## YOSHINAGA MasahikoProfessor

### Shapes and discrete structures

Department of Mathematics, Mathematics

 Theme I am interested in the shape (of spaces) via discrete structures (e.g. lattice points, posets, rooty system). Field Algebraic Geometry, Singularity, Combinatorics Keyword Hyperplane arrangements, Enumerative combinatorics, Posets, Polytopes, Root system, Magnitude homology

Department of Mathematics, Mathematics

YOSHINAGA Masahiko

Professor

Who is the researcher you respect most and why?

I admire Prof. Kyoji Saito and Prof. Hiroaki Terao. Prof. Saito was my Ph. D. supervisor. It was great experience that I attended his seminar and saw his way of thinking and visions. I learned many things, however, among others the importance of thinking thoroughly with strong motivations. Prof. Terao influenced my career at several levels. The first impact was his celebrated result, the so-called “Terao’s factorization theorem”. The beauty of the factorization theorem was the starting point of my research. One of my dreams is to get such a beautiful result one day.

What is the research theme that you are currently focusing on?

I am interested in the shapes of geometric figures and spaces. The motivation to my research is to understand shapes via discrete structures. Here, “understanding shapes” means to catch characteristic properties of shape, such as “a donut has a hole, but a sphere does not”. The phrase “Via discrete structure” is difficult to explain, however, the simplest example is “counting the number of points”. I mainly focus on hyperplane arrangements and related spaces to attain transparent understanding of the relationship between shapes and discrete structures.

Please briefly introduce us to the big project you have been tackling.

“Magnitude homology” is a research topic that I recently started and would like to study more in depth in future. Let me briefly describe what is magnitude homology. Magnitude is a numerical invariant of a metric space introduced in a few years ago, which is connected to several important notions, e.g., measures of biodiversity, the public facility location problem in urban engineering, entropy in statistical mechanics or information theory, the Euler characteristic in topology etc. The magnitude homology, recently introduced by Hepworth, Willerton, Leinster and Shulman, is a refinement (“categorification” in mathematics) of magnitude. I am particularly interested in the fact that the magnitude homology is defined for both discrete and continuous spaces. I expect that the magnitude homology enables us to study spaces without distinguishing discrete and continuous.