1. Zeroth \(\mathbb{A}^1\)-homology of smooth proper varieties

New York J. Math. 28 (2022), 824–834. arXiv: 2101.04951

This was the first paper I wrote during my master's program. The \(\mathbb{A}^1\)-homology is a computationally difficult object defined via a universal property. However, I realized that in the smooth and proper case, it can actually be computed formally using just this universality. Writing this paper gave me confidence that I could do meaningful research of my own.

2. Steinberg symbols and reciprocity sheaves

Annals of K-Theory 7:4 (2022), 695–730. arXiv: 2108.04163

My master's thesis. I was inspired by the result of Hu and Kriz, which explains the Steinberg relation \((a, 1-a) = 0\) in \(K\)-theory through \(\mathbb{A}^1\)-homotopy theory. Motivated by this, I tried to find an analogue for \(\Omega^q\). At the time, there was no established category of motives suited to additive symbols, so the formulation I arrived at is somewhat unnatural.

3. A motivic construction of the de Rham-Witt complex (with H. Miyazaki)

Journal of Pure and Applied Algebra 228 (6), 107602. arXiv: 2301.05846

The first part of my Ph.D. thesis. I was fascinated by the idea that Milnor \(K\)-theory can be seen as a tensor product of the motive of \(\mathbb{G}_m\). Based on this perspective, we developed a motivic interpretation of the de Rham-Witt complex using the theory of reciprocity sheaves.

4. Blow-up invariance of cohomology theories with modulus

Advances in Mathematics, Volume 458, Part B (2024), 109967. arXiv: 2306.14803

The second part of my Ph.D. thesis. I proved the blow-up invariance for cohomology with modulus, a conjecture proposed by Kahn, Miyazaki, Saito, and Yamazaki, under mild assumptions. Since counterexamples exist if these assumptions are removed, I believe this is the best result one can hope for. It was a problem I had long wanted to solve, and surprisingly, one flash of insight led to the solution.

5. Quadratic residues and domino tilings (with Y. Kamio and T. Nakazawa)

Preprint. arXiv: 2311.13597

My first elementary mathematics paper. Mr. Nakazawa, an acquaintance working in industry, made a striking observation: a resemblance between the Kasteleyn formula for domino tilings and Eisenstein’s formula related to quadratic residues. I formulated this as a precise identity and proved it jointly with Yuhi. I'm deeply happy to have discovered such a beautiful formula in my lifetime.

6. On iterated circumcenter sequences (with S. Kanda)

Preprint. arXiv: 2407.19767

Together with my brilliant friend Shuho, we solved an elementary problem listed in the Open Problems Garden. Shuho came up with a sharp and effective strategy for the proof, and I carefully carried it out step by step. It was fascinating to see the mysterious sequence known as the Lyness cycle naturally appear in the process.

7. Irreducibility of polynomials defining parabolic parameters of period 3 (with Y. Murakami, K. Sano, K. Takehira)

Preprint. arXiv: 2408.04850

A paper on number-theoretic phenomena arising in dynamical systems. I read a preprint written by the other three authors and became intrigued by the Morton–Vivaldi conjecture presented in it. I reached out to them and was kindly invited to join the team. We had lively discussions every day through chat, making the collaboration both productive and enjoyable. In the end, we managed to completely resolve the case of period 3, which exceeded our initial expectations.

8. A note on the Erdős conjecture about square packing (with J. Baek and T. Ueoro)

Preprint. arXiv: 2411.07274

Together with my friend Takahiro, who works in industry, we partially solved an elementary problem from the Erdős Problems collection. Takahiro, a gold medalist in the International Mathematical Olympiad, often solves difficult puzzles with brilliant intuition. In this case, he devised a clever proof for the case of 5 squares, and I generalized it to arbitrary numbers. After posting the paper on arXiv, we learned that Jineon had independently discovered a proof as well, so it became a joint paper among the three of us.

9. Isosceles trapezoids of unit area with vertices in sets of infinite planar measure

Preprint. arXiv: 2501.01914

I had been thinking about various problems from the Erdős Problems collection, hoping to solve one, and happened to hit upon an idea that led to a solution. I made a video in Japanese explaining the proof, which unexpectedly received a lot of attention from a general audience.

10. Motivic homotopy theory with ramification filtrations (with H. Miyazaki and S. Saito)

Preprint. arXiv: 2504.02223

We finally realized our dream of constructing the motivic homotopy category of modulus pairs and studied its basic properties. To be honest, I had not imagined that such a well-behaved category could actually be built. Thanks to the blow-up invariance theorem I proved in my Ph.D. thesis, we were able to obtain a strong result showing that the cohomology of reciprocity sheaves is representable — something I am very happy about.

11. Irrationality of the reciprocal sum of doubly exponential sequences

Preprint. arXiv: 2504.05933

I wanted to tackle another problem from the Erdős Problems list, but I ran into a significant obstacle. So I decided to write up the partial progress I had made. The conjecture I posed is extremely simple to state yet elusive and frustrating to grasp. It seems that solving it may require a completely new perspective.