In this talk, we will analyze the incompressible Euler equation in a time-dependent 2D fluid domain, whose interface evolution is governed by the law of linear elasticity without damping. Our main result asserts that the Cauchy problem is globally well-posed in the energy space for irrotational initial data without any smallness assumption. We also prove the continuity with respect to the initial data and the propagation of regularity. In the absence of parabolic regularization, a key ingredient in our analysis is a novel reduction to a nonlinear Schrödinger-type equation, allowing us to apply dispersive estimates. To carry this out, we develop new estimates for the Dirichlet-to-Neumann operator in low-regularity regimes through tools from classical harmonic analysis and paradifferential calculus. This is a joint work with Thomas Alazard and Chengyang Shao.

An adjunction between categories, a weakened notion of an equivalence, is omnipresent in mathematics. In particular, it provides a useful framework for studying various phenomena in algebraic topology. In this talk, I will introduce a method for constructing an adjunction of module categories. Given an algebra R and a left ideal J, the eigenring, introduced by Ore in 1932, is defined as a canonical subquotient algebra of R with respect to J. Recently, I gave a construction of an adjunction between the category of R-modules and the category of modules over the eigenring. By anology between monads and algebras, I also introduced the notion of eigenmonad and established a similar adjunction. This adjunction has interesting applications to polynomial functors, outer functors (on free groups), and the Habiro-Massuyeau category in quantum topology. In part I, I will present the general construction with some classical examples and an overview of the applications. In part II, I will give a detailed explanation of the applications to polynomial functors and the Habiro-Massuyeau category.