New York J. Math. 28 (2022), 824–834. arXiv: 2101.04951
We give an explicit formula for the zeroth \(\mathbb{A}^1\)-homology sheaf of a smooth proper variety. We also provide a simple proof of a theorem of Kahn-Sujatha which describes hom sets in the birational localization of the category of smooth varieties.
Annals of K-Theory 7:4 (2022), 695–730. arXiv: 2108.04163
We study multilinear symbols on fields taking values in reciprocity sheaves. We prove that any such symbol satisfying natural axioms automatically has Steinberg-type relations, which is a manifestation of the geometry of modulus pairs lying behind.
Journal of Pure and Applied Algebra 228 (6), 107602. arXiv: 2301.05846
We generalize Kahn-Saito-Yamazaki’s theory of reciprocity sheaves over a field to noetherian base schemes. We also prove an afnalogue of the Hasse-Arf theorem for reciprocity sheaves. As an application, we provide a new construction of the de Rham-Witt complex via reciprocity sheaves.
Advances in Mathematics, Volume 458, Part B (2024), 109967. arXiv: 2306.14803
In this paper, we study cohomology theories of \(\mathbb{Q}\)-modulus pairs, which are pairs \((X,D)\) consisting of a scheme \(X\) and a \(\mathbb{Q}\)-divisor \(D\). Our main theorem provides a sufficient condition for such a cohomology theory to be invariant under blow-ups with centers contained in the divisor. This yields a short proof of the blow-up invariance of the Hodge cohomology with modulus proved by Kelly-Miyazaki. We also define the Witt vector cohomology with modulus using the Brylinski-Kato filtration and prove its blow-up invariance.
Preprint. arXiv: 2311.13597
The formula for the number of domino tilings due to Kasteleyn and Temperley-Fisher is strikingly similar to Eisenstein’s formula for the Legendre symbol. We study the connection between these two concepts and prove a formula which expresses the Jacobi symbol in terms of domino tilings.
Preprint. arXiv: 2407.19767
An iterated circumcenter sequence (ICS) in dimension \(d\) is a sequence of points in \(\mathbb{R}^d\) where each point is the circumcenter of the preceding \(d+1\) points. The purpose of this paper is to completely determine the parameter space of ICSs and its subspace consisting of periodic ICSs. In particular, we prove Goddyn’s conjecture on periodic ICSs, which was independently proven recently by Ardanuy. We also prove the existence of a periodic ICS in any dimension.
Preprint. arXiv: 2408.04850
Morton and Vivaldi defined the polynomials whose roots are parabolic parameters for a one-parameter family of polynomial maps. We call these polynomials delta factors. They conjectured that delta factors are irreducible for the family \(z\mapsto z^2+c\). One can easily show the irreducibility for periods 1 and 2 by reducing it to the irreducibility of cyclotomic polynomials. However, for periods 3 and beyond, this becomes a challenging problem. This paper proves the irreducibility of delta factors for the period 3 and demonstrates the existence of infinitely many irreducible delta factors for periods greater than 3.
Preprint. arXiv: 2411.07274
Let \(f(n)\) denote the maximum total length of the sides of n squares packed inside a unit square. Erdős conjectured that \(f(k^2+1)=k\). We show that the conjecture is true if we assume that the sides of the squares are parallel to the sides of the unit square.
Preprint. arXiv: 2501.01914
Paul Erdős posed the question of whether every measurable planar set of infinite Lebesgue measure contains the four vertices of an isosceles trapezoid of unit area. In this paper, we provide an affirmative answer to this question. Additionally, we present affirmative solutions to similar questions by Erdős concerning isosceles triangles and right-angled triangles.
Preprint. arXiv: 2504.02223
The aim of this paper is to connect two important and apparently unrelated theories: motivic homotopy theory and ramification theory. We construct motivic homotopy categories over a qcqs base scheme \(S\), in which cohomology theories with ramification filtrations are representable. Every such cohomology theory enjoys basic properties such as the Nisnevich descent, the cube-invariance, the blow-up invariance, the smooth blow-up excision, the Gysin sequence, the projective bundle formula and the Thom isomorphism. In case \(S\) is the spectrum of a perfect field, the cohomology of every reciprocity sheaf is upgraded to a cohomology theory with a ramification filtration represented in our categories. We also address relations of our theory with other non-\(\mathbb{A}^1\)-invariant motivic homotopy theories such as the logarithmic motivic homotopy theory of Binda, Park, and Østvær and the theory of motivic spectra of Annala-Iwasa.
Preprint. arXiv: 2504.05933
We show that sequences of positive integers whose ratios \(a_n^2/a_{n+1}\) lie within a specific range are almost uniquely determined by their reciprocal sums. For instance, the Sylvester sequence is uniquely characterized as the only sequence with \( a_n^2/a_{n+1}\in [2/3,4/3] \) whose reciprocal sum is equal to \(1\). This result has applications to irrationality problems. We prove that for almost every real number \(\alpha > 1\), sequences asymptotic to \(\alpha^{2^n}\) have irrational reciprocal sums. Furthermore, our observations provide heuristic insight into an open problem by Erdős and Graham.