Spring 2018

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 University

Speakers:
Registration: There will be a catered lunch.  To help out the organizers with the headcount, please
fill out a short registration form.

 10-11 Brian Osserman: Multiplication maps and limit linear series
 11:30-12:30 
 Eric Larson: A proof of the maximal rank conjecture

 12:30-2 Lunch

 2-3 David Jensen: The Kodaira dimensions of moduli spaces of curves

 3:30-4:30 Anand Patel: Branched covers and the Steinitz problem


Abstracts:  

David Jensen: The Kodaira dimensions of moduli spaces of curves
We 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 conjecture
We will discuss the recent proof of the maximal rank conjecture using methods of interpolation.

Brian Osserman: Multiplication maps and limit linear series
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 
spaces, 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 problem
A 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.


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