# Research

## Gene tree discordance

A species tree is a rooted binary tree with unique species names on the leaves, representing the evolutionary relationship between the species:

      .
/ \
/   \
+     +
/     / \
/     /   \
A     B   C


When attempting to reconstruct the evolutionary tree using any one gene, it’s possible to obtain discordant trees with conflicting topologies:

      .            .            .            .
/ \          / \          / \          / \
/   \        /   \        /   \        /   \
+     +      +     +      +     +      +     +
/     / \    /     / \    /     / \    /     / \
/     /   \  /     /   \  /     /   \  /     /   \
A     B   C  A     B   C  B     A   C  C     A   B


Resolving this discordance quickly using clustering is ongoing work and the topic of the following preprint:

## Trait evolution

Although most people imagine evolution occurring gradually, other modes of evolution are possible, too. Indeed, there’s evidence that some traits evolve in sudden bursts,’’ and then remain static for a long time. To infer which of these two models is correct for an inputted dataset and species tree, we developed a fast, information-theoretic method which can handle very noisey data. As application to a dataset (obtained by Jay McEntee over many summers…) is available here:

• Punctuated evolution in the learned songs of African sunbirds, with Jay McEntee et al., Proc Roy Soc B 288(1963) 2021 (preprint)
• Statistical model, fitting and model selection techniques are described in the supplementary information

This application was described in this very nice video:

Physical and social traits (wingspan, song pitch) can evolve in a variety of ways.

## Stochastic particle systems and the Keller-Segel model

My PhD focused on a system of singularly-interacting stochastic particles of varying masses which experience inelastic collisions. This system can be viewed as 2D overdamped gravity. A preprint is available:

• Coalescing particle systems and applications to nonlinear Fokker-Planck equations, with Ibrahim Fatkullin, CMS 16(2) 2018 (preprint)

When scaled appropriately, the hydrodynamic limit of the empirical mass density converges to the solution $\pho$ of a nonlinear Fokker-Planck equation, such as the Keller-Segel model of chemotaxis: $$\begin{cases} \partial_t \rho &= \nabla \cdot (\mu \nabla \rho - \chi \rho \nabla c)\\ \Delta c &= -\rho \end{cases},$$ or its generalization, the multispecies Keller-Segel model. These PDEs blow up in finite time and form Dirac-type singularities. Such blow-ups correspond to coalescence in the particle system.

Using a combination of analytical estimates and grid-particle numerical methods, this particle system can be simulated quickly, thereby allowing for the numerical solution of some nonlinear Fokker-Planck PDEs. Here’s a coarsening system of ~40k pairwise interacting particles, approximating blow-up in a many-species Keller-Segel system:

This numerical method can be used to solve the original PDE near the blow-up time and observe various properties of the solution, e.g., see the linear change in the system’s second moment: