Title

On the K-moduli of Calabi–Yau fiber spaces over curves

Date

2026.5.13 (Wed) 16:45-17:45 (16:15- tea in Rm105)

Place

Room 110, Faculty of Science Bldg. No. 3, Kyoto University

Speaker

Masafumi Hattori (Kyoto University)

Abstract

　Deformations of algebraic varieties are an important concept, and moduli are geometric objects that realize the totality of such deformations as a single unified space. Once a moduli space is constructed, elusive phenomena—such as how varieties degenerate and how invariants behave in families—can be visualized as concrete geometric information. However, the construction of moduli spaces for higher-dimensional varieties has long been a difficult problem.
　On the other hand, K-stability is an algebro-geometric notion introduced in Kähler geometry, and it is expected to characterize varieties that admit “good” metrics. Yusuke Odaka proposed the so-called K-moduli conjecture, which predicts that moduli spaces of algebraic varieties can be constructed using the notion of K-stability. The philosophy is that, by allowing only K-stable degenerations (that is, degenerations admitting good metrics), one should be able to construct moduli spaces for higher-dimensional varieties.
　The K-moduli conjecture has been solved in the cases of negative curvature, Fano varieties (positive curvature), and Calabi-Yau varieties (zero curvature). However, for higher-dimensional varieties in which these curvature properties are mixed, the situation is still not well understood. 　In this talk, we focus on Calabi-Yau fiber spaces, namely varieties whose fibers have zero curvature while the base direction has positive curvature, and discuss how the K-moduli conjecture should be formulated in such “mixed” situations. We will also explain, time permitting, the geometric intuition underlying this problem and the difficulties that are currently encountered.

Title

Linear Variance of First-Passage Percolation on the Book Graph

Date

2026.4.22 (Wed)　16:45-17:45 (16:15- tea in 2nd floor common room)

Place

Rm110, Research Institute for Mathematical Sciences, Kyoto University

Speaker

Noe Kawamoto (Kyoto University)

Abstract

　We consider first-passage percolation (FPP) on the book graph with multiple pages, where upper-half planes are glued along the common axis. FPP was introduced by Hammersley and Welsh in 1965 as a model of fluid flow through a random medium. In the model, a non-negative random variable \( t_e \) is assigned on each edge of the graph, independently of the others. The passage time of a path is defined as the sum of the \( t_e \)'s over edges traversed by the path. Our interest is in the infimum of the passage times over all finite paths from \( o \) to \( ne_1 \), which is defined by \( T(0,ne_1) \). In this talk, we prove that when the number of pages of the book graph is sufficiently large, the variance of \( T(0,ne_1) \) is of order \( n \), which is markedly different from the conjectured behavior on two-dimentional integer lattice, where the variance is of order \( n^{2/3} \). This talk is based on joint work with Tzu-Han Chou (NUS) and Wai-Kit Lam (NTU).