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An overview of design theory
时间  Datetime
2019-07-05 16:00 — 17:00 
地点  Venue
Large Conference Room(706)
报告人  Speaker
Eiichi Bannai
单位  Affiliation
Kyushu University (emeritus)
邀请人  Host
张跃辉
报告摘要  Abstract

The purpose of design theory is for a given space M to find ”good” finite subsets X which approximate the whole space M. The space M may be an infinite (continuous) space or a finite space. A spherical t-design is a finite subset X of the unit sphere M = S n−1 in Rn such that 1 |Sn−1| Z Sn−1 f(x)dσ(x) = 1 |X| X x∈X f(x) holds for any polynomials f(x) of degree at most t, where t is a positive integer, and the integral is on the sphere. A combinatorial t-design (or t-(v, k, λ) design) is a subset X of M = V k  (=the set of all k-element subsets of a finite set V of size v) such that |{B ∈ X | T ⊂ B}| = λ(= a constant 0) holds for all T ∈ V t  , We first discuss what are the interesting problems in the theory of t-designs, i.e., the existence and construction problems, optimal designs (=designs of the smallest size), Fisher type lower bound, tight designs, and so on. We note that there exist close similarities among the studies of spherical t-designs, combinatorial t-designs, as well as for t-designs in various other spaces M. We also note that we can study many different situations in a uniform way through the viewpoint of algebraic combinatorics, although each individual case has its own interesting specific features. We also obtain various generalizations of the t-design concept, and study these generalized concepts also for various spaces M. At the end, we will discuss t-designs (unitary t-designs) in the unitary group U(d), including our current work in progre。