Algebraic Geometry near-Boston Conference "Recent progress on the Maximal Rank Conjecture"
When: March 24, 2018, 10am-4:30pm
Where: Room 101, Mathematics Department, Bromfield Pearson Bldg, Tufts UniversitySpeakers: - David Jensen (U Kentucky)
- Eric Larson (MIT)
- Brian Osserman (UC Davis)
- Anand Patel (Oklahoma State)
There will be a catered lunch. To help out the organizers with the headcount, pleaseRegistration: fill out a short registration form.
Abstracts: David Jensen: The Kodaira dimensions of moduli spaces of curvesWe discuss new methods for studying tropicalizations of linear series, and how these methods can be applied to outstanding cases of the Strong Maximal Rank Conjecture of Aprodu and Farkas. As a consequence, we will see that the moduli spaces of curves of genus 22 and 23 are of general type. This is joint work with Sam Payne. Eric Larson: A proof of the maximal rank conjectureWe will discuss the recent proof of the maximal rank conjecture using methods of interpolation. Brian Osserman: Various
theorems and conjectures involve studying ranks of multiplication maps
of linear series on curves. Such questions can have a range of implications,
in directions including smoothness of linear series moduli Multiplication
maps and limit linear seriesspaces, higher-rank Brill-Noether theory, and the geometry of moduli spaces of curves. We describe an approach to such questions by degenerating to a chain of genus-1 curves, and using ideas from Eisenbud-Harris limit linear series, and from the related construction of "linked linear series." This is joint work with Fu Liu, Montserrat Teixidor i Bigas, and Naizhen Zhang. Anand Patel: Branched covers and the Steinitz problemA basic and old Folklore Problem in number theory asks: If O_K is the ring of integers of a number field K, is every rank d > 1 projective O_K-module isomorphic to the ring of integers in a field extension L/K? Every rank d projective module over a Dedekind domain like O_K is isomorphic to a direct sum of a rank d-1 free module and a well defined ideal class I. The class I is sometimes called the Steinitz class of the module, and hence the folklore problem is often restated: Which Steinitz classes are realized by degree d field extensions? Although every Steinitz class is expected to be realizable, a proof of this statement seems far beyond the reach of current methods. This is probably why I made no progress on it. In joint work with Anand Deopurkar, however, we solve the (complex) geometric version of the Folklore Problem. I will speak about these things and more. Parking information: On weekends, guests can park in the lot adjacent to 574 Boston Ave, and in a small parking lot on the corner of Dearborn Rd and Boston Ave, without fear of ticketing. If those are full, there is pay-parking on Boston Ave. |