TBD

Arithmetic of Elliptic curves, one of fundamental research themes in modern number theory, is encoded in the special values of elliptic L-functions. For example, Mazur-Rubin heuristics on the distribution of the special L-values predict the behavior of ranks of elliptic curves. The average version of the heuristics was proved by several researchers including myself several years ago. In the talk, I will present how to use the dynamics of continued fractions to study the problem and introduce an approach to study the original problem, i.e., non-average version, after introducing a classical result, so-called Lochs' theorem, which compares entropies of two distinct dynamical systems.

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PPT Slide: English

De novo mutations provide a powerful source of information for identifying risk genes associated with phenotypes under selection, such as autism spectrum disorder (ASD), obsessive-compulsive disorder (OCD), congenital heart disease, and schizophrenia (SCZ). However, identifying de novo variants is costly, as it requires trio-based sequencing to obtain parental genotypes. To address this limitation, we propose a method to infer inheritance class using only offspring genetic data. In our new integrated model, we evaluate variation in case and control samples, attempt to distinguish de novo mutations from inherited variation, and incorporate this information into a gene-based association framework. We validate our method through ASD gene identification, demonstrating that it provides a robust and powerful approach for identifying risk genes.