From June 24-July 6, Isaac Craig (Ph.D. Candidate, Math) traveled to Japan to participate in a two-week intensive course on the theory of Homotopy (H) Principles. The seminar, hosted by the Mathematical Sciences Research Institute, took place at the Tambara Institute (through the University of Tokyo) in the Gunma Prefecture, Japan. Isaac joined thirty students from around the world to learn about and discuss the theory of H-Principles, a method traditionally used for solving partial differential equations. The seminar looked into the importance of H-principles in geometry and physics, such as its application to sympletic contact geometry and foliation theory.
The summer school specifically focused on using H-principles to classify how different types of mathematical objects (like spheres and donuts) can be situated in space, up to some equivalence (e.g. 鈥渘icely鈥 moving one object to the other). Isaac suggests two examples. In the first, H-principles can be applied to demonstrate that when parallel parking a vehicle, any parking space that is larger than a given car (even by a millimeter) is parallel parkable. Or, for a more field-specific example, H-principles help physicists and mathematicians address a phenomenon called sphere eversion, in which a sphere is turned inside out. H-principles demonstrate that by allowing self-intersections, a sphere can be turned inside-out without tearing or creasing, .
Each day Isaac and his peers took a bus from Tokyo to the Tambara Institute, located on a nature preserve in Gunma. Students took part in lectures on most days. When not in lectures Isaac was able to hike around the nature preserve and tour the shrines, hot springs, and waterfalls around Nikko.
Isaac also recently defended his preliminary qualification exams in November after receiving his Master's in May with the thesis titled, 鈥淥n Fibering 3-Manifolds."