Kenny Xie L&S Math & Physical Sciences
Mean Field Theory for Carrying Capacity Dynamics for Carrying Capacity
How do we let mathematical models of evolution feel the squeeze of competition? Current phylogenetic “birth-death” frameworks treat every lineage as if it lives alone; they falter once growth slows because space or resources run out. My project tackles the open challenge of mean-field interacting multi-type birth-death processes: proving that those models still behave predictably when each individual’s birth and death rates grow in proportion to the average population size instead of being artificially capped. Achieving this will require replacing classical Lipschitz bounds with ones that depend on Wasserstein distances, weaving together probability, optimal-transport geometry, and fixed-point analysis. A successful proof will extend the powerful “propagation-of-chaos” machinery to density-dependent interactions and open the door to fast, likelihood-based inference for real-world problems—from how antibodies evolve inside germinal centres to how viruses spread through communities—giving evolutionary trees the mathematics they need to breathe under crowding.
Message To Sponsor
I would like to express my heartfelt gratitude for your generous support of my SURF fellowship. This summer has been a transformative period for me, both as a scholar and as a person. Immersing myself fully in research allowed me to engage deeply with advanced questions, challenge myself to think independently, and develop the persistence needed to follow an idea through obstacles to insight. Beyond technical growth, the experience gave me confidence in my ability to contribute to a broader community of inquiry. I discovered how rewarding it is to exchange ideas with mentors and peers, and I learned how to position my own work within the ongoing conversations of my field. These moments strengthened my sense of identity as a researcher and affirmed my commitment to pursuing questions that matter.