Publishing type：Research paper (scientific journal)  

<jats:p>In this paper, we deal with two classes of Diophantine equations, $x^2+y^2+z^2+k_3xy+k_1yz+k_2zx=(3+k_1+k_2+k_3)xyz$ and $x^2+y^4+z^4+2xy^2+ky^2z^2+2xz^2=(7+k)xy^2z^2$, where $k_1,k_2,k_3,k$ are nonnegative integers. The former is known as the Markov Diophantine equation if $k_1=k_2=k_3=0$, and the latter is a Diophantine equation recently studied by Lampe if $k=0$. We give algorithms to enumerate all positive integer solutions to these equations, and discuss the structures of the generalized cluster algebras behind them.</jats:p>

DOI： 10.37236/11420

Open Access

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Informa {UK} Limited  

DOI： 10.1080/00927872.2023.2173763

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Cellule {MathDoc}/{CEDRAM}  

DOI： 10.5802/aif.3596

Open Access

Publisher：{SPIE}  

DOI： 10.2969/aspm/08810491

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Oxford University Press ({OUP})  

<jats:title>Abstract</jats:title><jats:p>In the present paper, we first give a characterization for Bongartz completion in $\tau $-tilting theory via $c$-vectors. Motivated by this characterization, we give the definition of Bongartz completion in cluster algebras using $c$-vectors. Then we prove the existence and uniqueness of Bongartz completion in cluster algebras. We also prove that Bongartz completion admits certain commutativity. We give two applications for Bongartz completion in cluster algebras. As the first application, we prove the full subquiver of the exchange quiver (or known as oriented exchange graph) of a cluster algebra $\mathcal A$ whose vertices consist of the seeds of $\mathcal A$ containing particular cluster variables is isomorphic to the exchange quiver of another cluster algebra. As the second application, we prove that in a $Y$-pattern over a universal semifield, each $Y$-seed (up to a $Y$-seed equivalence) is uniquely determined by the negative $y$-variables in this $Y$-seed.</jats:p>

DOI： 10.1093/imrn/rnac205

Authorship：Lead author   Language：English   Publishing type：Internal/External technical report, pre-print, etc.  

DOI： 10.48550/arXiv.2512.04547

Language：English   Publishing type：Internal/External technical report, pre-print, etc.  

DOI： 10.48550/arXiv.2507.06900

Language：English  

In this paper, we study positive integer solutions to a generalized form of
the Markov equation, given as $x^2 + y^2 + z^2 + k(yz + zx + xy) = (3 +
3k)xyz$. This equation extends the classical Markov equation $x^2 + y^2 + z^2 =
3xyz$. We generalize the concept of Cohn triples for the classical Markov
equation to the generalized Markov equations. Using this, we provide a
generalization of the uniqueness theorem of Markov numbers that are prime
powers.

arXiv

Other Link： http://arxiv.org/pdf/2312.07329v1

In this paper, we demonstrate that the lattice structure of a set of clusters
in a cluster algebra of finite type is anti-isomorphic to the torsion lattice
of a certain Geiss-Leclerc-Schr\"oer (GLS) path algebra and to the $c$-Cambrian
lattice. We prove this by explicitly describing the exchange quivers of cluster
algebras of finite type. Specifically, we prove that these quivers are
anti-isomorphic to those formed by support $\tau$-tilting modules in GLS path
algebras and to those formed by $c$-clusters consisting of almost positive
roots.

arXiv

Other Link： http://arxiv.org/pdf/2211.08935v2

Event date： 2026.2

Language：Japanese   Presentation type：Oral presentation (invited, special)  

Venue：九州大学   Country：Japan  

Event date： 2026.2

Language：English   Presentation type：Oral presentation (invited, special)  

Venue：Kavli IPMU  

Event date： 2026.1

Language：Japanese   Presentation type：Oral presentation (invited, special)  

Venue：富山大学   Country：Japan  

Event date： 2025.12

Language：Japanese   Presentation type：Public lecture, seminar, tutorial, course, or other speech  

Venue：オンライン   Country：Japan  

Event date： 2025.9

Language：Japanese   Presentation type：Oral presentation (invited, special)  

Venue：東京大学   Country：Japan