New results on envy-free distributions

Abstract: Similarly to the growth of Applied Topology, the uses and applications of Geometry are now expanding into scientific, computational, and engineering domains. First, we'll review the recent history of this burgeoning Applied Geometry area. I'll mention a couple of collaborations, developing and implementing algorithms inspired by the marked length spectrum that classify complex networks (with Eliassi-Rad and Torres) and analyzing digital images using a variant of curve-shortening flow (with Velazquez Richards). Then, I'll present joint work with Evangelista and Ruiz Pantaleón on computational Poisson geometry and its applications to learning symbolic expressions of Hamiltonian systems. We developed and released two Python packages that perform symbolic and numerical computation of objects in Poisson geometry. We then used them to train neural networks (hybrids with CNN and LSTM components) that learn symbolic expressions of Hamiltonian vector fields. Finally, I'll briefly mention the theoretical limitations of computationally analyzing Hamiltonian dynamics. I recently constructed an example of a Hamiltonian flow on the 4-sphere that is Turing complete. Therefore the most general cases of Hamiltonian learning problems are undecidable.

geometric topology

Audience: researchers in the topic

( video )

GEOTOP-A seminar

Series comments: Web-seminar series on Applications of Geometry and Topology