uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 19: Roth’s Theorem II: Fourier Analytic Proof in the Integers


Massachusetts Institute of Technology

addressCountry: US

addressLocality: Cambridge

addressRegion: Massachusetts

type: Organization

Yufei Zhao

honorificPrefix: Prof.

type: Person

keywords:
Mathematics
Probability and Statistics
Applied Mathematics
Discrete Mathematics

cs: Matematika

de: Mathematik

fi: matematiikka

uk: Математика

en: Mathematics

fr: Mathématiques

da: Matematik

es: Matemáticas

nl: Wiskunde

de: Mathematik, Naturwissenschaften

uk: Математика, природничі науки

en: Mathematics, Natural Sciences

MIT OpenCourseWare

type: Service

type: WebContent

@language: en

cs: přihlášení není vyžadováno

de: keine Anmeldung erforderlich

uk: вхід без авторизації

en: no login required

nl: geen login vereist

name: Lecture 2: Forbidding a Subgraph I: Mantel’s Theorem and Turán’s Theorem

cs: Video

de: Video

fi: Video

uk: Відео

en: Video

fr: Vidéo

da: Video

nl: Video

es: Video


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 20: Roth’s Theorem III: Polynomial Method and Arithmetic Regularity


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 21: Structure of Set Addition I: Introduction to Freiman’s Theorem


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 22: Structure of Set Addition II: Groups of Bounded Exponent and Modeling Lemma


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 23: Structure of Set Addition III: Bogolyubov’s Lemma and the Geometry of Numbers


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 24: Structure of Set Addition IV: Proof of Freiman’s Theorem


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 25: Structure of Set Addition V: Additive Energy and Balog-Szemerédi-Gowers Theorem


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 26: Sum-Product Problem and Incidence Geometry


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 3: Forbidding a Subgraph II: Complete Bipartite Subgraph


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 4: Forbidding a Subgraph III: Algebraic Constructions


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 5: Forbidding a Subgraph IV: Dependent Random Choice


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 6: Szemerédi’s Graph Regularity Lemma I: Statement and Proof


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 7: Szemerédi’s Graph Regularity Lemma II: Triangle Removal Lemma


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 8: Szemerédi’s Graph Regularity Lemma III: Further Applications


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 9: Szemerédi’s Graph Regularity Lemma IV: Induced Removal Lemma


keywords:
Mathematics
Probability and Statistics

uk: Математика, природничі науки

description: In this course, we will mostly be studying Fourier analysis of Boolean functions, which is a useful tool in theoretical computer science, combinatorics, and more. We will start with basic concepts such as influences, noise sensitivity, and hypercontractivity and some basic results in the area.

uk: вхід без авторизації

name: Topics in Combinatorics: Analysis of Boolean Functions

cs: Kurz

de: Kurs

fi: Kurssi

uk: Курс

en: Course

fr: Cours

da: Kursus

nl: Cursus

es: Curso


Graph Theory and Additive Combinatorics

name: Lecture 1: A bridge between graph theory and additive combinatorics

name: Lecture 10: Szemerédi’s Graph Regularity Lemma V: Hypergraph Removal and Spectral Proof

name: Lecture 11: Pseudorandom Graphs I: Quasirandomness

name: Lecture 12: Pseudorandom Graphs II: Second Eigenvalue

name: Lecture 13: Sparse Regularity and the Green-Tao Theorem

name: Lecture 14: Graph Limits I: Introduction

name: Lecture 15: Graph Limits II: Regularity and Counting

name: Lecture 16: Graph Limits III: Compactness and Applications

name: Lecture 17: Graph Limits IV: Inequalities between Subgraph Densities

name: Lecture 18: Roth’s Theorem I: Fourier Analytic Proof over Finite Field

name: Lecture 19: Roth’s Theorem II: Fourier Analytic Proof in the Integers

uk: Математика, природничі науки

description:
This course examines classical and modern developments in graph theory and additive combinatorics, with a focus on topics and themes that connect the two subjects. The course also introduces students to current research topics and open problems.

This course was previously numbered 18.217.

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 2: Forbidding a Subgraph I: Mantel’s Theorem and Turán’s Theorem


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 20: Roth’s Theorem III: Polynomial Method and Arithmetic Regularity


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 21: Structure of Set Addition I: Introduction to Freiman’s Theorem


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 22: Structure of Set Addition II: Groups of Bounded Exponent and Modeling Lemma


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 23: Structure of Set Addition III: Bogolyubov’s Lemma and the Geometry of Numbers


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 24: Structure of Set Addition IV: Proof of Freiman’s Theorem


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 25: Structure of Set Addition V: Additive Energy and Balog-Szemerédi-Gowers Theorem


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 26: Sum-Product Problem and Incidence Geometry


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 3: Forbidding a Subgraph II: Complete Bipartite Subgraph


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 4: Forbidding a Subgraph III: Algebraic Constructions


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 5: Forbidding a Subgraph IV: Dependent Random Choice


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 6: Szemerédi’s Graph Regularity Lemma I: Statement and Proof


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 7: Szemerédi’s Graph Regularity Lemma II: Triangle Removal Lemma


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 8: Szemerédi’s Graph Regularity Lemma III: Further Applications


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 9: Szemerédi’s Graph Regularity Lemma IV: Induced Removal Lemma


Probabilistic Methods in Combinatorics

name: Threshold for a Random Graph to Contain a Triangle

name: Bounded Differences Inequality (aka Azuma-Hoeffding Inequality)

name: Existence of Graphs with High Girth and High Chromatic Number

name: Extremal Set Theory: Intersecting Families

uk: Математика, природничі науки

description: This course is a graduate-level introduction to the probabilistic methods, a fundamental and powerful technique in combinatorics and theoretical computer science. The essence of the approach is to show that some combinatorial object exists and prove that a certain random construction works with positive probability. The course focuses on methodology as well as combinatorial applications.

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


Crossing Number Inequality

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


Extremal Set Theory: Sperner's Theorem

uk: Математика, природничі науки

uk: вхід без авторизації


Independent Sets and Turán's Theorem

uk: Математика, природничі науки

uk: вхід без авторизації


Large Bipartite Subgraph

uk: Математика, природничі науки

uk: вхід без авторизації


Linearity of Expectations

uk: Математика, природничі науки

uk: вхід без авторизації


Lower Bounds to Ramsey Numbers

uk: Математика, природничі науки

uk: вхід без авторизації


Markov, Chebyshev, and Chernoff

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


Geometry and Quantum Field Theory

keywords:
Science
Physics
Mathematics
Applied Mathematics
Quantum Mechanics
Algebra and Number Theory
Topology and Geometry

de: Physik

uk: Фізика

en: Physics

uk: Математика, природничі науки

de: Physik, Astronomie

fi: fysiikka, tähtitiede

uk: Фізика, Астрономія

en: Physics, Astronomy

fr: Physique, Astronomie

da: Fysik, Astronomi

es: Física, Astronomía

nl: Natuurkunde, Astronomie

description: The main goal of this course is to present the basic ideas of quantum field theory (QFT) in a completely rigorous and mathematical way. Topics of the course include generalities on classical and quantum mechanics and field theory, 0-dimensional QFT, 1-dimensional QFT, *d*\-dimensional QFT, supergeometry and field theories with fermions, and an introduction to 2-dimensional conformal field theory.

uk: вхід без авторизації


keywords:
Mathematics
Applied Mathematics
Linear Algebra
Differential Equations

name: Linear Partial Differential Equations, Midterm Exam Solution

uk: Математика, природничі науки

description: This course provides students with the basic analytical and computational tools of linear partial differential equations (PDEs) for practical applications in science engineering, including heat / diffusion, wave, and Poisson equations. Analytics emphasize the viewpoint of linear algebra and the analogy with finite matrix problems. Numerics focus on finite-difference and finite-element techniques to reduce PDEs to matrix problems. The [Julia Language](http://julialang.org) (a free, open-source environment) is introduced and used in homework for simple examples.

uk: вхід без авторизації

name: Linear Partial Differential Equations: Analysis and Numerics


uk: Математика, природничі науки

description: This is a resource related to Midterm Exam.

uk: вхід без авторизації

name: Linear Partial Differential Equations, Midterm Exam

cs: Cvičení

de: Übung

fi: Harjoitus

uk: Практика

en: Drill and Practice

fr: Apprentissage par Répétition

da: Øvelse

nl: Oefenen

es: Práctica


uk: Математика, природничі науки

description: This is a resource related to Midterm Exam Solution.

uk: вхід без авторизації


Linear Partial Differential Equations

Matthew Hancock

honorificPrefix: Dr.

keywords:
Mathematics
Applied Mathematics
Differential Equations

uk: Математика, природничі науки

description: This course covers the classical partial differential equations of applied mathematics: diffusion, Laplace/Poisson, and wave equations. It also includes methods and tools for solving these PDEs, such as separation of variables, Fourier series and transforms, eigenvalue problems, and Green's functions.

uk: вхід без авторизації


final.pdf

uk: Математика, природничі науки

uk: вхід без авторизації


finalprep.pdf

uk: Математика, природничі науки

description: This resource contains an outline of material 1 expected to know for final and some extra practice problems for the final.

uk: вхід без авторизації


finalsoln.pdf

uk: Математика, природничі науки

description: This file contains the solutions to the questions asked in final exam.

uk: вхід без авторизації


given.pdf

uk: Математика, природничі науки

description: This resource contains final exam questions.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This resource contains some review problems based on heat.

uk: вхід без авторизації


uk: Математика, природничі науки

description: Solutions to the review problems.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This resource contains test problems.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This resource contains test problems from a previous year.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This resource contains solutions to the test from a previous year based on heat pr.

uk: вхід без авторизації


uk: Математика, природничі науки

description: Solutions to the test problems.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This file contains the solutions to the questions asked in the test.

uk: вхід без авторизації


keywords:
Mathematics
Applied Mathematics
Algebra and Number Theory
Discrete Mathematics

uk: Математика, природничі науки

description: This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic.

uk: вхід без авторизації

name: Undergraduate Seminar in Discrete Mathematics


keywords:
Mathematics
Applied Mathematics
Differential Equations
Mathematical Analysis

uk: Математика, природничі науки

description: Advanced Analytic Methods in Science and Engineering is a comprehensive treatment of the advanced methods of applied mathematics. It was designed to strengthen the mathematical abilities of graduate students and train them to think on their own.

uk: вхід без авторизації


uk: Математика, природничі науки

description: Examination problems with solutions.

name: Advanced Analytic Methods in Science and Engineering

uk: вхід без авторизації


uk: Математика, природничі науки

description: The focus of the course is the concepts and techniques for solving the partial differential equations (PDE) that permeate various scientific disciplines. The emphasis is on nonlinear PDE. Applications include problems from fluid dynamics, electrical and mechanical engineering, materials science, quantum mechanics, etc.

uk: вхід без авторизації

name: Advanced Partial Differential Equations with Applications


18.306 Problem List

uk: Математика, природничі науки

description: A list of problems on partial differential equations that may be assigned in problem sets or exams.

uk: вхід без авторизації


Exam Number 01

uk: Математика, природничі науки

description: Exam on partial differential equations (PDEs), dispersive waves and modulation, discontinuous coefficients in linear 1st order PDEs, and initial values for a kinematic wave.

uk: вхід без авторизації


Exam Number 02

uk: Математика, природничі науки

description: Exam on partial differential equations, Green’s function for the heat equation in the infinite line, Green’s function for the 1-D heat equation, and heat equation half plane Dirichelet signaling.

uk: вхід без авторизації


Integral Equations

keywords:
Science
Physics
Mathematics
Applied Mathematics
Theoretical Physics
Differential Equations
Calculus
Mathematical Analysis

uk: Математика, природничі науки

uk: Фізика, Астрономія

description: This course emphasizes concepts and techniques for solving integral equations from an applied mathematics perspective. Material is selected from the following topics: Volterra and Fredholm equations, Fredholm theory, the Hilbert-Schmidt theorem; Wiener-Hopf Method; Wiener-Hopf Method and partial differential equations; the Hilbert Problem and singular integral equations of Cauchy type; inverse scattering transform; and group theory. Examples are taken from fluid and solid mechanics, acoustics, quantum mechanics, and other applications.

uk: вхід без авторизації


keywords:
Communication
Mathematics
Probability and Statistics
Social Science
Applied Mathematics
Discrete Mathematics

de: Sozialwissenschaften/Soziologie

uk: Суспільні науки та соціологія

en: Social Sciences/Sociology

uk: Математика, природничі науки

de: Rechts-, Wirtschafts- und Sozialwissenschaften

uk: Право, економіка та суспільні науки

en: Law, Economics and Social Sciences

description: This course is an introduction to discrete applied mathematics. Topics include probability, counting, linear programming, number-theoretic algorithms, sorting, data compression, and error-correcting codes. This is a [Communication Intensive in the Major](https://registrar.mit.edu/registration-academics/academic-requirements/communication-requirement/ci-m-subjects) (CI-M) course, and thus includes a writing component.

uk: вхід без авторизації

name: Principles of Discrete Applied Mathematics


18.310 Exam 1 practice questions

uk: Суспільні науки та соціологія

uk: Математика, природничі науки

uk: Право, економіка та суспільні науки

description: This file contains the information regarding 18.310 exam1 practice questions.

uk: вхід без авторизації


18.310 Exam 2 practice questions

uk: Суспільні науки та соціологія

uk: Математика, природничі науки

uk: Право, економіка та суспільні науки

description: This file contains the information regarding 18.310 exam2 practice questions.

uk: вхід без авторизації


Principles of Applied Mathematics

uk: Математика, природничі науки

description: 18.311 Principles of Continuum Applied Mathematics covers fundamental concepts in continuous applied mathematics, including applications from traffic flow, fluids, elasticity, granular flows, etc. The class also covers continuum limit; conservation laws, quasi-equilibrium; kinematic waves; characteristics, simple waves, shocks; diffusion (linear and nonlinear); numerical solution of wave equations; finite differences, consistency, stability; discrete and fast Fourier transforms; spectral methods; transforms and series (Fourier, Laplace). Additional topics may include sonic booms, Mach cone, caustics, lattices, dispersion, and group velocity.

uk: вхід без авторизації


Combinatorial Analysis

keywords:
Mathematics
Applied Mathematics
Discrete Mathematics

name: Combinatorial Analysis, Practice Final Exam

name: Combinatorial Analysis, Solutions to Practice Final Exam

name: Combinatorial Analysis, Practice Midterm 1

name: Combinatorial Analysis, Solutions to Practice Midterm 1

name: Combinatorial Analysis, Practice Midterm 2

name: Combinatorial Analysis, Solutions to Practice Midterm 2

uk: Математика, природничі науки

description: This course analyzes combinatorial problems and methods for their solution. Topics include: enumeration, generating functions, recurrence relations, construction of bijections, introduction to graph theory, network algorithms, and extremal combinatorics.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This resource contains the information regarding Practice Final Exam.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This resource contains the information regarding Solutions to Practice Final Exam.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This resource contains the information regarding Practice Midterm 1.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This resource contains the information regarding Solutions to Practice Midterm 1.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This resource contains the information regarding Practice Midterm 2.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This resource contains the information regarding Solutions to Practice Midterm 2.

uk: вхід без авторизації


keywords:
Mathematics
Algebra and Number Theory
Topology and Geometry
Discrete Mathematics

uk: Математика, природничі науки

description: This is a graduate-level course in combinatorial theory. The content varies year to year, according to the interests of the instructor and the students. The topic of this course is hyperplane arrangements, including background material from the theory of posets and matroids.

uk: вхід без авторизації

name: Combinatorial Theory: Hyperplane Arrangements


uk: Математика, природничі науки

description: This course serves as an introduction to major topics of modern enumerative and algebraic combinatorics with emphasis on partition identities, young tableaux bijections, spanning trees in graphs, and random generation of combinatorial objects. There is some discussion of various applications and connections to other fields.

uk: вхід без авторизації

name: Combinatorial Theory: Introduction to Graph Theory, Extremal and Enumerative Combinatorics


Topics in Algebraic Combinatorics

keywords:
Mathematics
Algebra and Number Theory
Linear Algebra
Discrete Mathematics

uk: Математика, природничі науки

description: The course consists of a sampling of topics from algebraic combinatorics. The topics include the matrix-tree theorem and other applications of linear algebra, applications of commutative and exterior algebra to counting faces of simplicial complexes, and applications of algebra to tilings.

uk: вхід без авторизації


Geometric Combinatorics

keywords:
Engineering
Computer Science
Mathematics
Algorithms and Data Structures
Algebra and Number Theory
Topology and Geometry
Discrete Mathematics

cs: Technické vědy

de: Ingenieurwissenschaften

fi: tekniikka

uk: Інженерія

en: Engineering Sciences

fr: Ingénierie

da: Ingeniørvidenskab

es: Ingeniería

nl: Engineering

cs: Informatika

de: Informatik

fi: tietojenkäsittelytiede

uk: Комп'ютерні науки

en: Computer Science

fr: Informatique

da: Datalogi

es: Ciencias de la Computación

nl: Informatica

uk: Математика, природничі науки

description: This course offers an introduction to discrete and computational geometry. Emphasis is placed on teaching methods in combinatorial geometry. Many results presented are recent, and include open (as yet unsolved) problems.

uk: вхід без авторизації


keywords:
Mathematics
Applied Mathematics
Computation

uk: Математика, природничі науки

description: This class covers the mathematics of inverse problems involving waves, with examples taken from reflection seismology, synthetic aperture radar, and computerized tomography.

uk: вхід без авторизації

name: Topics in Applied Mathematics: Waves and Imaging


Wavelets, Filter Banks and Applications

keywords:
Engineering
Mathematics
Electrical Engineering
Applied Mathematics
Signal Processing
Algebra and Number Theory
Differential Equations

de: Elektrotechnik und Informationstechnik

uk: Електротехніка та інформаційні технології

en: Electrical Engineering and Information Engineering

uk: Математика, природничі науки

description: Wavelets are localized basis functions, good for representing short-time events. The coefficients at each scale are filtered and subsampled to give coefficients at the next scale. This is Mallat's pyramid algorithm for multiresolution, connecting wavelets to filter banks. Wavelets and multiscale algorithms for compression and signal/image processing are developed. Subject is project-based for engineering and scientific applications.

uk: вхід без авторизації


Introduction to Numerical Analysis

keywords:
Engineering
Computer Science
Mathematics
Systems Engineering
Algorithms and Data Structures
Numerical Simulation
Linear Algebra
Differential Equations
Mathematical Analysis

uk: Комп'ютерні науки

uk: Математика, природничі науки

description: This course analyzed the basic techniques for the efficient numerical solution of problems in science and engineering. Topics spanned root finding, interpolation, approximation of functions, integration, differential equations and direct and iterative methods in linear algebra.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This course analyzed the basic techniques for the efficient numerical solution of problems in science and engineering. Topics spanned root finding, interpolation, approximation of functions, integration, differential equations, direct and iterative methods in linear algebra.

uk: вхід без авторизації


Introduction to Numerical Methods

keywords:
Engineering
Computer Science
Mathematics
Systems Engineering
Algorithms and Data Structures
Numerical Simulation
Linear Algebra

uk: Комп'ютерні науки

uk: Математика, природничі науки

description: This course offers an advanced introduction to numerical analysis, with a focus on accuracy and efficiency of numerical algorithms. Topics include sparse-matrix/iterative and dense-matrix algorithms in numerical linear algebra (for linear systems and eigenproblems), floating-point arithmetic, backwards error analysis, conditioning, and stability. Other computational topics (e.g., numerical integration or nonlinear optimization) are also surveyed.

uk: вхід без авторизації


keywords:
Engineering
Computer Science
Mathematics
Systems Engineering
Algorithms and Data Structures
Numerical Simulation
Differential Equations
Mathematical Analysis

uk: Комп'ютерні науки

uk: Математика, природничі науки

description: This graduate-level course is an advanced introduction to applications and theory of numerical methods for solution of differential equations. In particular, the course focuses on physically-arising partial differential equations, with emphasis on the fundamental ideas underlying various methods.

uk: вхід без авторизації

name: Numerical Methods for Partial Differential Equations


Parallel Computing

keywords:
Engineering
Computer Science
Mathematics
Applied Mathematics
Computation
Algorithms and Data Structures
Theory of Computation
Computer Design and Engineering
Linear Algebra

uk: Комп'ютерні науки

uk: Математика, природничі науки

description: This is an advanced interdisciplinary introduction to applied parallel computing on modern supercomputers. It has a hands-on emphasis on understanding the realities and myths of what is possible on the world's fastest machines. We will make prominent use of the [Julia Language](http://julialang.org/), a free, open-source, high-performance dynamic programming language for technical computing.

uk: вхід без авторизації


Infinite Random Matrix Theory

keywords:
Science
Physics
Mathematics
Probability and Statistics
Theoretical Physics
Algebra and Number Theory
Linear Algebra
Differential Equations

uk: Математика, природничі науки

uk: Фізика, Астрономія

description: In this course on the mathematics of infinite random matrices, students will learn about the tools such as the Stieltjes transform and Free Probability used to characterize infinite random matrices.

uk: вхід без авторизації


Nonlinear Dynamics I: Chaos

keywords:
Mathematics
Computation
Linear Algebra
Differential Equations

uk: Математика, природничі науки

description: This course provides an introduction to nonlinear dynamics and chaos in dissipative systems. The content is structured to be of general interest to undergraduates in engineering and science.

uk: вхід без авторизації


Midterm: At-home portion

uk: Математика, природничі науки

description: This resource contains information regarding take-home portion of the midterm.

uk: вхід без авторизації


Midterm Practice Problems

uk: Математика, природничі науки

description: This file contains information regarding overview of the midterm and some suggested problems.

uk: вхід без авторизації


keywords:
Science
Physics
Mathematics
Applied Mathematics
Classical Mechanics

uk: Математика, природничі науки

uk: Фізика, Астрономія

description: This course introduces the basic ideas for understanding the dynamics of continuum systems, by studying specific examples from a range of different fields. Our goal will be to explain the general principles, and also to illustrate them via important physical effects. A parallel goal of this course is to give you an introduction to mathematical modeling.

uk: вхід без авторизації

name: Nonlinear Dynamics II: Continuum Systems


Interfacial Phenomena

keywords:
Engineering
Mathematics
Mechanical Engineering
Applied Mathematics
Fluid Mechanics

de: Maschinenbau/Verfahrenstechnik

uk: Машинобудування/Технологія

en: Mechanical Engineering / Process Engineering

uk: Математика, природничі науки

description: This graduate-level course covers fluid systems dominated by the influence of interfacial tension. The roles of curvature pressure and Marangoni stress are elucidated in a variety of fluid systems. Particular attention is given to drops and bubbles, soap films and minimal surfaces, wetting phenomena, water-repellency, surfactants, Marangoni flows, capillary origami and contact line dynamics.

uk: вхід без авторизації


Random Walks and Diffusion

keywords:
Engineering
Mathematics
Mechanical Engineering
Probability and Statistics
Applied Mathematics
Transport Processes
Fluid Mechanics
Chemical Engineering

uk: Машинобудування/Технологія

uk: Хімічна інженерія/Інженерія хімічних процесів

en: Chemical Engineering/Chemical Process Engineering

uk: Математика, природничі науки

description: This graduate-level subject explores various mathematical aspects of (discrete) random walks and (continuum) diffusion. Applications include polymers, disordered media, turbulence, diffusion-limited aggregation, granular flow, and derivative securities.

uk: вхід без авторизації


uk: Машинобудування/Технологія

uk: Математика, природничі науки

description: This resource provides solutions to exam 1 on random walks and diffusion.

uk: вхід без авторизації


uk: Машинобудування/Технологія

uk: Математика, природничі науки

description: An exam on topics multivariate normal random walk, student random walk, and the largest step.

uk: вхід без авторизації


uk: Машинобудування/Технологія

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Машинобудування/Технологія

uk: Математика, природничі науки

description: This resource provides solutions to exam 2 on random walks and diffusion.

uk: вхід без авторизації


uk: Машинобудування/Технологія

uk: Математика, природничі науки

description: An exam including problems on electrochemical?equilibrium, first passage?of?a?set?of?random?walkers, and escape?from?a?symmetric?trap.

uk: вхід без авторизації


uk: Машинобудування/Технологія

uk: Математика, природничі науки

description: This resource contains midterm exam.

uk: вхід без авторизації


uk: Машинобудування/Технологія

uk: Математика, природничі науки

description: This resource contains solution to the midterm exam.

uk: вхід без авторизації


Mathematical Methods in Nanophotonics

keywords:
Engineering
Science
Physics
Mathematics
Nanotechnology
Applied Mathematics
Atomic, Molecular, Optical Physics
Quantum Mechanics
Linear Algebra

uk: Математика, природничі науки

uk: Фізика, Астрономія

description:
Find out what solid-state physics has brought to Electromagnetism in the last 20 years. This course surveys the physics and mathematics of nanophotonics—electromagnetic waves in media structured on the scale of the wavelength.

Topics include computational methods combined with high-level algebraic techniques borrowed from solid-state quantum mechanics: linear algebra and eigensystems, group theory, Bloch's theorem and conservation laws, perturbation methods, and coupled-mode theories, to understand surprising optical phenomena from band gaps to slow light to nonlinear filters.

Note: An earlier version of this course was published on OCW as 18.325 Topics in Applied Mathematics: Mathematical Methods in Nanophotonics, Fall 2005.

uk: вхід без авторизації


2005 Midterm

uk: Математика, природничі науки

uk: Фізика, Астрономія

description: Midterm exam on 2d metallic electromagnetic cavities, reflected and transmitted light, Hermitian matrix, and band diagrams.

uk: вхід без авторизації


2007 Midterm

uk: Математика, природничі науки

uk: Фізика, Астрономія

description: Midterm exam on hexagons, band diagrams, and operators.

uk: вхід без авторизації


2008 Midterm

uk: Математика, природничі науки

uk: Фізика, Астрономія

description: Midterm exam on waveguide gaps, symmetry, and projected bands.

uk: вхід без авторизації


Nonlinear Dynamics and Chaos

keywords:
Mathematics
Applied Mathematics
Topology and Geometry
Differential Equations

uk: Математика, природничі науки

description: This graduate level course focuses on nonlinear dynamics with applications. It takes an intuitive approach with emphasis on geometric thinking, computational and analytical methods and makes extensive use of demonstration software.

uk: вхід без авторизації


uk: Математика, природничі науки

description: Exam Number 1 for 18.385j/2.036j.

uk: вхід без авторизації


uk: Математика, природничі науки

description: Exam Number 2 for 18.385j/2.036j

uk: вхід без авторизації


uk: Математика, природничі науки

description: This resource consists of problem statement on balancing a broom.

uk: вхід без авторизації


uk: Математика, природничі науки

description: Balancing a Broom

uk: вхід без авторизації


uk: Математика, природничі науки

description: This resource consists of Coastline Fractal construction.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This resource consists of problem set for deriving equation for pendulum.

uk: вхід без авторизації


uk: Математика, природничі науки

description: Variable Length Pendulum

uk: вхід без авторизації


Theory of Computation

keywords:
Engineering
Computer Science
Mathematics
Computation
Theory of Computation
Discrete Mathematics

name: 18.404J F2006 Sample Final Exam Solutions

name: Lecture 2: Nondeterminism, Closure Properties, Regular Expressions → Finite Automata

name: Lecture 5: CF Pumping Lemma, Turing Machines

name: Lecture 7: Decision Problems for Automata and Grammars

name: Lecture 1: Introduction, Finite Automata, Regular Expressions

name: Lecture 14: P and NP, SAT, Poly-Time Reducibility

name: Lecture 17: Space Complexity, PSPACE, Savitch's Theorem

name: Lecture 19: Games, Generalized Geography

name: Lecture 22: Provably Intractable Problems, Oracles

name: Lecture 23: Probabilistic Computation, BPP

name: Lecture 24: Probabilistic Computation (cont.)

name: Lecture 25: Interactive Proof Systems, IP

name: Lecture 3: Regular Pumping Lemma, Finite Automata → Regular Expressions, CFGs

name: Lecture 6: TM Variants, Church-Turing Thesis

uk: Комп'ютерні науки

uk: Математика, природничі науки

description: This course emphasizes computability and computational complexity theory. Topics include 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.

uk: вхід без авторизації


uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


Lecture 10: Computation History Method

uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


Lecture 11: Recursion Theorem and Logic

uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


Lecture 12: Time Complexity

uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


Lecture 15: NP-Completeness

uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


Lecture 16: Cook-Levin Theorem

uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


Lecture 18: PSPACE-Completeness

uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


Lecture 20: L and NL, NL = coNL

uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


Lecture 21: Hierarchy Theorems

uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


Lecture 26: coNP ⊆ IP

uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


Lecture 8: Undecidability

uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


Lecture 9: Reducibility

uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


18.404J F2006 Sample Final Exam

uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


18.404J F2001 Sample Midterm Exam

uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


18.404J F2006 Sample Midterm Exam

uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


Lecture 4: Pushdown Automata, CFG ↔ PDA

uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: вхід без авторизації


Advanced Complexity Theory

keywords:
Engineering
Computer Science
Mathematics
Computation
Theory of Computation

uk: Комп'ютерні науки

uk: Математика, природничі науки

description: This graduate-level course focuses on current research topics in computational complexity theory. Topics include: 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.

uk: вхід без авторизації


keywords:
Engineering
Computer Science
Mathematics
Algorithms and Data Structures
Algebra and Number Theory
Discrete Mathematics

uk: Комп'ютерні науки

uk: Математика, природничі науки

description: In this course, we will present the theory of Probabilistically Checkable Proofs (PCPs), and prove some fundamental consequences of it as well as more recent advances. More specifically, the first half of the course will be devoted to the (algebraic) proof of the basic PCP Theorem and basic relation to approximation problems. We will then move on to more advanced topics, such as hardness amplification, the long-code framework, the Unique-Games Conjecture and its implications, and the 2-to-2 Games Theorem.

uk: вхід без авторизації

name: Topics in Theoretical Computer Science: Probabilistically Checkable Proofs


Algorithmic Aspects of Machine Learning

keywords:
Engineering
Computer Science
Mathematics
Applied Mathematics
Algorithms and Data Structures
Theory of Computation
Discrete Mathematics

uk: Комп'ютерні науки

uk: Математика, природничі науки

description: This course is organized around algorithmic issues that arise in machine learning. Modern machine learning systems are often built on top of algorithms that do not have provable guarantees, and it is the subject of debate when and why they work. In this class, we focus on designing algorithms whose performance we can rigorously analyze for fundamental machine learning problems.

uk: вхід без авторизації


Behavior of Algorithms

keywords:
Engineering
Computer Science
Mathematics
Computation
Algorithms and Data Structures
Theory of Computation
Mathematical Analysis

uk: Комп'ютерні науки

uk: Математика, природничі науки

description: This course is a study of Behavior of Algorithms and covers an area of current interest in theoretical computer science. The topics vary from term to term. During this term, we discuss rigorous approaches to explaining the typical performance of algorithms with a focus on the following approaches: smoothed analysis, condition numbers/parametric analysis, and subclassing inputs.

uk: вхід без авторизації


keywords:
Engineering
Computer Science
Mathematics
Applied Mathematics
Algorithms and Data Structures
Theory of Computation
Topology and Geometry
Linear Algebra

uk: Комп'ютерні науки

uk: Математика, природничі науки

description: This course covers a collection of geometric techniques that apply broadly in modern algorithm design.

uk: вхід без авторизації

name: Topics in Theoretical Computer Science: An Algorithmist's Toolkit


Error-Correcting Codes Laboratory

keywords:
Engineering
Computer Science
Mathematics
Computation
Algorithms and Data Structures
Cryptography
Theory of Computation

uk: Комп'ютерні науки

uk: Математика, природничі науки

description: This course introduces students to iterative decoding algorithms and the codes to which they are applied, including Turbo Codes, Low-Density Parity-Check Codes, and Serially-Concatenated Codes. The course will begin with an introduction to the fundamental problems of Coding Theory and their mathematical formulations. This will be followed by a study of Belief Propagation--the probabilistic heuristic which underlies iterative decoding algorithms. Belief Propagation will then be applied to the decoding of Turbo, LDPC, and Serially-Concatenated codes. The technical portion of the course will conclude with a study of tools for explaining and predicting the behavior of iterative decoding algorithms, including EXIT charts and Density Evolution.

uk: вхід без авторизації


keywords:
Engineering
Science
Mathematics
Biological Engineering
Biology
Computation
Computational Biology
Molecular Biology

de: Biotechnologie

uk: Біотехнологія

en: Biotechnology

de: Biologie

uk: Біологія

en: Biology

uk: Математика, природничі науки

description: This course introduces the basic computational methods used to understand the cell on a molecular level. It covers subjects such as the sequence alignment algorithms: dynamic programming, hashing, suffix trees, and Gibbs sampling. Furthermore, it focuses on computational approaches to: genetic and physical mapping; genome sequencing, assembly, and annotation; RNA expression and secondary structure; protein structure and folding; and molecular interactions and dynamics.

uk: вхід без авторизації

name: Introduction to Computational Molecular Biology


Combinatorial Optimization

keywords:
Engineering
Computer Science
Mathematics
Systems Engineering
Computation
Algorithms and Data Structures
Theory of Computation
Systems Optimization

uk: Комп'ютерні науки

en: Mathematics

uk: Математика, природничі науки

description: Combinatorial Optimization provides a thorough treatment of linear programming and combinatorial optimization. Topics include network flow, matching theory, matroid optimization, and approximation algorithms for NP-hard problems.

uk: вхід без авторизації


Quantum Computation

keywords:
Engineering
Science
Computer Science
Physics
Mathematics
Computation
Algorithms and Data Structures
Cryptography
Theory of Computation
Quantum Mechanics
Discrete Mathematics

uk: Комп'ютерні науки

uk: Математика, природничі науки

en: Mathematics, Natural Sciences

uk: Фізика, Астрономія

description: This course provides an introduction to the theory and practice of quantum computation. Topics covered include: physics of information processing, quantum logic, quantum algorithms including Shor's factoring algorithm and Grover's search algorithm, quantum error correction, quantum communication, and cryptography.

uk: вхід без авторизації


midterm.pdf

uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: Фізика, Астрономія

description: This resource deals with midterm exam.

uk: вхід без авторизації

en: no login required


midtermsol.pdf

uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: Фізика, Астрономія

description: This resource contains solutions of midterm exam.

uk: вхід без авторизації


uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: Фізика, Астрономія

description: Quiz problems.

uk: вхід без авторизації


uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: Фізика, Астрономія

description: Solutions to quiz problems.

uk: вхід без авторизації


Probability and Random Variables

keywords:
Mathematics
Probability and Statistics
Discrete Mathematics

name: 18.440 Practice Midterm 2 Partial Solutions

uk: Математика, природничі науки

description: This course introduces students to probability and random variables. Topics include distribution functions, binomial, geometric, hypergeometric, and Poisson distributions. The other topics covered are uniform, exponential, normal, gamma and beta distributions; conditional probability; Bayes theorem; joint distributions; Chebyshev inequality; law of large numbers; and central limit theorem.

uk: вхід без авторизації


18.440 Final Exam Solutions

uk: Математика, природничі науки

description: This resource contains information related to final exam solutions.

uk: вхід без авторизації


Fall 2012 18.440 Final Exam Solutions

uk: Математика, природничі науки

description: This resource contains information related to fall 2012 final exam solutions.

uk: вхід без авторизації


Spring 2014 18.440 Final Exam Solutions

uk: Математика, природничі науки

description: This resource contains information related to spring 2014 final exam solutions.

uk: вхід без авторизації


18.440 Final Exam

uk: Математика, природничі науки

description: This resource contains information related to final exam.

uk: вхід без авторизації


Fall 2012 18.440 Final Exam

uk: Математика, природничі науки

description: This resource contains information related to fall 2012 final exam.

uk: вхід без авторизації


Spring 2014 18.440 Final Exam

uk: Математика, природничі науки

description: This resource contains information related to spring 2014 final exam.

uk: вхід без авторизації


18.440 Midterm 1, Fall 2009

uk: Математика, природничі науки

description: This resource contains information related to midterm 1, fall 2009.

uk: вхід без авторизації


18.440 Midterm 1, Spring 2011

uk: Математика, природничі науки

description: This resource contains information related to midterm 1, spring 2011.

uk: вхід без авторизації


18.440 Midterm 1 Solutions, Spring 2011

uk: Математика, природничі науки

description: This resource contains information related to midterm 1 solutions, spring 2011.

uk: вхід без авторизації


18.440 Midterm 1, Spring 2014

uk: Математика, природничі науки

description: This resource contains information related to midterm 1, spring 2014.

uk: вхід без авторизації


18.440 Midterm 1 Solutions, Spring 2014

uk: Математика, природничі науки

description: This resource contains information related to midterm 1 solutions, spring 2014.

uk: вхід без авторизації


18.440 Midterm 2, Fall 2009

uk: Математика, природничі науки

description: This resource contains information related to midterm 2, fall 2009.

uk: вхід без авторизації


18.440 Midterm 2 Solutions, Fall 2009

uk: Математика, природничі науки

description: This resource contains information related to midterm 2 solutions, fall 2009.

uk: вхід без авторизації


18.440 Midterm 2, Fall 2012

uk: Математика, природничі науки

description: This resource contains information related to midterm 2, fall 2012.

uk: вхід без авторизації


18.440 Midterm 2 Solutions, Fall 2012

uk: Математика, природничі науки

description: This resource contains information related to midterm 2 solutions, fall 2012.

uk: вхід без авторизації


18.440 Midterm 2, Spring 2014

uk: Математика, природничі науки

description: This resource contains information related to midterm 2, spring 2014.

uk: вхід без авторизації


18.440 Midterm 2 Solutions, Spring 2014

uk: Математика, природничі науки

description: This resource contains information related to midterm 2 solutions, spring 2014.

uk: вхід без авторизації


18.440 Midterm 2 Solutions, Fall 2011

uk: Математика, природничі науки

description: This resource contains information related to midterm 2 solutions, fall 2011.

uk: вхід без авторизації


18.440 Midterm 2, Spring 2011

uk: Математика, природничі науки

description: This resource contains information related to midterm 2, spring 2011.

uk: вхід без авторизації


18.440 Midterm 2 Solutions, Spring 2011

uk: Математика, природничі науки

description: This resource contains information related to midterm 2 solutions, spring 2011.

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18.440 Practice Final Exam

uk: Математика, природничі науки

description: This resource contains information related to practice final exam.

uk: вхід без авторизації


18.440 Practice Final Exam Solutions

uk: Математика, природничі науки

description: This resource contains information related to practice final exam solutions.

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18.440 Practice Midterm 1 Solutions

uk: Математика, природничі науки

description: This resource contains information related to practice midterm 1 solutions.

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18.440 Practice Midterm Two

uk: Математика, природничі науки

description: This resource contains information related to practice midterm two.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This resource contains information related to practice midterm 2 partial solutions.

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18.440 Practice Midterm 1

uk: Математика, природничі науки

description: This resource contains information related to practice midterm 1.

uk: вхід без авторизації


Statistics for Applications

keywords:
Mathematics
Probability and Statistics
Applied Mathematics
Discrete Mathematics

uk: Математика, природничі науки

description:
This course offers a broad treatment of statistics, concentrating on specific statistical techniques used in science and industry. Topics include: hypothesis testing and estimation, confidence intervals, chi-square tests, nonparametric statistics, analysis of variance, regression, and correlation.

OCW offers an earlier version of this course, from Fall 2003. This newer version focuses less on estimation theory and more on multiple linear regression models. In addition, a number of Matlab examples are included here.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This resource contains practice test on sample mean, sample standard deviation and Matlab commands.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This resource contains problem test on Poisson distribution and variance analysis.

uk: вхід без авторизації


name: Statistics for Applications Exam 1 Solution

name: Statistics for Applications Exam 2 Solution

name: Statistics for Applications Exam 3 Solution

uk: Математика, природничі науки

description: This course is a broad treatment of statistics, concentrating on specific statistical techniques used in science and industry. Topics include: hypothesis testing and estimation, confidence intervals, chi-square tests, nonparametric statistics, analysis of variance, regression, correlation, decision theory, and Bayesian statistics.

uk: вхід без авторизації


Statistics for Applications Exam 1

uk: Математика, природничі науки

description: This is a file that contains information regarding exam 1.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This is a file that contains information regarding exam 1 solution.

uk: вхід без авторизації


Statistics for Applications Exam 2

uk: Математика, природничі науки

description: This is a file that contains information regarding exam 2.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This is a file that contains information regarding exam 2 solution.

uk: вхід без авторизації


Statistics for Applications Exam 3

uk: Математика, природничі науки

description: This is a file that contains information regarding exam 3.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This is a file that contains information regarding exam 3 solution.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This is a file that contains information regarding exam 3 table.

uk: вхід без авторизації


Introduction to Stochastic Processes

keywords:
Mathematics
Probability and Statistics

uk: Математика, природничі науки

description: This course is an introduction to Markov chains, random walks, martingales, and Galton-Watsom tree. The course requires basic knowledge in probability theory and linear algebra including conditional expectation and matrix.

uk: вхід без авторизації


uk: Математика, природничі науки

description:
This graduate-level course focuses on one-dimensional nonparametric statistics developed mainly from around 1945 and deals with order statistics and ranks, allowing very general distributions.

For multidimensional nonparametric statistics, an early approach was to choose a fixed coordinate system and work with order statistics and ranks in each coordinate. A more modern method, to be followed in this course, is to look for rotationally or affine invariant procedures. These can be based on empirical processes as in computer learning theory.

Robustness, which developed mainly from around 1964, provides methods that are resistant to errors or outliers in the data, which can be arbitrarily large. Nonparametric methods tend to be robust.

uk: вхід без авторизації

name: Topics in Statistics: Nonparametrics and Robustness


keywords:
Engineering
Science
Computer Science
Artificial Intelligence
Mathematics
Probability and Statistics
Cognitive Science
Algorithms and Data Structures

de: Psychologie

uk: Психологія

en: Psychology

uk: Комп'ютерні науки

uk: Математика, природничі науки

uk: Право, економіка та суспільні науки

description: The main goal of this course is to study the generalization ability of a number of popular machine learning algorithms such as boosting, support vector machines and neural networks. Topics include Vapnik-Chervonenkis theory, concentration inequalities in product spaces, and other elements of empirical process theory.

uk: вхід без авторизації

name: Topics in Statistics: Statistical Learning Theory


Probability and Random Variables

name: Spring 2014 18.440 Midterm Exam 1 Solutions

name: Spring 2016 18.600 Midterm Exam 1 Solutions

name: Spring 2017 18.600 Midterm Exam 1 Solutions

name: Spring 2018 18.600 Midterm Exam 1 Solutions

name: Spring 2019 18.600 Midterm Exam 1 Solutions

name: Fall 2012 18.440 Midterm Exam 1 Solutions

name: Fall 2011 18.440 Midterm Exam 1 Solutions

name: Spring 2011 18.440 Midterm Exam 1 Solutions

name: Spring 2014 18.440 Midterm Exam 2 Solutions

name: Spring 2016 18.600 Midterm Exam 2 Solutions

name: Spring 2017 18.600 Midterm Exam 2 Solutions

name: Spring 2018 18.600 Midterm Exam 2 Solutions

name: Spring 2019 18.600 Midterm Exam 2 Solutions

name: Fall 2012 18.440 Midterm Exam 2 Solutions

name: Fall 2011 18.440 Midterm Exam 2 with Solutions

name: Spring 2011 18.440 Midterm Exam 2 Solutions

name: Fall 2019 18.600 Midterm Exam 1 Solutions

name: Fall 2019 18.600 Midterm Exam 2 Solutions

name: 18.600 Practice Final Exam Partial Solutions

de: Mathematik

uk: Математика, природничі науки

description: This course introduces students to probability and random variables. Topics include distribution functions, binomial, geometric, hypergeometric, and Poisson distributions. The other topics covered are uniform, exponential, normal, gamma and beta distributions; conditional probability; Bayes theorem; joint distributions; Chebyshev inequality; law of large numbers; and central limit theorem.

uk: вхід без авторизації

en: Course


Spring 2011 18.440 Final Exam

de: Mathematik, Naturwissenschaften

uk: Математика, природничі науки

uk: вхід без авторизації


Spring 2011 18.440 Final Exam Solutions

uk: Математика, природничі науки

de: keine Anmeldung erforderlich

uk: вхід без авторизації


Fall 2012 18.440 Final Exam

uk: Математика, природничі науки

uk: вхід без авторизації


Fall 2012 18.440 Final Exam Solutions

uk: Математика, природничі науки

uk: вхід без авторизації


Spring 2014 18.440 Final Exam

uk: Математика, природничі науки

uk: вхід без авторизації


Spring 2014 18.440 Final Exam Solutions

uk: Математика, природничі науки

uk: вхід без авторизації


Spring 2016 18.600 Final Exam

uk: Математика, природничі науки

uk: вхід без авторизації


Spring 2016 18.600 Final Exam Solutions

uk: Математика, природничі науки

uk: вхід без авторизації


Spring 2017 18.600 Final Exam

uk: Математика, природничі науки

uk: вхід без авторизації


Spring 2017 18.600 Final Exam Solutions

uk: Математика, природничі науки

uk: вхід без авторизації


Spring 2018 18.600 Final Exam

uk: Математика, природничі науки

uk: вхід без авторизації


Spring 2018 18.600 Final Exam Solutions

uk: Математика, природничі науки

uk: вхід без авторизації


Spring 2019 18.600 Final Exam

uk: Математика, природничі науки

uk: вхід без авторизації


Spring 2019 18.600 Final Exam Solutions

uk: Математика, природничі науки

uk: вхід без авторизації


Fall 2019 18.600 Final Exam

uk: Математика, природничі науки

uk: вхід без авторизації


Fall 2019 18.600 Final Exam Solutions

uk: Математика, природничі науки

uk: вхід без авторизації


Fall 2012 18.440 Midterm Exam 1

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


Spring 2014 18.440 Midterm Exam 1

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Spring 2014 18.440 Midterm Exam 1 Solutions


Spring 2016 18.600 Midterm Exam 1

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Spring 2016 18.600 Midterm Exam 1 Solutions


Spring 2017 18.600 Midterm Exam 1

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Spring 2017 18.600 Midterm Exam 1 Solutions


Spring 2018 18.600 Midterm Exam 1

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Spring 2018 18.600 Midterm Exam 1 Solutions


Spring 2019 18.600 Midterm Exam 1

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Spring 2019 18.600 Midterm Exam 1 Solutions


Fall 2011 18.440 Midterm Exam 1

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Fall 2011 18.440 Midterm Exam 1 Solutions


Spring 2011 18.440 Midterm Exam 1

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Spring 2011 18.440 Midterm Exam 1 Solutions


Fall 2012 18.440 Midterm Exam 2

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


Spring 2014 18.440 Midterm Exam 2

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Spring 2014 18.440 Midterm Exam 2 Solutions


Spring 2016 18.600 Midterm Exam 2

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Spring 2016 18.600 Midterm Exam 2 Solutions


Spring 2017 18.600 Midterm Exam 2

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Spring 2017 18.600 Midterm Exam 2 Solutions


Spring 2018 18.600 Midterm Exam 2

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Spring 2018 18.600 Midterm Exam 2 Solutions


Spring 2019 18.600 Midterm Exam 2

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Spring 2019 18.600 Midterm Exam 2 Solutions


uk: Математика, природничі науки

uk: вхід без авторизації

name: Fall 2011 18.440 Midterm Exam 2 with Solutions


Spring 2011 18.440 Midterm Exam 2

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Spring 2011 18.440 Midterm Exam 2 Solutions


Fall 2019 18.600 Midterm Exam 1

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Fall 2019 18.600 Midterm Exam 1 Solutions


Fall 2019 18.600 Midterm Exam 2

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Fall 2019 18.600 Midterm Exam 2 Solutions


18.600 Practice Final Exam

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: 18.600 Practice Final Exam Partial Solutions


18.600 Fall 2019 story sheet

uk: Математика, природничі науки

uk: вхід без авторизації


Statistics for Applications

name: Lecture 4: Parametric Inference (cont.) and Maximum Likelihood Estimation

name: Lecture 5: Maximum Likelihood Estimation (cont.)

name: Lecture 6: Maximum Likelihood Estimation (cont.) and the Method of Moments

name: Lecture 7: Parametric Hypothesis Testing

name: Lecture 8: Parametric Hypothesis Testing (cont.)

name: Lecture 9: Parametric Hypothesis Testing (cont.)

name: Lecture 11: Parametric Hypothesis Testing (cont.) and Testing Goodness of Fit

name: Lecture 12: Testing Goodness of Fit (cont.)

name: Lecture 19: Principal Component Analysis

name: Lecture 2: Introduction to Statistics (cont.)

name: Lecture 20: Principal Component Analysis (cont.)

name: Lecture 22: Generalized Linear Models (cont.)

name: Lecture 23: Generalized Linear Models (cont.)

name: Lecture 24: Generalized Linear Models (cont.)

uk: Математика, природничі науки

description: This course offers an in-depth the theoretical foundations for statistical methods that are useful in many applications. The goal is to understand the role of mathematics in the research and development of efficient statistical methods.

uk: вхід без авторизації


Lecture 1: Introduction to Statistics

uk: Математика, природничі науки

uk: вхід без авторизації

de: Video

en: Video


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


Lecture 13: Regression

uk: Математика, природничі науки

uk: вхід без авторизації


Lecture 14: Regression (cont.)

uk: Математика, природничі науки

uk: вхід без авторизації


Lecture 15: Regression (cont.)

uk: Математика, природничі науки

uk: вхід без авторизації


Lecture 17: Bayesian Statistics

uk: Математика, природничі науки

uk: вхід без авторизації


Lecture 18: Bayesian Statistics (cont.)

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


Lecture 21: Generalized Linear Models

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


Lecture 3: Parametric Inference

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


Mathematical Statistics

keywords:
Mathematics
Probability and Statistics
Social Science
Game Theory

uk: Суспільні науки та соціологія

uk: Математика, природничі науки

uk: Право, економіка та суспільні науки

description: This course provides students with decision theory, estimation, confidence intervals, and hypothesis testing. It introduces large sample theory, asymptotic efficiency of estimates, exponential families, and sequential analysis.

uk: вхід без авторизації

de: Kurs


Mathematical Statistics Mid-term Exam 1

uk: Суспільні науки та соціологія

uk: Математика, природничі науки

uk: Право, економіка та суспільні науки

description: This resource contains information regarding mathematical statistics, mid-term exam 1.

uk: вхід без авторизації


Mathematical Statistics Mid-term Exam 2

uk: Суспільні науки та соціологія

uk: Математика, природничі науки

uk: Право, економіка та суспільні науки

description: This resource contains information regarding mathematical statistics, mid-term exam 2.

uk: вхід без авторизації


Mathematics of Machine Learning

keywords:
Engineering
Computer Science
Artificial Intelligence
Mathematics
Probability and Statistics
Applied Mathematics
Data Mining
Algorithms and Data Structures
Discrete Mathematics

uk: Комп'ютерні науки

uk: Математика, природничі науки

description:
Broadly speaking, Machine Learning refers to the automated identification of patterns in data. As such it has been a fertile ground for new statistical and algorithmic developments. The purpose of this course is to provide a mathematically rigorous introduction to these developments with emphasis on methods and their analysis.

You can read more about Prof. Rigollet's work and courses [on his website](http://www-math.mit.edu/~rigollet/).

uk: вхід без авторизації


Linear Algebra

keywords:
Mathematics
Linear Algebra

uk: Математика, природничі науки

description: This course offers a rigorous treatment of linear algebra, including vector spaces, systems of linear equations, bases, linear independence, matrices, determinants, eigenvalues, inner products, quadratic forms, and canonical forms of matrices. Compared with [_18.06 Linear Algebra_](/courses/18-06sc-linear-algebra-fall-2011/), more emphasis is placed on theory and proofs.

uk: вхід без авторизації


Algebra I

keywords:
Mathematics
Algebra and Number Theory
Linear Algebra

uk: Математика, природничі науки

description: This undergraduate level Algebra I course covers groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups.

uk: вхід без авторизації


Algebra II

uk: Математика, природничі науки

description: This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.

uk: вхід без авторизації


Modern Algebra

keywords:
Mathematics
Algebra and Number Theory

uk: Математика, природничі науки

description: This undergraduate course focuses on traditional algebra topics that have found greatest application in science and engineering as well as in mathematics.

uk: вхід без авторизації


18.703 Modern Algebra, Exam 1

uk: Математика, природничі науки

description: This resource contains information exam 1.

uk: вхід без авторизації


18.703 Modern Algebra, Exam 2

uk: Математика, природничі науки

description: This resource contains information exam 2.

uk: вхід без авторизації


18.703 Modern Algebra, Exam 3

uk: Математика, природничі науки

description: This resource contains information exam 3.

uk: вхід без авторизації


keywords:
Mathematics
Computation
Algebra and Number Theory
Topology and Geometry

uk: Математика, природничі науки

description: In this undergraduate level seminar series, topics vary from year to year. Students present and discuss the subject matter, and are provided with instruction and practice in written and oral communication. Some experience with proofs required. The topic for fall 2008: Computational algebra and algebraic geometry.

uk: вхід без авторизації

name: Seminar in Algebra and Number Theory: Computational Commutative Algebra and Algebraic Geometry


uk: Математика, природничі науки

description: This is a seminar for mathematics majors, where the students present the lectures. No prior experience giving lectures is necessary.

uk: вхід без авторизації

name: Seminar in Algebra and Number Theory: Rational Points on Elliptic Curves


Commutative Algebra

uk: Математика, природничі науки

description: In this course students will learn about Noetherian rings and modules, Hilbert basis theorem, Cayley-Hamilton theorem, integral dependence, Noether normalization, the Nullstellensatz, localization, primary decomposition, DVRs, filtrations, length, Artin rings, Hilbert polynomials, tensor products, and dimension theory.

uk: вхід без авторизації


Noncommutative Algebra

uk: Математика, природничі науки

description: Noncommutative algebra studies algebraic phenomena involving multiplication for which commutativity law fails, such as product of matrices in linear algebra; such phenomena arise in various disciplines ranging from quantum physics to number theory.

uk: вхід без авторизації


Introduction to Representation Theory

uk: Математика, природничі науки

description: The goal of this course is to give an undergraduate-level introduction to representation theory (of groups, Lie algebras, and associative algebras). Representation theory is an area of mathematics which, roughly speaking, studies symmetry in linear spaces.

uk: вхід без авторизації


Algebraic Geometry

keywords:
Mathematics
Algebra and Number Theory
Topology and Geometry

uk: Математика, природничі науки

description: This course covers the fundamental notions and results about algebraic varieties over an algebraically closed field. It also analyzes the relations between complex algebraic varieties and complex analytic varieties.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This is the first semester of a two-semester sequence on Algebraic Geometry. The goal of the course is to introduce the basic notions and techniques of modern algebraic geometry. It 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. This course is an introduction to the language of schemes and properties of morphisms.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This course provides an introduction to the language of schemes, properties of morphisms, and sheaf cohomology. Together with 18.725 Algebraic Geometry, students gain an understanding of the basic notions and techniques of modern algebraic geometry.

uk: вхід без авторизації


uk: Математика, природничі науки

description: The main aims of this seminar will be to go over the classification of surfaces (Enriques-Castelnuovo for characteristic zero, Bombieri-Mumford for characteristic p), while working out plenty of examples, and treating their geometry and arithmetic as far as possible.

uk: вхід без авторизації

name: Topics in Algebraic Geometry: Algebraic Surfaces


uk: Математика, природничі науки

description: The topics for this course vary each semester. This semester, the course aims to introduce techniques for studying intersection theory on moduli spaces. In particular, it covers the geometry of homogeneous varieties, the Deligne-Mumford moduli spaces of stable curves and the Kontsevich moduli spaces of stable maps using intersection theory.

uk: вхід без авторизації

name: Topics in Algebraic Geometry: Intersection Theory on Moduli Spaces


keywords:
Mathematics
Algebra and Number Theory
Topology and Geometry
Linear Algebra

uk: Математика, природничі науки

description: Double affine Hecke algebras (DAHA), also called Cherednik algebras, and their representations appear in many contexts: integrable systems (Calogero-Moser and Ruijsenaars models), algebraic geometry (Hilbert schemes), orthogonal polynomials, Lie theory, quantum groups, etc. In this course we will review the basic theory of DAHA and their representations, emphasizing their connections with other subjects and open problems.

uk: вхід без авторизації

name: Double Affine Hecke Algebras in Representation Theory, Combinatorics, Geometry, and Mathematical Physics


Lie Groups and Lie Algebras I

uk: Математика, природничі науки

description: This course is the first half of the year-long introductory graduate sequence 18.745/18.755 on Lie groups and Lie algebras. Topics include foundations of the theory of Lie groups and Lie algebras; theorems of Engel and Lie; the universal enveloping algebra, the Poincare-Birkhoff-Witt theorem; free Lie algebras; the Campbell-Hausdorff formula; classification and structure of finite dimensional complex simple Lie algebras; their finite dimensional representations; and the Weyl character formula.

uk: вхід без авторизації


Introduction to Lie Groups

keywords:
Mathematics
Algebra and Number Theory
Topology and Geometry
Differential Equations

uk: Математика, природничі науки

description:
This course is devoted to the theory of Lie Groups with emphasis on its connections with Differential Geometry. The text for this class is [_Differential Geometry, Lie Groups and Symmetric Spaces_](http://www.ams.org/bookstore?fn=20&arg1=gsmseries&item=GSM-34) by Sigurdur Helgason (American Mathematical Society, 2001).

Much of the course material is based on Chapter I (first half) and Chapter II of the text. The text however develops basic Riemannian Geometry, Complex Manifolds, as well as a detailed theory of Semisimple Lie Groups and Symmetric Spaces.

uk: вхід без авторизації


Lie Groups and Lie Algebras II

uk: Математика, природничі науки

description: This course is the second half of the year-long introductory graduate sequence [*Lie Groups and Lie Algebras I*](https://ocw.mit.edu/courses/18-745-lie-groups-and-lie-algebras-i-fall-2020/) *& Lie Groups and Lie Algebras II*. Topics include classical groups, Haar measure on locally compact groups, the representation-theoretic understanding of the hydrogen atom, representations of compact (in particular, finite) groups, the Peter-Weyl theorem with proof, maximal tori, Cartan and Iwasawa decompositions, classification of real reductive Lie groups, topology of Lie groups, proof of the third fundamental theorem of Lie theory, Levi decomposition, Ado's theorem, Borel subgroups, and flag manifolds.

uk: вхід без авторизації


Representations of Lie Groups

uk: Математика, природничі науки

description: The goal of this course is to give an introduction to the representation theory of compact and non-compact Lie groups. It will rely on some material from [*18.745 Lie Groups and Lie Algebras I*](https://ocw.mit.edu/courses/18-745-lie-groups-and-lie-algebras-i-fall-2020/) and [*18.755 Lie Groups and Lie Algebras II*](https://ocw.mit.edu/courses/18-755-lie-groups-and-lie-algebras-ii-spring-2024/), but full familiarity with this material is not required. Topics include continuous representations; algebras of measures on a Lie group; K-finite, smooth, and analytic vectors; admissible representations; unitary representations; the Harish-Chandra admissibility and analyticity theorems; (g,K)-modules; infinitesimal equivalence and realizations; Harish-Chandra’s globalization theorem; representations of SL(2,R), the Chevalley restriction theorem; the Chevalley-Shepard-Todd theorem; Kostant’s theorem; Harish-Chandra isomorphism; Bernstein-Gelfand-Gelfand category O; projective functors; classification of irreducible Harish-Chandra bimodules; the nilpotent cone; the Duflo-Joseph theorem; Duflo’s primitive ideal theorem; the Borel-Weil theorem; Springer resolution; Beilinson-Bernstein localization; D-modules; and classification of representations of a semisimple real group in terms of the orbits of the complexified maximal compact subgroup on the flag variety.

uk: вхід без авторизації


Topics in Lie Theory: Tensor Categories

uk: Математика, природничі науки

description: This course will give a detailed introduction to the theory of tensor categories and review some of its connections to other subjects (with a focus on representation-theoretic applications). In particular, we will discuss categorifications of such notions from ring theory as: module, morphism of modules, Morita equivalence of rings, commutative ring, the center of a ring, the centralizer of a subring, the double centralizer property, graded ring, etc.

uk: вхід без авторизації


Theory of Numbers

name: Theory of Numbers,  Exam Final Guidelines

name: Theory of Numbers, Exam 1 Practice Solutions

name: Theory of Numbers, Exam Final Practice Solutions

name: Theory of Numbers, Exam 2 Practice Solutions

uk: Математика, природничі науки

description: This course is an elementary introduction to number theory with no algebraic prerequisites. Topics covered include primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, and partitions.

uk: вхід без авторизації


Theory of Numbers, Exam 1 Guidelines

uk: Математика, природничі науки

description: This file contains information regarding Guidelines for Midterm1.

uk: вхід без авторизації


Theory of Numbers,  Exam 2 Guidelines

uk: Математика, природничі науки

description: This file contains information regarding Guidelines for Midterm 2.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This file contains information regarding Guidelines for Final exam .

uk: вхід без авторизації


uk: Математика, природничі науки

description: This file contains information regarding Solutions to practice problems for Midterm 1.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This file contains information regarding Solution to Practice Questions for Final Exam.

uk: вхід без авторизації


Theory of Numbers, Exam 1 Practice

uk: Математика, природничі науки

description: This file contains information regarding Practice problems for Midterm1.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This file contains information regarding Practice Questions for Midterm 2.

uk: вхід без авторизації


Theory of Numbers, Exam Final Practice

uk: Математика, природничі науки

description: This file contains information regarding Practice Questions for Final Exam.

uk: вхід без авторизації


Theory of Numbers, Exam 2 Practice

uk: Математика, природничі науки

uk: вхід без авторизації


Introduction to Arithmetic Geometry

uk: Математика, природничі науки

description: This course is an introduction to arithmetic geometry, a subject that lies at the intersection of algebraic geometry and number theory. Its primary motivation is the study of classical [Diophantine problems](http://en.wikipedia.org/wiki/Diophantine_equation) from the modern perspective of algebraic geometry.

uk: вхід без авторизації


Elliptic Curves

keywords:
Engineering
Computer Science
Mathematics
Computation
Algebra and Number Theory

uk: Комп'ютерні науки

uk: Математика, природничі науки

description: This course is a computationally focused introduction to elliptic curves, with applications to number theory and cryptography. While this is an introductory course, we will (gently) work our way up to some fairly advanced material, including an overview of the proof of Fermat's last theorem.

uk: вхід без авторизації


Number Theory I

uk: Математика, природничі науки

description: This is the first semester of a one-year graduate course in number theory covering standard topics in algebraic and analytic number theory. At various points in the course, we will make reference to material from other branches of mathematics, including topology, complex analysis, representation theory, and algebraic geometry.

uk: вхід без авторизації


Number Theory II: Class Field Theory

uk: Математика, природничі науки

description: This course is the continuation of [_18.785 Number Theory I_](/courses/18-785-number-theory-i-fall-2019/). It begins with an analysis of the quadratic case of Class Field Theory via Hilbert symbols, in order to give a more hands-on introduction to the ideas of Class Field Theory. More advanced topics in number theory are discussed in this course, such as Galois cohomology, proofs of class field theory, modular forms and automorphic forms, Galois representations, and quadratic forms.

uk: вхід без авторизації


Topics in Algebraic Number Theory

uk: Математика, природничі науки

description: This course is a first course in algebraic number theory. Topics to be covered include number fields, class numbers, Dirichlet's units theorem, cyclotomic fields, local fields, valuations, decomposition and inertia groups, ramification, basic analytic methods, and basic class field theory. An additional theme running throughout the course will be the use of computer algebra to investigate number-theoretic questions; this theme will appear primarily in the problem sets.

uk: вхід без авторизації


finalexam.pdf

uk: Математика, природничі науки

description: This file contains the questions asked in the final exam.

uk: вхід без авторизації


midterm.pdf

uk: Математика, природничі науки

description: This file contains the questions asked in the midterm exam.

uk: вхід без авторизації


Topics in Algebraic Number Theory

uk: Математика, природничі науки

description: This course provides an introduction to algebraic number theory. Topics covered include dedekind domains, unique factorization of prime ideals, number fields, splitting of primes, class group, lattice methods, finiteness of the class number, Dirichlet's units theorem, local fields, ramification, discriminants.

uk: вхід без авторизації


Project Laboratory in Mathematics

uk: Математика, природничі науки

description:
_Project Laboratory in Mathematics_ is a course designed to give students a sense of what it's like to do mathematical research. In teams, students explore puzzling and complex mathematical situations, search for regularities, and attempt to explain them mathematically. Students share their results through professional-style papers and presentations.

This course site was created specifically for educators interested in offering students a taste of mathematical research. This site features extensive description and commentary from the instructors about why the course was created and how it operates.

uk: вхід без авторизації


Geometry and Topology in the Plane

keywords:
Mathematics
Topology and Geometry

uk: Математика, природничі науки

description: This course introduces students to selected aspects of geometry and topology, using concepts that can be visualized easily. We mix geometric topics (such as hyperbolic geometry or billiards) and more topological ones (such as loops in the plane). The course is suitable for students with no prior exposure to differential geometry or topology. Think of it as a moderate hike, overlooking various parts of the geometry and topology landscape. Bits are flat, bits are uphill, there are occasional rocky parts (may be different for everyone), but none that are designed to be cliff faces. The class moves from one topic to the next quickly, so it’s important to learn continuously.

uk: вхід без авторизації


Introduction to Topology

uk: Математика, природничі науки

description: This course introduces topology, covering topics fundamental to modern analysis and geometry. It also deals with subjects like topological spaces and continuous functions, connectedness, compactness, separation axioms, and selected further topics such as function spaces, metrization theorems, embedding theorems and the fundamental group.

uk: вхід без авторизації


Seminar in Topology

uk: Математика, природничі науки

description: This course is a seminar in topology. The main mathematical goal is to learn about the fundamental group, homology and cohomology. The main non-mathematical goal is to obtain experience giving math talks.

uk: вхід без авторизації


Algebraic Topology I

uk: Математика, природничі науки

description: This is a course on the singular homology of topological spaces. Topics include: Singular homology, CW complexes, Homological algebra, Cohomology, and Poincare duality.

uk: вхід без авторизації


Algebraic Topology II

uk: Математика, природничі науки

description: This is the second part of the two-course series on algebraic topology. Topics include basic homotopy theory, obstruction theory, classifying spaces, spectral sequences, characteristic classes, and Steenrod operations.

uk: вхід без авторизації


Graduate Topology Seminar: Kan Seminar

uk: Математика, природничі науки

description: This is a literature seminar with a focus on classic papers in Algebraic Topology. It is named after the late MIT professor [Daniel Kan](http://math.mit.edu/about/history/obituaries/kan.php). Each student gives one or two talks on each of three papers, chosen in consultation with the instructor, reads all the papers presented by other students, and writes reactions to the papers. This course is useful not only to students pursuing algebraic topology as a field of study, but also to those interested in symplectic geometry, representation theory, and combinatorics.

uk: вхід без авторизації


uk: Математика, природничі науки

description: The goal of this course is to describe some of the tools which enter into the proof of Sullivan's conjecture.

uk: вхід без авторизації

name: Topics in Algebraic Topology: The Sullivan Conjecture


Differential Geometry

uk: Математика, природничі науки

description: This course is an introduction to differential geometry. The course itself is mathematically rigorous, but still emphasizes concrete aspects of geometry, centered on the notion of curvature.

uk: вхід без авторизації


Geometry of Manifolds

keywords:
Mathematics
Topology and Geometry
Linear Algebra
Differential Equations

uk: Математика, природничі науки

description: Geometry of Manifolds analyzes topics such as the differentiable manifolds and vector fields and forms. It also makes an introduction to Lie groups, the de Rham theorem, and Riemannian manifolds.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This is a second-semester graduate course on the geometry of manifolds. The main emphasis is on the geometry of symplectic manifolds, but the material also includes long digressions into complex geometry and the geometry of 4-manifolds, with special emphasis on topological considerations.

uk: вхід без авторизації


Topics in Geometry: Dirac Geometry

keywords:
Mathematics
Topology and Geometry
Linear Algebra

uk: Математика, природничі науки

description: This is an introductory (i.e. first year graduate students are welcome and expected) course in generalized geometry, with a special emphasis on Dirac geometry, as developed by Courant, Weinstein, and Severa, as well as generalized complex geometry, as introduced by Hitchin. Dirac geometry is based on the idea of unifying the geometry of a Poisson structure with that of a closed 2-form, whereas generalized complex geometry unifies complex and symplectic geometry. For this reason, the latter is intimately related to the ideas of mirror symmetry.

uk: вхід без авторизації


Topics in Geometry: Mirror Symmetry

keywords:
Science
Physics
Mathematics
Particle Physics
Topology and Geometry
Linear Algebra

de: Physik

en: Physics

uk: Математика, природничі науки

uk: Фізика, Астрономія

en: Physics, Astronomy

description: This course will focus on various aspects of mirror symmetry. It is aimed at students who already have some basic knowledge in symplectic and complex geometry (18.966, or equivalent). The geometric concepts needed to formulate various mathematical versions of mirror symmetry will be introduced along the way, in variable levels of detail and rigor.

uk: вхід без авторизації


Seminar in Geometry

keywords:
Mathematics
Topology and Geometry
Linear Algebra
Differential Equations
Calculus
Mathematical Analysis

uk: Математика, природничі науки

description: In this course, students take turns in giving lectures. For the most part, the lectures are based on Robert Osserman's classic book _A Survey of Minimal Surfaces_, Dover Phoenix Editions. New York: Dover Publications, May 1, 2002. ISBN: 0486495140.

uk: вхід без авторизації


keywords:
Mathematics
Probability and Statistics
Applied Mathematics
Linear Algebra
Calculus
Mathematical Analysis

uk: Математика, природничі науки

description: This course is an introduction to the basics of random matrix theory, motivated by engineering and scientific applications.

uk: вхід без авторизації

name: Random Matrix Theory and Its Applications


keywords:
Engineering
Computer Science
Electrical Engineering
Algorithms and Data Structures
Theory of Computation
Telecommunications
Computer Networks

uk: Комп'ютерні науки

description:
We will discuss numerous research problems that are related to the internet. Sample topics include: routing algorithms such as BGP, communication protocols such as TCP, algorithms for intelligently selecting a resource in the face of uncertainty, bandwidth sensing tools, load balancing algorithms, streaming protocols, determining the structure of the internet, cost optimization, DNS-related problems, visualization, and large-scale data processing. The seminar is intended for students who are ready to work on challenging research problems. Each lecture will discuss:

*   methods used today
*   issues and problems
*   formulation of concrete problems
*   potential new lines of research

A modest amount of background information will be provided so that the importance and context of the problems can be understood. No previous study of the internet is required, but experience with algorithms and/or theoretical computer science at the graduate/research level is needed.

uk: вхід без авторизації

name: Topics in Theoretical Computer Science : Internet Research Problems


Simplicity Theory

keywords:
Mathematics
Algebra and Number Theory
Mathematical Logic

uk: Математика, природничі науки

description: This is an advanced topics course in model theory whose main theme is simple theories. We treat simple theories in the framework of compact abstract theories, which is more general than that of first order theories. We cover the basic properties of independence (i.e., non-dividing) in simple theories, the characterization of simple theories by the existence of a notion of independence, and hyperimaginary canonical bases.

uk: вхід без авторизації


Topics in Combinatorial Optimization

keywords:
Engineering
Computer Science
Mathematics
Systems Engineering
Applied Mathematics
Computation
Theory of Computation
Systems Optimization

uk: Комп'ютерні науки

uk: Математика, природничі науки

description: In this graduate-level course, we will be covering advanced topics in combinatorial optimization. We will start with non-bipartite matchings and cover many results extending the fundamental results of matchings, flows and matroids. The emphasis is on the derivation of purely combinatorial results, including min-max relations, and not so much on the corresponding algorithmic questions of how to find such objects. The intended audience consists of Ph.D. students interested in optimization, combinatorics, or combinatorial algorithms.

uk: вхід без авторизації


uk: Математика, природничі науки

description: This course is a seminar intended for undergraduate students who enjoy solving challenging mathematical problems, and to prepare them for the Putnam Competition. All students officially registered in the class are required to participate in the [William Lowell Putnam Mathematical Competition](https://en.wikipedia.org/wiki/William_Lowell_Putnam_Mathematical_Competition).

uk: вхід без авторизації

name: Mathematical Problem Solving (Putnam Seminar)


keywords:
Mathematics
Applied Mathematics
Linear Algebra
Calculus

name: Lecture 4 Part 1: Gradients and Inner Products in Other Vector Spaces

name: Lecture 4 Part 2: Nonlinear Root Finding, Optimization, and Adjoint Gradient Methods

name: Lecture 5 Part 1: Derivative of Matrix Determinant and Inverse

name: Lecture 5 Part 2: Forward Automatic Differentiation via Dual Numbers

name: Lecture 5 Part 3: Differentiation on Computational Graphs

name: Lecture 6 Part 1: Adjoint Differentiation of ODE Solutions

name: Lecture 6 Part 2: Calculus of Variations and Gradients of Functionals

name: Lecture 7 Part 1: Derivatives of Random Functions

name: Lecture 7 Part 2: Second Derivatives, Bilinear Forms, and Hessian Matrices

name: Lecture 8 Part 1: Derivatives of Eigenproblems

name: Lecture 8 Part 2: Automatic Differentiation on Computational Graphs

name: Lecture 1 Part 2: Derivatives as Linear Operators

name: Lecture 1 Part 1: Introduction and Motivation

name: Lecture 2 Part 1: Derivatives in Higher Dimensions: Jacobians and Matrix Functions

name: Lecture 2 Part 2: Vectorization of Matrix Functions

name: Lecture 3 Part 1: Kronecker Products and Jacobians

name: Lecture 3 Part 2: Finite-Difference Approximations

uk: Математика, природничі науки

description:
We all know that calculus courses such as [*18.01 Single Variable Calculus*](https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/) and [*18.02 Multivariable Calculus*](https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/) cover univariate and vector calculus, respectively. Modern applications such as machine learning and large-scale optimization require the next big step, "matrix calculus" and calculus on arbitrary vector spaces.

This class covers a coherent approach to matrix calculus showing techniques that allow you to think of a matrix holistically (not just as an array of scalars), generalize and compute derivatives of important matrix factorizations and many other complicated-looking operations, and understand how differentiation formulas must be reimagined in large-scale computing.

uk: вхід без авторизації

name: Matrix Calculus for Machine Learning and Beyond


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 4 Part 1: Gradients and Inner Products in Other Vector Spaces


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 4 Part 2: Nonlinear Root Finding, Optimization, and Adjoint Gradient Methods


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 5 Part 1: Derivative of Matrix Determinant and Inverse


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 5 Part 2: Forward Automatic Differentiation via Dual Numbers


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 5 Part 3: Differentiation on Computational Graphs


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 6 Part 1: Adjoint Differentiation of ODE Solutions


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 6 Part 2: Calculus of Variations and Gradients of Functionals


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 7 Part 1: Derivatives of Random Functions


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 7 Part 2: Second Derivatives, Bilinear Forms, and Hessian Matrices


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 8 Part 1: Derivatives of Eigenproblems


uk: Математика, природничі науки

uk: вхід без авторизації

name: Lecture 8 Part 2: Automatic Differentiation on Computational Graphs


Topics in Mathematics of Data Science

keywords:
Engineering
Computer Science
Mathematics
Probability and Statistics
Applied Mathematics
Data Mining

uk: Комп'ютерні науки

uk: Математика, природничі науки

description: This is a mostly self-contained research-oriented course designed for undergraduate students (but also extremely welcoming to graduate students) with an interest in doing research in theoretical aspects of algorithms that aim to extract information from data. These often lie in overlaps of two or more of the following: Mathematics, Applied Mathematics, Computer Science, Electrical Engineering, Statistics, and / or Operations Research.

uk: вхід без авторизації


keywords:
Mathematics
Probability and Statistics
Business
Finance
Applied Mathematics

name: Lecture 1: Introduction, Financial Terms and Concepts

name: Lecture 10: Regularized Pricing and Risk Models

name: Lecture  19: Black-Scholes Formula, Risk-neutral Valuation

name: Lecture 20: Option Price and Probability Duality

name: Lecture 21: Stochastic Differential Equations

name: Lecture 24: HJM Model for Interest Rates and Credit

name: Lecture 26: Introduction to Counterparty Credit Risk

de: Betriebswirtschaftslehre

uk: Бізнес-адміністрування

en: Business Administration

uk: Математика, природничі науки

uk: Право, економіка та суспільні науки

de: Wirtschaftswissenschaften

uk: Бізнес та економіка

en: Business and Economics

description: The purpose of the class is to expose undergraduate and graduate students to the mathematical concepts and techniques used in the financial industry. Mathematics lectures are mixed with lectures illustrating the corresponding application in the financial industry. MIT mathematicians teach the mathematics part while industry professionals give the lectures on applications in finance.

uk: вхід без авторизації

name: Topics in Mathematics with Applications in Finance


uk: Бізнес-адміністрування

uk: Математика, природничі науки

uk: Право, економіка та суспільні науки

en: Law, Economics and Social Sciences

uk: Бізнес та економіка

uk: вхід без авторизації


uk: Бізнес-адміністрування

uk: Математика, природничі науки

uk: Право, економіка та суспільні науки

uk: Бізнес та економіка

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Lecture 11: Time Series Analysis II

uk: Бізнес-адміністрування

uk: Математика, природничі науки

uk: Право, економіка та суспільні науки

uk: Бізнес та економіка

uk: вхід без авторизації