30 problems

Consider the stochastic Fisher–KPP equation and the genealogies of sampled individuals. Bolthausen–Sznitman genealogy conjecture. After suitable scaling, these genealogies are conj…

Consider stochastic wavefronts with population-size parameter $N$. In the semi-pushed regime, genealogies are expected to exhibit multiple mergers on suitable time scales; in the f…

Macroscopic-event jump conjecture. The macroscopic events captured by $\Psi$ should give rise to jumps in the forward-in-time diffusion process.

Consider the noisy $\lfloor N^\gamma\rfloor$-BRW and the corresponding genealogical process of its population as $N$ tends to infinity. In the related model, there is a critical va…

Let the noisy $\lfloor N^\gamma\rfloor$-BRW be a branching-selection particle system, with the selection regime determined by whether selection favors the fittest or the luckiest i…

Let $N$ be the population size and consider a suitably adjusted version of the diploid Moran model described in the source, with overlapping generations and no selfing. Let the mod…

Let $k\geq1$ particles be sampled uniformly without replacement from the population alive at a large time in a critical continuous-time multitype Bienaymé–Galton–Watson process wit…

Let $n>2$ be fixed, and sample $n$ genes at time-step $0$. Let $(\Pi_k^{N,n})_{k\in\mathbb{Z}_+}$ be the ancestral process taking values in the space $\mathcal{E}_n$ of partitions…

Let $(Y_t)$ be a well-behaved Markov process on a compact Polish space $E$, with stationary distribution $\lambda$ and continuous transition densities $(q_t)$ with respect to…

Limiting conditional genealogy conjecture. If $y_*=0$, then the conditional genealogy behaves like scenario (ii); if $y_*\in(0,1]$, then the conditional genealogy behaves like scen…

Let a population expand into new territory as a travelling wave, and consider genealogies in a model with local population density tending to infinity. Bolthausen–Sznitman genealog…

Consider a branching Brownian motion in the semipushed regime, with … A time-changed $\operatorname{Beta}(2-\alpha,\alpha)$-coalescent is the conjectured limiting genealogy after t…

Consider the fixed-population branching-selection particle system in which each generation particles independently produce exactly $k$ children, after which only the $N$ rightmost…

Let $(\vec{X}^N_t)_{0\leq t<\infty}$ be a Fleming–Viot process, and let the ancestries of its particles be traced backwards in time, producing a coalescent process. Kingman coalesc…

Clustering and star-shaped genealogy conjecture. At a typical time, the majority of the population is close to the leftmost particle, and the genealogy of the population is given b…

Let $k$ ancestral lineages be sampled from near the wavefront, and let $\pi$ be the stationary distribution of a single ancestral lineage relative to the wavefront. Assume that the…

Let $B$ be the wave parameter and let $b1$ denote the corresponding parameter, with $B=2$ corresponding to $\alpha=3/2$. Consider the genealogical trees of a sample of $A$ alleles…

Let $\hat \ell_n:=\ell_n-\bar \ell_n$ be the total internal branch length of an $n$-coalescent, where $\ell_n$ is its total length and $\bar \ell_n$ is its total external branch le…

Extension conjecture. These results can be further extended to regularly varying coalescents.

Let $gamma>0$ and $N\in\mathbb{N}$. Let $(\Pi^N_t)_{t\geq 0}$ denote the genealogical partition process of the $N$-particle branching-selection model, and let $(\rho_N)$ be a seque…

A branching-selection particle system is a population process in which particles branch independently and a selection rule retains only some newborn particles for the next generati…

Let $\Xi\in\mathcal{X}$ be the finite measure governing a general $\Xi$-coalescent, and let $\psi^\Xi_n$ denote the quantity controlling the additional sample-size requirement for…

Brunet et al.'s conjecture. Genealogies in models of this type are described by the Bolthausen–Sznitman coalescent.

The seed-bank coalescent is a coalescent structure arising as the limiting genealogy of the seed-bank model described in the paper. Universality conjecture. The seed-bank coalescen…

Fix $N\in\mathbb{N}$, $\beta>0$, and $\varepsilon\in(0,1)$, and let … For the genealogy of a sample from this population, the time to the most recent common ancestor is the coalesc…