This three-year, $270,315 National Science Foundation Division of Mathematical Sciences Project Grant will support research and outreach activities focused on curves with rare geometric and arithmetic properties. Specifically, the principal investigator at Colorado State University will evaluate Galois actions, cohomological obstructions, and supersingular reductions of curves through three research projects. The first will study Galois group actions on the cohomology of curves with automorphisms to further understanding of the absolute Galois group of rational numbers. The second will prove certain genus four curves have infinitely many primes of supersingular reduction, centered on cyclic covers of the projective line. The third will count supersingular curves in families with automorphisms, relating to conjectures on their existence and generalizing mass formulas. A unifying theme is demonstrating interplay between geometric and arithmetic techniques for such curves. The grant also supports the Vantage seminar and related training to develop open problems in number theory and arithmetic geometry. This award falls under the National Science Foundation's Mathematical and Physical Sciences program to increase scientific knowledge and understanding in these fields.