Grant: Subatomic Physics Envelope - Individual

These are very exciting times in physics. The world's most powerful particle accelerator, the Large Hadron Collider (LHC) at CERN, allows us to explore the properties of elementary particles with a precision like never before. The discovery of the Higgs boson in 2012, a particle predicted nearly 60 years ago, and the last piece of a beautiful theoretical framework known as the Standard Model, is a triumph of human ingenuity. Even though the success of the LHC seemed to be hard or impossible to match, in 2015 the Laser Interferometer Gravitational-wave Observatory (LIGO) detected the first gravitational wave (a ripple in space-time produced by the violent merger of black holes). This phenomenon was predicted a century ago and thought to be impossible to measure due to how small the ripples are once they reach Earth. While the physical phenomena in both cases are quite different, (protons colliding in the former, black holes in the latter), they share a similarity. The corresponding theories for each are easy to write down but extracting predictions from them involves massively complicated computations. Such predictions are crucial if we are to maximize the full potential of today's experiments and future technology. When elementary particles collide, there is no certain outcome. Instead, different outcomes are observed with different probabilities. Over nearly twenty years, my research team has been developing tools to compute these probabilities. In the process, we found that very abstract areas of mathematics have become central. Two simple examples are: the problem of cataloging all different ways of placing non-collinear points on a plane and the problem of classifying all tree-like graphs (points joined by line segments of different lengths). Both problems are related to how gluons, particles responsible for keeping quarks confined in a proton, are allowed to interact with each other. The former falls into an area of mathematics called projective geometry, a subject studied for hundreds of years, while the latter in tropical geometry, formally introduced in the 1990s. Surprising, a large set of tools developed for elementary particles can also be used for studying the merging of black holes. This topic has been an active and vibrant area of research for the past five years. In this proposal, my team, composed of graduate students and postdoctoral fellows, will explore the structures underlying our theories of elementary particles and of gravity to understand the origin of the novel mathematical structures and at the same time develop more efficient computational techniques. Physics and mathematics have a history of cross-fertilization; this project aims to advance the mathematical area of tropical geometry by using physical insights. At the same time, tropical geometry provides a window into theoretical universes with never-before-seen properties whose study can help us understand the reason why our universe is the way it is.