NSF Impact Statement for 2015-16

Ian Agol's Impact Statement:

In 2015-16, a program was organized on the topic of geometric structures on 3-manifolds. This field has continued to progress after the program. I’ll highlight some further progress.

  • During the program, Kahn and Wright began a collaboration to construct immersed surfaces in cusped hyperbolic 3-manifolds.

    • This resulted in the paper:  https://arxiv.org/abs/1809.07211, just published this year in Duke math journal. This complements the results of Cooper-Futer proved during the program which created immersed surfaces, but with less control on the geometry. Kahn was a program member and Wright a school member during that year.
  • During the program, Alan Reid asked Ian Agol about arithmetic manifolds which virtually fiber on a congruence cover. Some examples were produced, and eventually these examples were generalized in collaboration with Matthew Stover (who attended some of the program activities) in the paper https://arxiv.org/abs/1912.10283
  • Another topic considered during the program was the relation between veering triangulations and pseudo-Anosov flows. A conjecture formulated during the program by Agol, that veering triangulations give pseudo-Anosov flows without perfect fits (a notion formulated by program member Sergio Fenley) has been proven by Saul Schleimer and Henry Segerman (Schleimer participated in a program workshop).
  • During the program, Maryam Mirzakhani continued revising her preprint with Alex Eskin (a math school member that year). The extensive revision was finally published in 2018, and Eskin was awarded the 2019 Breakthrough Prize largely for this work.
  • The L-space conjecture, that irreducible non-L-spaces have taut foliations, was considered during the program and has continued as an important conjecture connecting gauge theory and geometric structures.

These are just a sample of the results stemming from the program that Ian is aware of.