The colloquium typically meets Mondays at 4:00 PM in Room 617 on the sixth floor of Wachman Hall.
The colloquium is preceded by tea starting at 3:30 in the Faculty Lounge, adjacent to Room 617. Click on title for abstract.
|Jennifer Balakrishnan, Boston University|
Let C be a smooth projective curve of genus at least 2 defined over the rational numbers. It was conjectured by Mordell in 1922 and proved by Faltings in 1983 that C has finitely many rational points. However, Faltings' proof does not give an algorithm for finding these points.
In the case when the Jacobian of C has rank less than its genus, the Chabauty--Coleman method can often be used to find the rational points of C, using the construction of p-adic line integrals. In certain cases of higher rank, p-adic heights can often be used to find rational or integral points on C. I will describe these "quadratic Chabauty" techniques (part of Kim's nonabelian Chabauty program) and will highlight some recent examples where the techniques have been used: this includes a 1700-year old problem of Diophantus originally solved by Wetherell and the problem of the "cursed curve", the split Cartan modular curve of level 13. This talk is based on joint work with Amnon Besser, Netan Dogra, Steffen Mueller, Jan Tuitman, and Jan Vonk.
Genevieve Walsh, Tufts University
When is a group the fundamental group of a 3-manifold? What properties of 3-manifold groups can we extract to better understand other groups? A group is coherent if every finitely generated subgroup is finitely presented, and incoherent otherwise. A group algebraically fibers if it admits a map to the integers with finitely generated kernel. The fundamental groups of closed hyperbolic 3-manifolds are coherent, and they virtually algebraically fiber. We will discuss the geometry and importance of these notions, and develop techniques to find witnesses to incoherence and algebraic fibers. We apply these techniques to large classes of groups, including many free by free, surface by surface and surface by free groups. Any new work mentioned is joint work with Rob Kropholler.
Jean-Christophe Mourrat, Courant Institute, NYU
Kavita Ramanan, Brown University
Lai-Sang Young, Courant Institute, NYU
Marc Culler, University of Illinois at Chicago
Vlad Vicol, Courant Institute, NYU