Random plane geometry

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The goal of this proposal is to create and study a modern version of plane geometry that involves uncertainty. The route google maps chooses for our tip is not a straight line or a curve. Can we still understand it mathematically? What is the right language to use? Probabilists have worked on such questions before. What is a random function like? This was answered by Bachelier, Einstein, and Wiener in separate works: it is Brownain motion, curve we see, e.g. in stock price fluctuations. The strength of the concept of Brownian motion comes from its universality: deep down, most random curves look like Brownian motion: random walks, stock prizes, diffusion of particles, even surfaces of two-dimensional grwoth. With J. Ortmann and D. Dauvergne we construrced the directed landscape, an object that plays the role of Brownian motion in the world of plane geometry. It is expected to be universal object just like Brownian motion: every random plane geometry should look like the directed landscape. Our 2019 paper (about 70 citations) was invited to Acta Mathematica. This opens up new lines of research. 1. Statistical and real-world applications. The distributions that arise in random plane geometry have been observed in experiments using liquid crystals; they also accurately describe coffee stains. But any two-dimensional data, e.g. maps, brain surface measurements, distances in cities, shortest paths for flights in turbulent environment should have a random geometric structure. Caveat: the presence of heavy-tailed randomness will change exponents. I will collect planar data sets to understand which exponents arise in nature, and develop mathematics for all exponent classes. The theory tells us what measurements to take to classify geographic data in a new way. 2. Universality. We know only a few models in which the directed landscape can be proven to be the limit. I have been working with students to find more examples and prove universality in some limiting cases. We don't even have any good bounds on the exponents in simple cases. 3. Mathematical applications. Just like ordinary geometry, random plane geometry is often hiding behind the surface. One of these problems is one-dimensional heat flow in a spacetime random environment. The corresponding Kardar-Parisi-Zhang equation lent its name to this entire research area in physics. Recently, I showed that this equation converges to the directed landscape. With a student, we are working to extend such results to general directed polymers. In another project, we work to understand two-dimensional random heat flow (the parabolic Anderson model) and how it is connected to the directed landscape. 4 Particle systems. The simple particle system called tasep is deeply equivalent to random geometry. It is used by biologists to understand traffic jams in protein synthesis inside cells. We work to show that second-class particles in tasep behave very similarly to geodesics in the directed landscape.