Harvard/MIT Algebraic Geometry Seminar
Fall 2015  Spring 2016
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 2131.
Schedule

Feb 162016
SC 507Harvard
Dhruv Ranganathan
Nonarchimedean geometry, logarithmic structures, and spaces of curves in toric varieties
abstract±
I will discuss the relationship between moduli spaces of logarithmic stable maps to toric varieties and the polyhedral geometry of their Birkovich skeletons. In genus 0, the latter can be interpreted as a parameter space for tropical curves, and this gives explicit control on the geometry of logarithmic stable maps. Concerning higher genus, I will discuss some recent ideas, building on work of AbramovichWise, Olsson, and Ulirsch, that point to a relationship between the virtual fundamental class and the tropical realizability problem.

Feb 232016
2131MIT
Marc Hoyois (MIT)
A^{1}homotopical classification of principal Gbundles
abstract±
Let k be an infinite field and G an isotropic reductive kgroup. If X is a smooth affine kvariety, then locally trivial Gtorsors over X are classified by maps X → BG in the A^{1}homotopy category. I will discuss the proof of this statement and some applications. This is joint work with Aravind Asok and Matthias Wendt.

March 12016
SC 507Harvard
Andrei Negut
From braids to sheaves, or how to define geometric knot invariants
abstract±
Soergel bimodules form a categorification of the type A_{n} Hecke algebra, which has been known to give rise to 3variable polynomial knot invariants. These invariants are notoriously difficult to compute, that is, until recent work of Elias and Hogancamp developed new techniques to attack the problem via a construction they call "categorical diagonalization". We observe that this construction is equivalent to a pair of adjoint functors from D^{b}(Coh(P^{n})) into the category of Soergel bimodules. Iterating the construction n times allows one to construct a pair of adjoint functors from D^{b}(Coh(FHilb_{n})) into the category of Soergel bimodules, where FHilb_{n} denotes the flag Hilbert scheme of n points on the affine plane. These functors achieve two things: the left adjoint allows one to categorify the maximal commutative subalgebra of projectors inside the Hecke algebra, and the right adjoint allows to associate to any braid a sheaf on the flag Hilbert scheme. This addresses conjectures of GorskyOblomkovRasmussenShende and GorskyNeguț, since the Euler characteristic of such a sheaf is expected to be the 3variable knot invariant. Joint work with Eugene Gorsky and Jacob Rasmussen.

March 82016
2131MIT
Chenyang Xu
Compact moduli spaces of smoothable Kstable Fano varieties
abstract±
(joint with Chi Li, Xiaowei Wang) In this talk, I will discuss the existence of proper moduli spaces which parametrize smoothable Kstable Fano varieties. The proof relies on the solution of YTDconjecture by ChenDonaldsonSun and Tian. Beyond that, we prove the uniqueness of the degeneration and other algebraic properties which we need to construct good quotient spaces as the moduli.

March 152016
2131MIT
Arend Bayer
BrillNoether via wall crossing in the derived category
abstract±
This talk I will explain applications of Bridgeland stability conditions on surfaces to questions purely within algebraic geometry. I will explain some of the underlying mechanism at the hand of reproving BrillNoether for curves on K3 surfaces. Other applications include birational geometry of moduli spaces, or slopestability of (restrictions of) LazarsfeldMukai bundles.

March 222016
SC 507Harvard
Hélène Esnault
Various aspects of Lefschetz theorems
abstract±
We review various forms of Lefschetz theorems, in particular for ladic sheaves, explain what one hopes on the crystalline side.

March 292016
SC 507Harvard
Ben Davison (EPFL)
Beyond the Kontsevich Soibelman integrality conjecture
abstract±
I will discuss recent work with Sven Meinhardt, starting by explaining what the integrality conjecture is, before explaining the categorified analog of this conjecture, stated in terms of Hall algebras. The proofs of these conjectures makes concrete the link between DonaldsonThomas theory and quantum groups (specifically, Yangians).

April 52016
2131MIT
No seminar
No seminar
abstract±

April 122016
SC 507Harvard
Alex Isaev
Isolated hypersurface singularities and associated forms
abstract±
In our recent articles joint with M. Eastwood and J. Alper, it was conjectured that all rational GL_{n}invariant functions of homogeneous forms of degree d>2 on complex space C^{n} can be extracted, in a canonical way, from those of forms of degree n(d2) by means of assigning to every form with nonvanishing discriminant the socalled associated form. While this surprising statement is interesting from the point of view of classical invariant theory, its original motivation was the reconstruction problem for isolated hypersurface singularities, which is the problem of finding a constructive proof of the wellknown MatherYau theorem. Settling the conjecture is part of our program to solve the reconstruction problem for quasihomogeneous isolated hypersurface singularities. In my talk, I will give an overview of the recent progress on the conjecture. If time permits, I will further discuss the morphism that assigns to a nondegenerate form its associated form. This morphism is rather natural and deserves attention regardless of the conjecture. In particular, it leads to a natural contravariant.

April 192016
2131MIT
Jonathan Wise
Compactifying the Abel map with logarithmic geometry
Given a smooth curve with marked points and specified multiplicities for those points, the Abel map yields a point of the curve's Picard scheme. This map extends naturally over compact type curves and irreducible curves, but fails to extend over all stable curves. I will show how to use logarithmic geometry to modify the moduli space of curves in a modular way so that the Abel map extends over the boundary. Among other things, this yields a compact moduli space of meromorphic differentials with a virtual fundamental class.

April 262016
SC 507Harvard
Xinwen Zhu
Towards a padic nonabelian Hodge theory
abstract±
I will first give a brief review the nonabelian Hodge theory for complex manifolds and describe a conjectural padic analogue. Then I will discuss what we know so far and the following consequence of (the known part of) the theory: Let L be a padic local system on a (geometrically) connected algebraic variety over a number field. If it’s stalk at one point, regarded as a padic Galois representation, is geometric in the sense of FontaineMazur, then the stalk at every point is geometric. This is based on a joint work with Ruochuan Liu.

May 32016
2131MIT
Rahul Pandharipande
Moduli spaces of holomorphic/meromorphic differentials
abstract±
Let m=(m_{1},...,m_{n}) be a vector of integers summing to 2g2. A nonsingular
pointed curve [C,p_{1},...,p_{n}] corresponds to a differential with zero and pole multiplicities specified by m if the divisor ∑_{i} m_{i}*p_{i} yields the canonical bundle K_{C}. I will define a proper locus of twisted canonical divisors (joint work with G. Farkas) and discuss a proposed connection
of the fundamental class to Pixton's formulas (joint work with F. Janda, A. Pixton, and D. Zvonkine).
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.