KPPY 65

The adjacency algebra of an association scheme is defined over an arbitrary field. In general, it is always semisimple over a field of characteristic zero but not always semisimple over a field of positive characteristic. The structures of adjacency algebras over fields of positive characteristic have not been sufficiently studied. In this talk, we consider the structures of adjacency algebras of some P-polynomial schemes of class d with intersection numbers $c_i \not\equiv 0$ modulo $p$ for $1 \leq i \leq d$ over fields of positive characteristic $p$. The classes of these $P$-polynomial schemes include association schemes originating from Grassmann graphs, double Grassmann graphs, and all types of dual polar graphs. We will discuss the structures of the modular adjacency algebras of Grassmann graphs with particular parameters. This is based on a joint work with Masayoshi Yoshikawa (Nagano Prefecture Azusagawa Senior High School).

For a simple connected graph $G$, we define a canonical embedding which is an isometric embedding from $G$ to a product of graphs. And we define the isometric dimension and irreducible graph. In this talk, we will show some examples of the isometric dimension and we focus on decreasing of the isometric dimension of non-irreducible graphs. Finally, we calculate the isometric dimension of strongly regular graphs.

The multiple zeta value (abbreviated MZV) is a kind of generalization of the well-known Riemann zeta values $\zeta(n), n \in \mathcal{Z}_{>1}$, and is an important real value related to knot theory, quantum field theory and so on. It is known that the space of MZVs has the structure of a commutative algebra over $\mathcal{Q}$. This algebra is deeply connected with the theory of mixed Tate motives. In this talk, we consider the multiple Eisenstein series (abbreviated MES). The MES, defined by Gangl-Kaneko-Zagier in 2006, plays key role in understanding a connection between MZVs and modular forms for $\rm{SL}_2\mathcal{Z}$. One of the goal is to construct the multiple Eisenstein series algebra by making use of the Goncharov coproduct which gives a Hopf algebra structure in the MZV algebra. This is based on a joint work with Henrik Bachmann (Hamburg University).

We start by defining algebraic multiplier ideal sheaves with respect to divisors. With a divisor, one can get a plurisubharmonic function which can define analytic multiplier ideal sheaves. Naturally, these two sheaves are the same thing if they came from one divisor. And the sheaves are coherent. Using this theories, we introduce the algebraic analogue of multiplier ideal, called adjoint ideal sheaves, and define the analytic adjoint ideal sheaves attached to plurisubharmonic function. We suggest a analytic approach to proof of coherence of adjoint ideal sheaves.

Let $\Theta^*$ be the transitive closure of the Djokovic-Winkler's relation $\Theta$. $\Theta^*$ is an essential relation in metric graph theory which has many applications; we will present a number of them. In this talk, a history of isometric embedding and a general description of the cut method is presented. More precisely, a computation of the Wiener index and more distance base graph invariants is described. Finally, we present some open problems and conjectures in metric theory of graphs.