KPPY 83

A finite subset $X$ of a Euclidean space is called a $k$-distance set if there exists exactly $k$ distances between two distinct points in $X$. Then there is a natural corresponding between $k$-distance sets and edge colored complete graphs by $k$ colors. In this talk, I will introduce some topics related two-distance sets in a Euclidean space. In particular, we will focus on edge coloring problems of complete graphs

In this talk, we only consider codes in Hamming cube, and we discuss local average of these using Harmonic distribution.

Recently, L. Guth improved the restriction estimate for the surfaces with strictly positive Gaussian curvature in $\mathbb R^3$. In this talk, we consider the restriction problem for hyperbolic surfaces in $\mathbb R^3$ and improve the previous result using polynomial partitioning.

The prime counting function $\pi$ is of central importance in Number Theory. For a real number $x,$ the value of $\pi(x)$ is the number of primes that are less than or equal to $x.$ Properties of $\pi$ can often be understood and studied in terms of combinatorial probability, in particular with analogy to the theory of Markov chains. We apply Mathematica software that deals with arithmetic of large numbers. Using recent algorithms for computation of $\pi,$ this allows us to experiment with the behaviour of the prime counting function. We suggest a new simple approximation to its asymptotic growth and explain some relations to a theorem of Skewes and a question posed by Riemann. This is joint work with Enrique Garcia Moreno Esteva (University of Helsinki).