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Applied Algebra and Mathematical Crystallography Lab
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The lab uses mathematics to handle lattices and point configurations, and also works on crystal analysis.
AboutAbout the lab
Using algebra and number theory as tools, including for crystal analysis.
Prof. Tomiyasu's lab develops algorithms and applications for discrete structures such as lattices, rings of integers, and point configurations.
The lab's tools come from mathematics such as the geometry of numbers, algebraic number theory, discrete mathematics, and convex optimization.
The lab also handles problems with concrete applications in numerical analysis and in materials science dealing with atomic structure.
The lab's activities, including cross-disciplinary collaboration, take place in mathematical crystallography, where the geometry of numbers, algebraic computation, harmonic analysis, and the generation of packings and meshes can all be used.
Key terms
- Lattice
- A set of points arranged regularly in space. It is also used to represent the periodic arrangement of atoms in a crystal.
- Packing
- A way of arranging points so that they are packed almost uniformly on a space or surface.
- Golden-angle method
- A method of placing points while rotating by the golden angle each time, producing arrangements like the leaves or seeds of plants.
- Semidefinite programming relaxation
- A convex optimization method for finding the globally best solution to a problem of minimizing a quadratic expression.
NewsLab news
A paper generalizing the golden-angle method was published in Constructive Approximation
A paper on a method for determining the Bravais type of a lattice was published in Acta Cryst. A
An explanatory article on CONOGRAPH was published in the Journal of the Crystallographic Society of Japan
ResearchResearch themes
The lab develops algorithms and applications for discrete structures such as lattices, rings of integers, and point configurations, using mathematics such as the geometry of numbers, algebraic number theory, and convex optimization as tools. It works on computational methods that take into account observational errors or only a finite amount of data, and on mathematics that can be used to analyze the structure of materials.
*Illustrative imagePacking and Mesh Generation Methods for Surfaces and Manifolds
The lab extended the golden-angle method, known since the 18th century, to general surfaces and general dimensions. The resulting theory has interdisciplinary applications and is also relevant across problems in algebra, geometry, and analysis.
*Illustrative imageAlgebraic Computation and Fourier Analysis Accounting for Observational Error
Algebraic computation mainly deals with exact numbers, so data containing noise requires new theory and algorithms. The lab works on determining lattice symmetry in the presence of noise, determining lattices from diffraction data, and estimating the truncation error of Fourier transforms of finite data.
*Illustrative imageMathematics of Materials Structure Analysis
The lab uses semidefinite programming relaxation, a type of convex optimization, to analyze crystal structures and magnetic structures. In joint research with experimentalists, it showed that the global optimality of the solution can be guaranteed even when determining magnetic moments, atomic occupancies, and atomic species.
MembersFaculty
- ProfessorRyoko Tomiyasu
VoicesStudent voices
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Rendering of quasicrystal structure” LinKayser / CC BY-SA 4.0 (cropped)
- Packing and Mesh Generation Methods for Surfaces and Manifolds: “Subdivision surface of quadrilateral mesh level 3” Fredrik Orderud / Public domain (cropped)
- Algebraic Computation and Fourier Analysis Accounting for Observational Error: “Fourier crater 4155 h3” James Stuby based on NASA image / CC0 (cropped)
- Mathematics of Materials Structure Analysis: “X-ray diffraction pattern 3clpro” Jeff Dahl / CC BY-SA 3.0 (cropped)
- Related lab (Applied Algebra and Mathematical Crystallography Lab): “Simple Cubic Lattice (Brillouin zone)” Auro69 at en.wikipedia / Public domain (cropped)
- Related lab (Topology and Singularity Theory Lab): “Linkedin maps data visualization” Luc Legay / CC BY-SA 4.0 (cropped)
- Related lab (Topology and Singularity Theory Lab): “Bastard Weaver s Knot diagram-removebg-preview” Zaripov999 / CC BY-SA 4.0 (cropped)
- Related lab (Probability Theory and Random Point Processes Lab): “A boolean model” Tensorproduct / CC BY-SA 4.0 (cropped)
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