I will discuss joint work with Alex Bartel in which we study the frequency of failures of the Hasse principle in quadratic twist families of torsors under an abelian variety. The main technical ingredient is a result on the distribution of 2-Selmer groups in certain families of twists defined by Frobenian conditions.

I will discuss joint work with Alex Bartel in which we study the frequency of failures of the Hasse principle in quadratic twist families of torsors under an abelian variety. The main technical ingredient is a result on the distribution of 2-Selmer groups in certain families of twists defined by Frobenian conditions.

We study a density-dependent Markov jump process describing a population where each individual is characterized by a type, and reproduces at rates depending both on its type and on the population type distribution. We are interested in the empirical distribution of ancestral lineages in the population process. First, we exhibit a time-inhomogeneous Markov process, which allows to capture the behavior of a sampled lineage in the population process. This is achieved through a many-to-one formula, which relates the expected value of a functional evaluated over the lineages in the population process to the expectation of the functional evaluated along this time-inhomogeneous process. This provides a direct interpretation of the underlying survivorship bias. Second, we consider the large population regime, when the population size grows to infinity. Under classical assumptions, the population type distribution converges to a deterministic limit. Here, we focus on the empirical distribution of ancestral lineages in this large population limit, for which we establish a many-to-one formula. Using coupling arguments, we further quantify the approximation error which arises when sampling in this large population approximation instead of the finite-size population process.

In the first half of this talk, I will present the construction of the topological Fukaya category of a graded marked surface, as introduced by Haiden, Katzarkov, Kontsevich, and present some of its main properties.
In the second half, I will present a minimal model for the localization of this category at a spherical object corresponding to a simple closed curve on the surface. This new description will then be used to describe the indecomposable objects of the localization in terms of graded curves on the singular surface obtained by collapsing the simple closed curve to a point.

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