2012-10 Blog Entry List

代数分野：特別セミナー

日時　１１月１日木曜　１５：１５－１６：３０

ColorSymmetries Associated with Non-Periodic Structures

Ma. Louise Antonette N. De Las Peñas, PhD

Professor, Mathematics Department

Ateneo De Manila University Philippines

With the discovery of quasicrystals in 1984, the research field ofnon-periodic crystallography has grown and expanded in several directions.Structural problems continue to interest mathematicians and physicists.

In this talk, we discuss a method that allows the investigation of symmetriesof non-periodic structures via colorings of cyclotomic integers. In particular,our work looks at ideal colorings of M_n= Z[x_n] where x_n = e^2pi/nis a primitive nth root ofunity for values of n for which Z[x_n] is aprincipalideal domain and thus has class number one. The values of n are groupedinto classes with equal value of f(n),the Euler’s totient function. In the lecture, some results on color groups andcolor preserving groups will be presented.

The colorings of M_n may be manifested geometricallyas a vertex or tile coloring of a two dimensional tiling with n-foldrotational symmetry, which is non-periodic for f(n) > 2.  For suchcases, since M_n is dense on the plane, we choose a discretesubset of M_n for which we show the colors. The discovery ofquasiperiodic tilings such as the Penrose tiling, also raised the questionabout color symmetries of such tilings.

群論を応用して複数の分子からなる結晶構造を調べる研究のお話です。

　　　　　　　　　　　　　　　　　　　　　　　連絡先　秋山茂樹（4395）