**This is Part 1 of a two-semester course. May be used to count toward one Prosem - permission of DGS required
The backgrounds and mathematical sophistication of the students entering the LCS-IGERT graduate program will vary widely. A two-semester Mathematical Foundations sequence will provide all students with basic mathematical modeling and algorithmic tools, while still providing sufficient challenges for the most advanced. These two courses (course numbers to be announced soon), will be taught in a computer/media lab setting and will cover relevant aspects of a wide range of mathematical topics that are directly relevant to animal, human or machine communication, or that provide prerequisites for these topics. Examples of topics directly relevant to communication include information theory, game theory, and formal language theory. Examples of important topics include signal processing, machine learning, and probabilistic models. These two semesters obviously cannot substitute entirely for the dozen or more semesters that normally would be required to cover a similar range of topics. However, they can give students the ability to understand and implement algorithms from published descriptions, especially given appropriate libraries of basic functions, and to discuss alternative approaches with experts in a well-informed manner. It is clearly not the case that every LCS-IGERT students will use every mathematical or algorithmic topic from this course in his or her research. However, applications are often unexpected, and fortune favors the prepared. In addition, this background will enable students to make sense of a wide range of courses and readings that might otherwise be inaccessible. Finally, the shared experience of this course will help IGERT students to establish a personal as well as conceptual basis for future collaborations. Each semester of this two-semester sequence will be co-taught by two faculty members. Because of the diversity of topics and of the students' backgrounds, the two-semester course sequence will be organized into a series of "modules", each designed to explicate a core mathematical and algorithmic topic. Each module will deal with specific problems of the type that IGERT students need to solve and will be as self-contained as possible, although of course one module will often require understanding of concepts and techniques taught in another. This is a "Mind"sector course.