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.

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.

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.

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.

Dedekind domains, unique factorization of ideals, splitting of primes. Lattice methods, finiteness of the class group, Dirichlet’s unit theorzem. 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.

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.

Basic arithmetics 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.

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.

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

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.

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.

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.

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.

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.

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.

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.

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, interchange of limit operations. Shows the utility of abstract concepts and teaches understanding and construction of proofs.

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

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

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

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.

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.

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

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.

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.

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

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

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

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

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.

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.

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.

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.

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.

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