbook: Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach, 1st edition.

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.

Textbook: Michael Sipser, Theory of Computation, 3rd edition.

The power and sources of randomness in computation. Connections and applications to computational complexity, computational learning theory, cryptography and combinatorics. Topics include: probabilistic proofs, uniform generation and approximate counting, Fourier analysis of Boolean functions, computational learning theory, expander graphs, pseudorandom generators, derandomization.

Textbook: Salil Vadhan, Pseudorandomness, 1st edition.
Textbook: Ryan O’Donnell, Analysis of Boolean Functions, 1st edition.

Covers discrete geometry and algorithms underlying the reconfiguration of foldable structures, with applications to robotics, manufacturing, and biology. Linkages made from one-dimensional rods connected by hinges: constructing polynomial curves, characterizing rigidity, characterizing unfoldable versus locked, protein folding. Folding two-dimensional paper (origami): characterizing flat foldability, algorithmic origami design, one-cut magic trick. Unfolding and folding three-dimensional polyhedra: edge unfolding, vertex unfolding, gluings, Alexandrov’s Theorem, hinged dissections.

Textbook: Erik Demaine and Joseph O’Rourke, Geometric Folding Algorithms: Linkages, Origami, Polyhedra, 1st edition.

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.

Textbook: Rajeev Motwani and Prabhakar Raghavan, Randomized Algorithms, 1st edition.
Textbook: William Feller, An Introduction to Probability Theory and its Applications, Volume 1, 3rd edition.

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.

Textbook: Thomas Cormen, Charles Leiserson, Ronald Rivest, and Clifford Stein, Introduction to Algorithms, 4th edition.

Applications of algebra to combinatorics. Topics include walks in graphs, the Radon transform, groups acting on posets, Young tableaux, electrical networks.

Range of topics in knot theory and \(3\)-manifolds. A particular emphasis will be on developing geometric intuition about algebraic invariants like the Alexander polynomial or homology. Instruction and practice in written and oral communication provided.

Textbook: Dale Rolfsen, Knots and Links, 1st edition.
Textbook: W. B. Raymond Lickorish, An Introduction to Knot Theory, 1st edition.
Textbook: Charles Livingston, Knot Theory, 1st edition.

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.

Textbook: James Munkres, Topology, 2nd edition.
Textbook: Allen Hatcher, Algebraic Topology, 1st edition.

Second part of a two-subject sequence. Covers variable coefficient elliptic, parabolic and hyperbolic partial differential equations. Emphasis on topics and applications of harmonic analysis. Topics include the interactions of convolution and differentiation with the Fourier transform, central limit theorem, the circle method; symmetry and its applications to stationary phase arguments, Weyl equidistribution, Sobolev inequalities, and dynamics; orthogonal projections and group homomorphisms in Fourier transform, geometric inequalities, and large sieve in analytic number theory; using dyadic scales in interpolation, Calderon-Zygmund theory, and the multipole algorithm; and bounding linear operators with the Hardy-Littlewood Sobolev inequality, the \(TT^{*}\) method, and the Strichartz inequality.

Provides a rigorous introduction to Lebesgue’s theory of measure and integration. Covers material that is essential in analysis, probability theory, and differential geometry.

Covers fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchanging of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs.

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.

Fundamentals of deep learning, including both theory and applications. Topics include neural net architectures (MLPs, CNNs, RNNs, graph nets, transformers), geometry and invariances in deep learning, backpropagation and automatic differentiation, learning theory and generalization in high-dimensions, and applications to computer vision, natural language processing, and robotics.

Textbook: Antonio Torralba, Phillip Isola, and William Freeman, Foundations of Computer Vision, 1st edition.

Provides an introduction to the design of digital systems and computer architecture. Emphasizes expressing all hardware designs in a high-level hardware description language and synthesizing the designs. Topics include combinational and sequential circuits, instruction set abstraction for programmable hardware, single-cycle and pipelined processor implementations, multi-level memory hierarchies, virtual memory, exceptions and I/O, and parallel systems.

Introduction to C and assembly language for students coming from a Python background (6.100A). Studies the C language, focusing on memory and associated topics including pointers, how different data structures are stored in memory, the stack, and the heap in order to build a strong understanding of the constraints involved in manipulating complex data structures in modern computational systems. Studies assembly language to facilitate a firm understanding of how high-level languages are translated to machine-level instructions.

Introduces fundamental principles and techniques of software development: how to write software that is safe from bugs, easy to understand, and ready for change. Topics include specifications and invariants; testing, test-case generation, and coverage; abstract data types and representation independence; design patterns for object-oriented programming; concurrent programming, including message passing and shared memory concurrency, and defending against races and deadlock; and functional programming with immutable data and higher-order functions. Includes weekly programming exercises and larger group programming projects.

Introduces fundamental concepts of programming. Designed to develop skills in applying basic methods from programming languages to abstract problems. Topics include programming and Python basics, computational concepts, software engineering, algorithmic techniques, data types, and recursion. Lab component consists of software design, construction, and implementation of design.

Develops foundational skills in programming and in computational modeling. Covers widely used programming concepts in Python, including mutability, function objects, and object-oriented programming. Introduces algorithmic complexity and some common libraries. Throughout, demonstrates using computation to help understand real-world phenomena; topics include optimization problems, building simulations, and statistical modeling.

Topics on the engineering of computer software and hardware systems: techniques for controlling complexity; strong modularity using client-server design, operating systems; performance, networks; naming; security and privacy; fault-tolerant systems, atomicity and coordination of concurrent activities, and recovery; impact of computer systems on society. Case studies of working systems and readings from the current literature provide comparisons and contrasts. Includes a single, semester-long design project. Students engage in extensive written communication exercises.

Textbook: Jerome Saltzer and M. Frans Kaashoek, Principles of Computer System Design: An Introduction, 1st edition.

Propositional and predicate logic. Zermelo-Fraenkel set theory. Ordinals and cardinals. Axiom of choice and transfinite induction. Elementary model theory: completeness, compactness, and Löwenheim-Skolem theorems. Gödel’s incompleteness theorem.

Quantitative introduction to the physics of planets, stars, galaxies and our universe, from origin to ultimate fate, with emphasis on the physics tools and observational techniques that enable our understanding. Topics include our solar system, extrasolar planets; our Sun and other “normal” stars, star formation, evolution and death, supernovae, compact objects (white dwarfs, neutron stars, pulsars, stellar-mass black holes); galactic structure, star clusters, interstellar medium, dark matter; other galaxies, quasars, supermassive black holes, gravitational waves; cosmic large-scale structure, origin, evolution and fate of our universe, inflation, dark energy, cosmic microwave background radiation, gravitational lensing, 21cm tomography.

Introduction to electromagnetism and electrostatics: electric charge, Coulomb’s law, electric structure of matter; conductors and dielectrics. Concepts of electrostatic field and potential, electrostatic energy. Electric currents, magnetic fields and Ampere’s law. Magnetic materials. Time-varying fields and Faraday’s law of induction. Basic electric circuits. Electromagnetic waves and Maxwell’s equations.

Textbook: Sen-Ben Liao, Peter Dourmashkin, and John Belcher, Introduction to Electricity and Magnetism, 1st edition.

Elementary mechanics, presented in greater depth than in 8.01. Newton’s laws, concepts of momentum, energy, angular momentum, rigid body motion, and non-inertial systems.

Textbook: Daniel Kleppner and Robert Kolenkow, An Introduction to Mechanics, 2nd edition.

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.

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.

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.

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.

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.

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’Hôpital’s rule. Improper integrals. Infinite series: geometric, \(p\)-harmonic, simple comparison tests, power series for some elementary functions.

A continuation of 21M.301, including chromatic harmony and modulation, a more extensive composition project, keyboard laboratory, and musicianship laboratory.

Explores Western diatonic music through regular composition and analysis assignments. Engages a broad range of historical periods, traditions, and individuals. Topics include rhythm and meter, harmony and counterpoint within a single key, and a brief overview of form and modulation. Individual skills are addressed through a variety of approaches, including the required piano and sight singing labs. Local musicians perform final composition projects. Students should be proficient in reading Western staff notation in at least one clef and have experience with key signatures and scales.

Accelerated half-semester study of the fundamentals of Western music. Requires ability to read Western staff notation in at least one clef. Coverage includes intervals, triads, major and minor keys, basic musical analysis over a variety of idioms in Western music. Also emphasizes developing the ear, voice, and keyboard skills.

Practice in a particular compositional technique not normally covered in the Harmony and Counterpoint or Musical Composition sequences. Possible topics include Renaissance counterpoint, fugue, ragtime, or indeterminacy.

An introduction to diverse musical traditions of the world. Music from a wide range of geographical areas is studied in terms of structure, performance practice, social use, aesthetics, and cross-cultural contact. Includes music making, live demonstrations by guest artists, and ethnographic research projects.

Provides a broad overview of Western music from the Middle Ages to the 21st century, with emphasis on late baroque, classical, romantic, and modernist styles. Designed to enhance the musical experience by developing listening skills and an understanding of diverse forms and genres. Major composers and works placed in social and cultural contexts.

The first term streamlined sequence. Designed for students who have conversational skills without a corresponding level of literacy.

Textbook: Yan Liu, Jingjing Ji, Grace Wu, and Min-Min Liang, Modern Chinese for Heritage Beginners: Stories about US, 1st edition.

A survey of the scientific study of human nature, including how the mind works, and how the brain supports the mind. Topics include the mental and neural bases of perception, emotion, learning, memory, cognition, child development, personality, psychopathology, and social interaction. Consideration of how such knowledge relates to debates about nature and nurture, free will, consciousness, human differences, self, and society.

Discusses core principles including chemical bonding and molecular interactions, protein structure/function and basic thermodynamics, how information flows in the cell, genetics, tools for studying and manipulating genetic material, sequencing, cell biology, evolution, and how the body fights off harmful disease-causing agents. 7.012 synthesizes the core principles into a coherent whole by following the scientific narrative of recent major advances in medicine. Students then complete a final project, with a video and written components, in which they explore an additional medical breakthrough that they choose from a set of options.

Introduction to mathematical modeling of computational problems, as well as common algorithms, algorithmic paradigms, and data structures used to solve these problems. Emphasizes the relationship between algorithms and programming, and introduces basic performance measures and analysis techniques for these problems.

Pressing issues in archaeology as an anthropological science. Stresses the natural science and engineering methods archaeologists use to address these issues. Reconstructing time, space, and human ecologies provides one focus; materials technologies that transform natural materials to material culture provide another. Topics include 14C dating, ice core and palynological analysis, GIS and other remote sensing techniques for site location, organic residue analysis, comparisons between Old World and New World bronze production, invention of rubber by Mesoamerican societies, analysis and conservation of Dead Sea Scrolls.

Introduction to chemistry, with emphasis on basic principles of atomic and molecular electronic structure, thermodynamics, acid-base and redox equilibria, chemical kinetics, and catalysis. Introduction to the chemistry of biological, inorganic, and organic molecules.

Textbook: David Oxtoby, H. Pat Gillis, and Laurie Butler, Principles of Modern Chemistry, 8th edition.

Graduate class. Function spaces, additive set functions, outer measure; measurable functions, integration.

Textbook: Halsey Royden and Patrick Fitzpatrick, Real Analysis, 4th edition.

Independent study. Topics in analysis chosen from inverse and implicit function theorems, differentiation, integration, infinite series, series of functions, and elementary functional analysis.

Textbook: Robert G. Bartle, Introduction to Real Analysis, 4th edition.
Textbook: Walter Rudin, Principles of Mathematical Analysis, 3rd edition.

Complex number plane, analytic functions of a complex variable, integration, power series, calculus of residues, conformal representation, applications of analytic function theory.

Textbook: Ruel Vance Churchill and James Ward Brown, Complex Variables and Applications, 8th edition.

Basic concepts and techniques of real analysis, including proofs by induction and contradiction, the number system, functions of real variables, sets, series and sequences, cardinality, continuity, convergence, and elementary topology.

Textbook: Steven R. Lay, Analysis with an Introduction to Proof, 5th edition.

Topics in real-valued functions of several variables including directional derivatives, implicit functions, gradient, Taylor’s Theorem, maxima, minima, and Lagrange multipliers. Differential calculus of vector-valued functions including chain rule and Inverse Function Theorem. Multiple integrals, line integrals, surface integrals, Stokes’ and Green’s Theorems.

Textbook: Bruce H. Edwards and Ron Larson, Multivariable Calculus, 11th edition.

Introduction to topology including topics selected from: topological spaces, mappings, homeomorphisms, metric spaces, surfaces, knots, manifolds, separation properties, compactness and connectedness.

Textbook: Tom Richmond, General Topology, 1st edition.

Introduction to discrete topics. Development of skills in abstraction and generalization. Set theory, functions and relations, mathematical induction, elementary propositional logic, quantification, truth tables, validity; counting techniques, pigeonhole principle, permutations and combinations; recurrence relations and generating functions; elementary graph theory, isomorphisms, trees.

Systems of linear equations, matrix algebra, vector spaces, inner product spaces, linear transformations, eigenvectors, quadratic forms.

Textbook: Ron Larson, Elementary Linear Algebra, 8th edition.

Students will study articles in current mathematical journals or undertake independent investigations in mathematics. Written and oral presentations will be required.

Research for credit. Study of Zagier’s multiple zeta functions, combinatorial numbers, and Dirichlet series studied by Choi, Shimura, and others.

Research for credit. Study of analytical and complex methods and Dirichlet series studied by Choi.

Research for credit. Study of discrete logarithms, quadratic reciprocity, number-theoretic functions, and Dirichlet series studied by Choi.

Research for credit. Introduction to the Riemann, Hurwitz, and Lerch zeta functions; Dirichlet series; and Dirichlet \(L\)-functions and Dirichlet characters. Study of Dirichlet series studied by Choi.

Problem-solving tools and techniques, with an emphasis on mathematical reasoning, algorithmic techniques, and computational methods. Techniques and tools are applied to (research) areas of interest to enrolled students, in the context of a project involving program design and implementation. The course is taught jointly by mathematics and computer science faculty.

A study of object-oriented software development and programming concepts including inheritance, polymorphism, stack, queue, list, and introduction to recursion and their applications, including user-interface design.

A study of the algorithmic approach to the analysis of problems and their computational solutions, using a high-level structured language.

This is the first half of a year-long course in calculus-based physics suggested for students in the physical sciences and mathematics. Definitions, concepts, and problem solving will be emphasized. Topics include kinematics, dynamics, energy, conservation laws, rotation, harmonic motion, mechanical waves and thermodynamics.

Textbook: Bruce Sherwood and Ruth Chabay, Matters & Interactions, 4th edition.

Students perform physics experiments in mechanics and thermodynamics which stress the fundamental definitions and laws developed in the lecture course. Students gain experience in computerized data acquisition and data analysis using modern techniques and equipment.

Introductory survey of our universe; from observations of the sun, moon and stars in the sky to our understanding of planets, stars, galaxies and the overall characteristics of the cosmos.

Textbook: Jeffrey Bennett, Megan Donahue, Nicholas Schneider, and Mark Voit, The Cosmic Perspective, 9th edition.

The first half of the standard year-long general chemistry course sequence for science majors and minors.

Laboratory to accompany CHEM 120. One third of each meeting is spent reviewing material from the lecture and the remaining time is used to carry out laboratory investigations. Pre-lab lecture and laboratory meet once each week for three hours per week.

Introductory course in biology that emphasizes evolutionary patterns and processes, diversity of life (bacteria, archaea, protists, plants, fungi, and animals), ecological principles, and conservation and management.

Introductory laboratory in biology for science majors that emphasizes the experimental aspects of evolutionary patterns and processes, diversity of life (bacteria, archaea, protists, plants, fungi, and animals), ecological principles, and conservation and management.

Interdisciplinary writing course to be taken in the junior year. Students will read and write about challenging texts from a number of fields. Each student will produce a substantial research project appropriate to his or her chosen field.

Introductory study of fiction, poetry, and drama demonstrating techniques by which literary artists reflect human experience. Substantial student writing about literature will be required.

A survey of the political, social, cultural, and economic phases of American life since the Civil War.

Textbook: James Roark, Michael Johnson, Francois Furtenberg, Sarah Stage, and Sarah Igo, The American Promise: A History of the United States, 8th edition.