Nonlocal Interactions in Physics and Biology

As Dipierro and Valdinoci write, ‘the study of surfaces which minimize the perimeter is…probably one of the oldest problems in the mathematical literature’ . A minimal surface is defined by having zero mean curvature, and the study of these mathematical objects have many applications to physical sciences, applying techniques from geometry and analysis. Of particular interest to this project are nonlocal minimal surfaces, a very new concept arising this century. Roughly speaking, while classical minimal surfaces consider only interactions between nearby agents, nonlocal minimal surfaces consider the entire surface as one body, with every point non-trivially influencing every other point . This project will explore properties and applications of nonlocal minimal surfaces following and building onrecent work by Dipierro and Valdinoci. In particular, guiding questions include

• Regularity of the nonlocal minimal surfaces in dimension greater than 21
• Connection between surfaces of constant nonlocal mean curvature and long-range phasetransitionmodels
• Detection of optimal stickiness exponents
• Analysis of the stickiness phenomenon for non-standard kernel interactions
• Asymptotics of the contact angle in the nonlocal capillarity theory
• Analysis of moving and stationary fronts in domains with holes
• Effect of holes on the nonlocal minimal surfaces and their connection with spin models.

Once I am familiarised with the fundamentals of this specific area of analysis, I intend to choose which of these questions are most interesting and within my ability and begin to explore it by summarising and understanding the literature, and hopefully by exploring some novel cases.

Joshua Troy

University of Western Australia

Josh Troy is intrigued by the boundary between mathematics and reality. He finds great fascination in the surprising ability for beautiful mathematical structures, often motivated by fun problems, to describe and help understand the world through physics, modelling, optimisation, and more. Josh will pursue an Honours in Mathematical Physics in 2023 at the University of Western Australia and intends to stay close to the Physics Society and Maths Union to help share his passion for the two fields. In addition to the objective sciences, Josh loves to read and learn about all fields, as it is often in the intersections of fields that the toughest problems find their most creative solutions.
Aside from studying, Josh enjoys (non-combat) boxing, and struggles to get the right ratio of rice to water.

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