Abstract: The quadratic reciprocity law is a classical theorem in elementary number theory. It has a natural place in arithmetic topology as an analog of linking number. I will recall the work of Redei on triple symbols in the early 20th century and Morishita's work which connects it to Massey products. Finally, I will discuss a modern approach to the subject, based on my joint work with Morishita.

Abstract: In algebraic number theory, the product formula for Artin symbols is an important formula that connects local and global class field theory. In abelian extensions, the product of local Artin symbols (norm residue symbols) over all places equals the unit element. In this talk, I will discuss this product in non-abelian extensions of number fields using graph theory, and describe it as a product of certain commutators. As an application, some properties on fundamental units of real quadratic fields are obtained.

Abstract: We show a regularized determinant formula for the zeta functions of certain 3-dimensional Riemannian foliated dynamical systems, in terms of the infinitesimal operator induced by the flow acting on the reduced leafwise cohomologies. It is the formula conjectured by Deninger. The proof is based on relating dynamical spectral ξ-function, analogues of the ξ-function in analytic number theory, with the zeta function, by applying the distributional dynamical Lefschetz trace formula. This is joint with Jesús A. Álvarez López and Junhyeong Kim (https://arxiv.org/abs/2410.20758). I will start with the motivation from arithmetic topology and Deninger's program, and explain why and how the foliation theory and dynamical system appear naturally from the basic problem in number theory.

Abstract: It is a classical result that two polygons with the same area are scissors congruent. But this is not true for polyhedra. In fact a regular hexahedron and a regular tetrahedron of volume 1 are not scissors congruent. In order to prove this fact, the Dehn invariant of a polyhedron plays an important role. We will study the Dehn invariant and the volume of a hyperbolic ideal tetrahedron. [pdf for more details]

Abstract: The structure of algebraic units as a Galois module is fundamental in number theory. Investigating this structure presents challenges due to the complexities involved in understanding the arithmetic of number fields and the difficulty in classifying integral representations of finite groups. A number field is called p-rational if the Galois group of its maximal pro-p p-ramified extension is a free pro-p group. The concept of p-rationality is known to simplify various problems in number theory. In this talk, we present our recent results on the implications of existing theories on integral representations of finite groups ---such as factor equivalence, regulator constants, and Yakovlev diagrams--- on the structure of algebraic units in totally real p-rational fields. This talk is based on several joint works with Z. Bouazzaoui, D. Burns, A. Kumon, and C. Maire.

Abstract: We consider the Burde--de Rham theorem for finitely presented pro-p groups under the assumption that the total degrees of all relators are 0. We also give some concrete examples including higher-dimensional cases under Iwasawa theoretic conditions, and consider some cohomological interpretations. (Joint work with Yasushi Mizusawa and Yuji Terashima.)