## Departmental Ph.D. Thesis Exam – Carrie Clark

Wednesday, June 29, 2022 at 10:00 a.m.

PhD Candidate: Carrie Clark
Supervisor: Almut Burchard
Thesis title: Droplet formation in simple nonlocal aggregation models

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We interaction energies given by various kernels, and investigate how  these kernels drive the formation of multiple flocks within a larger population. We show that for a class of kernels having a “well-barrier” shape that the energy is minimized by a sequence of indicators of finitely many balls whose supports become infinitely far apart from one another. The dichotomy case of the concentration compactness principle is a key ingredient in our proof. We also consider a toy model which forbids points in the support of an admissible density from being within a certain range of distances from one another. We show in one dimensions, that no matter the width of this range the energy is minimized by the indicator of a union of well separated intervals of length 1 and one smaller interval. Finally, we also consider weakly repulsive kernels and show that Wasserstein $d_{\infty}$ local minimizers must saturate the density constraint.

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A draft of the thesis can be found here: thesis copy