KPPY 73

Let $\mathbb{F}_q^n$ be the $n$-dimensional vector space over $\mathbb{F}_q$. We denote the set of coordinates of $\mathbb{F}_q^n$ as $[n] = \{1, 2, \dots, n \}$. We will give a poset structure on $[n]$ and consider a relation on the set of ideals of the poset. A poset on $[n]$ with a relation on the set of ideals admits an association scheme if $(\mathbb{F}_q^n,\{R_{\bar{I}}\})$ becomes an association scheme where $\bar{I}$ varies over the set of ideals under the given equivalence relation and $(x,y)\in R_{\bar{I}}$ iff the ideal generated by the support of $x-y$ is equivalent to $I$.

For example, if we give the antichain on $[n]$ and consider the relation defined by two ideals to be equivalent if and only if they have same cardinality, then we obtain the usual Hamming space. It is a well known fact that a Hamming space admits a Hamming scheme.

In this talk, we will introduce some well known results regarding poset codes and give a definition of MacWilliam condition and strongly regular semilattice. We will then prove that if a semilattice is equipped with the relation where two ideals are defined to be equivalent to each other if and only if they are isomorphic as posets and satisfies the MacWilliam condition, then the semilattice admits an association scheme. In particular, a semilattice admits an association scheme if and only if it is a strongly regular semilattice.

For palindromic polynomials, gamma-positivity implies unimodality. Recently, the gamma-positivity of polynomials with combinatorial or geometric meanings have received considerable attention, because of the famous Gal's conjecture in geometry of polytopes.

In this talk, I will report several gamma-positive results arising from Permutation Patterns, including the q-gamma-positivity of a new q-Narayana polynomials studied by Cheng et al., the gamma-positive of descent polynomials on Separable permutations (or (3142, 2413)-avoiding permutations) and (3124, 4213)-avoiding permutations. This work was partially motivated by Gessel's gamma-positivity conjecture on double Eulerian polynomials.

A finite subset $X$ on the unit sphere $S^{n-1} \subseteq \mathbb{R}^n$ is called a spherical design of harmonic index $t$, if the following condition is satisfied: $\sum_{x \in X} f(x)=0$ for all real homogeneous harmonic polynomials $f(x_1, \ldots, x_n)$ of degree $t$. Also, for a subset $T$ of $\mathbb{N} = \{ 1,2, \ldots, \}$, a finite subset $X \subseteq S^{n-1}$ is called a spherical design of harmonic index $T$, if $X$ becomes a spherical design of harmonic index $t$ for each $t \in T$. This concept is a generalization for spherical designs of Delsarte-Goethals-Seidel (1977).

In the present talk I will introduce some basic knowledge about spherical designs of harmonic index $T$ as well as new results.

This is joint work with Ei. Bannai, Et. Bannai, Wei-Hsuan Yu, and Yan Zhu.

We give an informal introduction to real world problems in big data networks that are of interest to graph theorists.

One of the important themes in ergodic theory is the phenomena of recurrence, which concerns how the initial state returns to the original state. With the help of ergodic method initiated by Furstenberg, various recurrence results in dynamical systems provide a new tool to deduce the combinatorial structures inherent in large subsets of integers.

In this talk we will present some recurrence results in dynamics connected to van der Waerden Theorem and Szemeredi Theorem. I hope this might be a brief introduction to ergodic Ramsey theory. We will also discuss recent recurrence results involving prime numbers which allow new combinatorial applications. This is joint work with V. Bergelson, G. Kolesnik, M. Madritsch and R. Tichy.