Please tell us about your research.

As a faculty member in the Department of Pure and Applied Mathematics, my research primarily centers on lattice polytopes. I am particularly interested in Ehrhart polynomials—counting functions for lattice points within lattice polytopes —and I also study lattice polytopes as they appear in contexts such as commutative ring theory and algebraic geometry.

Lattice polytopes are a special class of polytopes. In two dimensions, there are triangles and quadrilaterals; in three dimensions, there are cubes and regular octahedrons—there are various polyhedral, which can be mathematically referred to as “d-dimensional polytopes.” Furthermore, a lattice polytope is a polytope that satisfies the condition that its vertices are lattice points.
These lattice polytopes appear in various fields of mathematics, but the study of Ehrhart polynomials for lattice polytopes is particularly important. A lattice polytope contains a certain number of lattice points, but as the polytope is scaled up by a factor of 3, or n, the number of lattice points within it also increases. The Ehrhart polynomial expresses this growth pattern mathematically.
As we explore the Ehrhart polynomial, we find interesting connections not only to polytopes but also to fields that seem completely unrelated. One example is the “magic square.” A magic square is a mathematical puzzle in which distinct natural numbers from 1 to n² are arranged in an n×n grid such that the sum of the numbers in each row, column, and diagonal is equal. Instead of solving it conventionally, there is an application that uses Ehrhart polynomials to map the numbers to lattice points of a polytope.

Other applications to different fields of mathematics include algebraic geometry and commutative ring theory. Algebraic geometry is one of the most prominent and challenging theories in university-level mathematics, and parts of it can be developed using the theory of Ehrhart polynomials for lattice polytopes. Similarly, commutative ring theory is a field deeply connected to algebraic geometry, and certain aspects of it can also be advanced in this way.

Ehrhart polynomials were first introduced in 1962 by the French mathematician, Eugène Ehrhart. I believe Ehrhart himself conceived of this formula simply as a way to summarize the number of lattice points within a lattice polytope.

However, discussions evolved from there, and as connections were made with various fields, new problems emerged. Even in the 21st century, there are still many unsolved problems, making it a field that is far from being fully explored. I find this aspect fascinating, and I intend to master the study of Ehrhart polynomials—a field known as “Ehrhart theory.”

It should be noted that Ehrhart theory is a very niche field within the broader realm of mathematics. I am likely the only person in Japan specializing in this research, and I am confident that I am at the forefront of the field in Japan.

Please tell us about how you first encountered Ehrhart theory.

I’ve loved and excelled at math since I was a child, and in 2005, I enrolled in the Department of Mathematics at the Faculty of Science, the University of Osaka. From the very beginning, I had my sights set on pursuing a doctoral program, though it wasn’t out of some lofty ambition—it was simply because I wanted to keep doing what I loved.

In the Department of Mathematics at the University of Osaka, students are assigned to a seminar group starting in their fourth year. I chose the seminar led by Professor Takayuki Hibi (retired in March 2022; currently Professor Emeritus at the University of Osaka), who specialized in the niche fields of computational commutative algebra and combinatorics.

The reason I chose his seminar was that research areas in the Department of Mathematics are primarily divided into three categories: “Algebra,” “Geometry,” and “Analysis.” If I had to choose among these, I thought “Algebra” would be best for me. On the other hand, within algebraic research, “Algebraic Geometry” and “Number Theory” are the most prominent fields, and nearly all professors specializing in algebra focus on one of those two. Since I was aiming to obtain a PhD, I reasoned that “it might be easier to become a leader in a niche field,” and so I decided to study under Professor Hibi.

My introduction to Ehrhart theory came during a seminar in my fourth year of undergraduate studies, where we read Professor Hibi’s book *Commutative Algebra and Combinatorics* (Springer Tokyo, 1995) in a reading group and learned the basics of lattice polytope theory and Ehrhart theory.

Earlier, I mentioned that Ehrhart polynomials are related to various areas of mathematics; one such area is Pick’s formula (Pick’s theorem). Pick’s formula calculates the area of a polygon with no holes, where all vertices lie on equidistant lattice points (integer coordinates). In other words, this concerns two-dimensional polygons, but Ehrhart theory comes into play when considering three-dimensional, four-dimensional, and d-dimensional cases. Essentially, Ehrhart theory allows us to prove Pick’s formula and derive a d-dimensional version of it. In that sense, Pick’s formula can be considered as the origin of Ehrhart theory.

I realized this after studying Ehrhart theory in its entirety, and I was deeply impressed, thinking, “Pick’s formula is amazing!” That was when I became fully immersed in Ehrhart theory. Subsequently, as part of the problem of characterizing Ehrhart polynomials of lattice polytopes, I tackled the characterization of cases where the normalized volume is 3 or less, and in March 2009, I completed my first co-authored paper*. This marks the starting point of my research.

※ Ehrhart polynomials of convex polytopes with small volumes.(EUROPEAN JOURNAL OF COMBINATORICS)

Undergraduate students rarely have the opportunity to learn about lattice polytopes or Ehrhart theory. In that sense, I consider myself very fortunate. Once I got started, Ehrhart theory proved to be incredibly fascinating, and as I threw myself into it with single-minded determination, by the time I reached PhD course, I had gained enough confidence to say that I knew more about Ehrhart polynomials than Professor Hibi himself.

At that point, there was almost nothing left for me to learn about Ehrhart theory in Japan. Naturally, my eyes began to turn overseas.