Harvard/MIT Algebraic Geometry Seminar
Fall 2016  Spring 2017
The seminar will
alternate between Harvard and MIT. When it is at Harvard, it
will be at 3:00 pm in room 507. When the seminar is at MIT, it will be at
3:00 pm in room 4153.
Schedule

Sept 132016
SC 507Harvard
Brooke Ullery
Measures of irrationality for hypersurfaces of large degree
abstract±
The gonality of a smooth projective curve is the smallest degree of a map from the curve to the projective line. There are a few different definitions that attempt to generalize the notion of gonality to higher dimensional varieties. The intuition is that the higher these numbers, the further the variety is from being rational. I will discuss some of these definitions, and present joint work with Lawrence Ein and Rob Lazarsfeld. Our main result is that if X is an ndimensional hypersurface of degree d at least (5/2)n, then any dominant rational map from X to P^{n} must have degree at least d1.

Sept 202016
4153MIT
Simion Filip
Hodge theory and its applications in Teichmuller dynamics
abstract±
The moduli space of Riemann surfaces equipped with a holomophic 1form carries an interesting action of the group SL(2,R) which shares some features with locally homogeneous spaces. Understanding this action provides insight into understanding dynamics on individual surfaces. Hodge theory, in particular techniques from variations of Hodge structures, play a role in understanding the dynamics in moduli space.
I will introduce the basic objects in the story and explain how concepts such as real multiplication or torsion points on Jacobians come into play. Time permitting, I will discuss questions in Hodge theory motivated by dynamics, in particular the concept of Lyapunov exponents associated to a variation of Hodge structures.

Sept 272016
SC 507 Harvard
Junliang Shen
Elliptic CalabiYau 3folds, Jacobi forms, and derived categories
abstract±
By physical considerations, Huang, Katz and Klemm conjectured in 2015 that topological string partition functions for elliptic CalabiYau 3folds are governed by certain Jacobi forms. This gives strong structure results for curve counting invariants of elliptic CY 3folds. I will explain a mathematical approach to proving (part of) the HKK Conjecture. Our method is to construct an involution on the derived category and use wallcrossing techniques. If time permits, I will also discuss the connection to the OberdieckPandharipande Conjecture concerning K3 surfaces. The talk is based on joint work with Georg Oberdieck.

Oct 42016
4153MIT
Brian Lehmann
Exceptional sets and Manin's Conjecture
abstract±
Manin's conjecture predicts the growth rate of points of bounded height after removing an exceptional set. We study the geometric consistency of the conjecture by analyzing the exceptional sets. This is joint work with Sho Tanimoto.

Oct 112016
SC 507Harvard
Robert Friedman
Deformations of cusp singularities
abstract±
Cusp singularities are a class of normal surface singularities with a rich geometry and deformation theory. In particular their smoothing components are very closely connected to the moduli of certain rational surfaces. In 1981, Looijenga gave a necessary condition for a cusp singularity to be smoothable and conjectured that this condition was also sufficient, a conjecture recently proved by GrossHackingKeel and P. Engel. This talk describes recent joint work with Engel. We define an invariant λ for every semistable Ktrivial model of a one parameter smoothing of a cusp singularity and show that all possible values of the invariant arise. Using this result, we characterize those rational double point singularities which are adjacent to cusp singularities.

Oct 182016
4153MIT
Claudiu Raicu
Cohomology of determinantal thickenings
abstract±
I will explain how to determine the cohomology groups of arbitrary equivariant thickenings of generic determinantal ideals, as well as the ranks of the maps in cohomology induced by inclusions of such thickenings. One application of this work is a concrete description of the linear functions that describe the CastelnuovoMumford regularity of sufficiently large powers and symbolic powers of determinantal ideals.

Oct 252016
SC 507Harvard
Ian Shipman
Ulrich bundles and a generalization
abstract±
Let (X,L) be a smooth, polarized variety. A vector bundle E on X is
called an Ulrich bundle if all cohomology of the bundles E(L),...,E( dim(X) L
) vanishes. I will give an introduction to Ulrich bundles, highlighting their
connection to several elementary problems of interest. Then I will describe
some of my work (joint with Y. Mustopa & R. Kulkarni) on a generalization of
Ulrich bundles and on the connection between Ulrich bundles and (higher rank)
BrillNoether theory.

Nov 12016
4153MIT
Emanuele Macri
TBD

Nov 82016
SC 507Harvard
Francois Greer
TBD

Nov 152016
4153MIT
TBD
TBD

Nov 222016
SC 507Harvard
Kenny Ascher
TBD

Nov 292016
4153MIT
Chenyang Xu
TBD

Dec 62016
4153MIT
Gabriele di Cerbo
Log birational boundedness of CalabiYau pairs
abstract±
I will discuss a joint work with Roberto Svaldi on
boundedness of CalabiYau pairs. Recent works in the minimal model
program suggest that pairs with trivial log canonical class should
satisfy some boundedness properties. I will show that CalabiYau pairs
which are not birational to a product are indeed log birationally
bounded. This result is the first step towards boundedness of
elliptically fibered CalabiYau manifolds.

Dec 132016
4153MIT
Mark Mclean
TBD
This seminar is organized by Joe Harris (Harvard), Bjorn Poonen (MIT), Davesh Maulik (MIT), Sam Raskin (MIT), Maksym Fedorchuk (BC), Dawei Chen (BC), Yaim Cooper
(Harvard), Anand Patel (BC), Aaron Pixton (Harvard). This seminar is supported in part by
grants from the NSF. Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the author(s)
and do not necessarily reflect the views of the National Science
Foundation.