On completion of 6.042J, students will be able to explain and apply the basic methods of discrete (noncontinuous) mathematics in computer science. They will be able to use these methods in subsequent courses in the design and analysis of algorithms, computability theory, software engineering, and computer systems.
The CSDM program, founded in 1999, has two weekly seminars. The main purpose of this program is intensive research, often in cooperation with short-term visitors and local people from academic and research institutions such as Princeton University, Rutgers University, or DIMACS. For more information, check out the pages Seminars and People.
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This field is one of the most vibrant and active areas of scientific study today. Starting half a century ago, even before computers existed, theoretical computer scientists set out to define mathematically the concept of "computation", and to study its power and limits. The theoretical discoveries of Alan Turing, John von Neumann (Institute faculty member, 1933-57), and their contemporaries led to the practical construction of the first stored-program computer at the IAS, followed by the computer "revolution" we are witnessing today.
The practical use of computers and, simultaneously, the unexpected mathematical depth of the abstract notion of "computation" have significantly altered and expanded theoretical computer science. In the last quarter-century, it has turned into a rich and beautiful field, making connections to other areas and attracting talented young scientists. More technical (but still popular) descriptions of various related aspects can be found here:
The "parent" disciplines from which Theoretical Computer Science and Discrete Mathematics evolved were once represented at the Institute by John von Neumann (1933-57) and Kurt Gdel (1953-76). After a considerable gap, the School of Mathematics began to explore the possibility of re-opening this stream of research in the early 1990s with a well-received series of lectures given by Michael Rabin and Richard Karp. The year 1993 marked the opening of a series of exploratory programs led by various leading researchers from all over the world. In the same year, the weekly seminar (now known as Theoretical Computer Science and Discrete Mathematics Seminar) was established.
The exploratory programs proved to be quite successful, both scientifically and educationally and also were well-received by the mathematical community outside the Institute. Thus, it was decided to go ahead, and in 1997-98 and 1998-99, Noga Alon and Avi Wigderson assumed the leadership of further programs in combinatorics and computational complexity.
A commitment to the permanent presence of Theoretical Computer Science and Discrete Mathematics was made by appointing Avi Wigderson to the newly created faculty position (1999). This event also marks the formal beginning of the TCS/DM special program.
Almost immediately (in the academic year of 2000-01) the CSDM Program was holding a Special Year on Computational Complexity that attracted well-known researchers in this field. As another part of the effort to get it established, Alexander Razborov was appointed to be in residence as a senior member in 2001-2006.
From its very beginning, the CSDM Program has been working in close collaboration with nearby academic establishments, as well as with industry research groups. The list of our principal collaborators in particular includes:
The CSDM Program puts special stress on educational aspects. Under the auspices of IAS/Park City Institute, in the summer of 2000, Avi Wigderson and Steven Rudich (Carnegie-Mellon) organized a Graduate Summer School in Computational Complexity. Every year the CSDM Program seeks postdoctoral applications from graduating Ph.D.s in the areas of Theoretical Computer Science and Discrete Mathematics.
Discrete mathematics is foundational material for computer science: Many areas of computer science require the ability to work with concepts from discrete mathematics, specifically material from such areas as set theory, logic, graph theory, combinatorics, and probability theory.
The material in discrete mathematics is pervasive in the areas of data structures and algorithms but appears elsewhere in computer science as well. For example, an ability to create and understand a proof is important in virtually every area of computer science, including (to name just a few) formal specification, verification, databases, and cryptography. Graph theory concepts are used in networks, operating systems, and compilers. Set theory concepts are used in software engineering and in databases. Probability theory is used in artificial intelligence, machine learning, networking, and a number of computing applications.
Note that this document is intended to be a high-level guide to the types of concepts and level of understanding expected for passing the Discrete Mathematics proficiency exam. It should not be taken as an exhaustive list of material that might appear on the exam. Students should be comfortable solving problems in the areas described in this document. These topics map roughly to the following sections of the text Discrete Mathematics with Graph Theory, Third Edition, by Goodaire and Parmenter, copies of which can be found on reserve at the Schow library.
Discrete Mathematics is the language of Computer Science. One needs to be fluent in it to work in many fields including data science, machine learning, and software engineering (it is not a coincidence that math puzzles are often used for interviews). We introduce you to this language through a fun try-this-before-we-explain-everything approach: first you solve many interactive puzzles that are carefully designed specifically for this online specialization, and then we explain how to solve the puzzles, and introduce important ideas along the way. We believe that this way, you will get a deeper understanding and will better appreciate the beauty of the underlying ideas (not to mention the self confidence that you gain if you invent these ideas on your own!). To bring your experience closer to IT-applications, we incorporate programming examples, problems, and projects in the specialization.
Mathematical thinking is crucial in all areas of computer science: algorithms, bioinformatics, computer graphics, data science, machine learning, etc. In this course, we will learn the most important tools used in discrete mathematics: induction, recursion, logic, invariants, examples, optimality. We will use these tools to answer typical programming questions like: How can we be certain a solution exists? Am I sure my program computes the optimal answer? Do each of these objects meet the given requirements?
Counting is one of the basic mathematically related tasks we encounter on a day to day basis. The main question here is the following. If we need to count something, can we do anything better than just counting all objects one by one? Do we need to create a list of all phone numbers to ensure that there are enough phone numbers for everyone? Is there a way to tell that our algorithm will run in a reasonable time before implementing and actually running it? All these questions are addressed by a mathematical field called Combinatorics.
As prerequisites we assume only basic math (e.g., we expect you to know what is a square or how to add fractions), basic programming in python (functions, loops, recursion), common sense and curiosity. Our intended audience are all people that work or plan to work in IT, starting from motivated high school students.
When you enroll in the course, you get access to all of the courses in the Specialization, and you earn a certificate when you complete the work. If you only want to read and view the course content, you can audit the course for free. If you cannot afford the fee, you can apply for financial aidOpens in a new tab.
Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions). Objects studied in discrete mathematics include integers, graphs, and statements in logic.[1][2][3] By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry. Discrete objects can often be enumerated by integers; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets[4] (finite sets or sets with the same cardinality as the natural numbers). However, there is no exact definition of the term "discrete mathematics".[5]
The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business.
Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in "discrete" steps and store data in "discrete" bits. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems.
In university curricula, discrete mathematics appeared in the 1980s, initially as a computer science support course; its contents were somewhat haphazard at the time. The curriculum has thereafter developed in conjunction with efforts by ACM and MAA into a course that is basically intended to develop mathematical maturity in first-year students; therefore, it is nowadays a prerequisite for mathematics majors in some universities as well.[6][7] Some high-school-level discrete mathematics textbooks have appeared as well.[8] At this level, discrete mathematics is sometimes seen as a preparatory course, like precalculus in this respect.[9] 152ee80cbc
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