Lecture 1. The Statistics of Topology and Algebra
1. Hilbert’s Sixteenth Problem
2. The Statistics of Smooth Functions
3. Statistics and the Topology of Periodic Functions and Trigonometric Polynomials
4. Algebraic Geometry of Trigonometric Polynomials

Lecture 2. Combinatorial Complexity and Randomness
1. Binary Sequences
2. Graph of the Operation of Taking Differences
3. Logarithmic Functions and Their Complexity
4. Complexity and Randomness of Tables of Galois Fields

Lecture 3. Random Permutations and Young Diagrams of Their Cycles
1. Statistics of Young Diagrams of Permutations of Small Numbers of Objects
2. Experimentation with Random Permutations of Larger Numbers of Elements
3. Random Permutations of 𝑝² Elements Generated by Galois Fields
4. Statistics of Cycles of Fibonacci Automorphisms

Lecture 4. The Geometry of Frobenius Numbers for Additive Semigroups
1. Sylvester’s Theorem and the Frobenius Numbers
2. Trees Blocked by Others in a Forest
3. The Geometry of Numbers
4. Upper Bound Estimate of the Frobenius Number
5. Average Values of the Frobenius Numbers
6. Proof of Sylvester’s Theorem
7. The Geometry of Continued Fractions of Frobenius Numbers
8. The Distribution of Points of an Additive Semigroup on the Segment Preceding the Frobenius Number

One of the traditional ways mathematical ideas and even new areas of mathematics are created is from experiments. One of the best-known examples is that of the Fermat hypothesis, which was conjectured by Fermat in his attempts to find integer solutions for the famous Fermat equation. This hypothesis led to the creation of a whole field of knowledge, but it was proved only after several hundred years.

This book, based on the author's lectures, presents several new directions of mathematical research. All of these directions are based on numerical experiments conducted by the author, which led to new hypotheses that currently remain open, i.e., are neither proved nor disproved. The hypotheses range from geometry and topology (statistics of plane curves and smooth functions) to combinatorics (combinatorial complexity and random permutations) to algebra and number theory (continuous fractions and Galois groups). For each subject, the author describes the problem and presents numerical results that led him to a particular conjecture. In the majority of cases there is an indication of how the readers can approach the formulated conjectures (at least by conducting more numerical experiments).