Chapter 1. Measure and integral
Chapter 2. High-dimensional geometry and measure concentration
Chapter 3. Fourier analysis
Chapter 4. Representations of finite groups
Chapter 5. Polynomials
Chapter 6. Topology

Mathematics++ is a concise introduction to six selected areas of 20th century mathematics providing numerous modern mathematical tools used in contemporary research in computer science, engineering, and other fields. The areas are: measure theory, high-dimensional geometry, Fourier analysis, representations of groups, multivariate polynomials, and topology. For each of the areas, the authors introduce basic notions, examples, and results. The presentation is clear and accessible, stressing intuitive understanding, and it includes carefully selected exercises as an integral part. Theory is complemented by applications—some quite surprising—in theoretical computer science and discrete mathematics. The chapters are independent of one another and can be studied in any order.