Harvard/MIT Algebraic Geometry Seminar
Fall 2017
Tuesdays at 3 pm
The Harvard/MIT Algebraic Geometry Seminar will alternate between MIT
(4237) and Harvard (Science Center Hall A).

Sep 122017
Hall AHarvard
Rohini Ramadas, Harvard
Dynamics on the moduli space of pointconfigurations on the Riemann Sphere
abstract±
Hurwitz correspondences are a special family of algebraic discrete dynamical systems on the moduli space M_{0,n} parametrizing configurations of n points on P^1. Hurwitz correspondences arise in the study of the topological dynamics of selfmaps of P^1.
Dynamical degrees are numerical invariants associated to algebraic discrete dynamical systems. I will discuss the dynamical degrees of Hurwitz correspondences.

Sep 192017
4237MIT
Giulia Sacca, MIT
Degenerations of hyperkahler manifolds
abstract±
The problem of understanding semistable degenerations of K3 surfaces has been greatly studied and is completely understood (KulikovPinkhamPersson). The aim of this talk is to presentjoint work with J. Kollár, R. Laza, and C. Voisin generalizing some of these results to higher dimensional hyperkähler (HK) manifolds. I will also present some applications, including a generalization of theorem of Huybrechts to possibly singular symplectic varieties and shortcuts to showing that certain HK manifolds are of a given deformation type.

Sep 262017
Hall AHarvard
Alina Marian, Northeastern
On the intersection theory of Hilbert schemes of points on a surface
abstract±
I will describe the structure of natural intersectiontheoretic invariants coming from Chern classes of tautological vector bundles over Hilbert schemes of points on a smooth projective surface. The subject has a long history; I will explain aspects of it, as well as recent contributions in joint work with D. Oprea and R. Pandharipande.

Oct 32017
4237MIT
Yuchen Liu, Yale
Kstability of cubic threefolds
abstract±
We prove the Kmoduli space of cubic threefolds is identical to their GIT moduli. More precisely, the K(semi,poly)stability of cubic threefolds coincide to the corresponding GIT stabilities, which could be explicitly calculated. In particular, this implies that all smooth cubic threefolds admit KählerEinstein metric as well as provides a precise list of singular KE ones. To achieve this, the main new ingredient is an estimate in dimension three of the normalized volumes of kawamata log terminal singularities introduced by Chi Li. This is a joint work with Chenyang Xu.

Oct 102017
SC 411Harvard
ManWai Cheung, Harvard
Quiver representations and theta functions
abstract±
Scattering diagrams theta functions and broken lines were developed in order to describe toric degenerations of CalabiYau varieties and construct mirror pairs. Later, GrossHackingKeelKontsevich unravel the relation of those objects with cluster algebras. In the talk, we will discuss how we can combine the representation theory with these objects. We will also see how the broken lines on scattering diagram give a stratification of quiver Grassmannians using this setting.

Oct 172017
Hall AHarvard
Dave Jensen, U Kentucky
Linear Systems on General Curves of Fixed Gonality
abstract±
The geometry of an algebraic curve is governed by its linear systems. While many curves exhibit bizarre and pathological linear systems, the general curve does not. This is a consequence of the BrillNoether theorem, which says that the space of linear systems of given degree and rank on a general curve has dimension equal to its expected dimension. In this talk, we will discuss a generalization of this theorem to general curves of fixed gonality. To prove this result, we use tropical and combinatorial methods. This is joint work with Dhruv Ranganathan, based on prior work of Nathan Pflueger.

Oct 242017
2449 MIT
David Hansen, Columbia
**TIME AND LOCATION CHANGE: TALK IS 1:302:30 AT MIT IN ROOM 2449**
Vanishing theorems in rigid analytic geometry
abstract±
I will review some classical cohomological vanishing theorems due to AndreottiFrankel and Artin, and then explain an analogous result in rigid analytic geometry. I will also explain how I was convinced that excellent rings deserve their name, and why Ofer Gabber is the master of us all.

Oct 312017
4237MIT
Nick Rozenblyum, U Chicago
Shifted symplectic structures and quantization
abstract±
Many moduli problems of interest, such as moduli spaces of coherent sheaves or principal Gbundles, come equipped with a natural symplectic structure. Moreover, quantization of these symplectic structures is closely related to counting problems in geometry and topology, such as DonaldsonThomas invariants
and their generalizations, as well as Feynman integration in physics. The theory of shifted symplectic structures introduced provides a vast generalization of algebraic symplectic geometry, provides a natural framework for constructing and studying such symplectic structures and their quantizations. I will give a brief overview of this theory and describe several geometric applications.

Nov 72017
Hall AHarvard
Matthew Morrow, CNRS
(Topological) Hochschild homology and and crystalline cohomology
abstract±
Cyclic homology and its variants (periodic cyclic homology, negative cyclic homology) were introduced by Alain Connes and Boris Tsygan in the 1980s by exploiting the action of the circle on Hochschild homology. For smooth varieties over a field of characteristic zero, classical theorems in the subject show that they encode the de Rham cohomology and the Hodge filtration; the trace map from Ktheory to cyclic homology is then a manifestation of the de Rham character. I will give an overview of this classical story and then pass to finite characteristic, in which a similar framework now exists — at the expense of working instead with topological Hochschild and cyclic homology. Joint work with Bhargav Bhatt and Peter Scholze.

Nov 142017
4237 MIT
Ananth Shankar, MIT
The pcurvature conjecture and monodromy about simple closed loops
abstract±
The GrothendieckKatz pcurvature conjecture is an analogue of the Hasse Principle for differential equations. It states that a set of arithmetic differential equations on a variety has finite monodromy if its pcurvature vanishes modulo p, for almost all primes p. We prove that if the variety is a generic curve, then every simple closed loop has finite monodromy.

Nov 212017
Hall AHarvard
Dhruv Ranganathan, MIT
Curves, maps, and singularities in genus one
abstract±
I will introduce a framework in which to study genus one curve singularities using logarithmic and tropical geometry, and its relationship to the moduli space of curves and maps. This builds on work of Speyer, Smyth, Viscardi, and VakilZinger and leads to the construction of nonsingular modular compactifications of the space elliptic curves in projective space, in particular recovering a construction of Vakil and Zinger. I will discuss an appropriate generalization of this result to the pairs case for toric targets. In this setting, there is a fruitful interplay between the geometry of the space of maps and a subtle theorem of Speyer in tropical geometry from 2005. Time permitting, I will explain how the same framework leads to modular factorizations of the rational maps among Smyth’s spaces of pointed elliptic curves. This is joint work with Keli SantosParker and Jonathan Wise.

Nov 282017
4237MIT
Valentijn Karemaker, U Penn
Dynamics of Belyi maps
abstract±
A (genus 0) Belyi map is a finite map from the projective line to itself, branched exactly at 0, 1, and infinity. Such maps can be described combinatorially by their generating systems. Assuming further that 0, 1, and infinity are both fixed points and the unique ramification points above 0, 1, and infinity respectively yields dynamical Belyi maps, since the resulting maps can be iterated and will therefore exhibit dynamical behaviour. In this talk, we will discuss several results on the dynamics, reductions, and monodromy of dynamical Belyi maps, and the interplay between these. (This is joint work with J. Anderson, I. Bouw, O. Ejder, N. Girgin, and M. Manes.)

Dec 52017
4237MIT
Felix Janda, U Michigan
Structures for GromovWitten invariants of quintic threefolds
abstract±
Quintic threefolds are the simplest examples of CalabiYau threefolds,
yet their higher genus GromovWitten theory is not very well
understood. Still, there are many predictions from physics, for example
saying that their generating functions lie in a finitely generated ring
(of modular forms). In my talk, I will survey these predictions, the mathematical progress,
and explain a new approach to the predictions using a new (logarithmic) moduli space.

Dec 122017
Hall AHarvard
Marco Gualtieri, U Toronto
Holomorphic symplectic Morita equivalence and the generalized
Kaehler potential
abstract±
Since the introduction of generalized Kaehler geometry in
1984 by Gates, Hull, and Rocek in the context of twodimensional
supersymmetric sigma models, we have lacked an understanding of
the degrees of freedom inherent in the geometry. In particular, the
description of a usual Kaehler structure in terms of a complex manifold
together with a local Kaehler potential function is not available for
generalized Kaehler structures, despite many positive indications in
the literature over the last two decades. I will explain recent work
showing that a generalized Kaehler structure may be viewed in terms of
a holomorphic Morita equivalence between Poisson manifolds; this
allows us to solve the problem of existence of a generalized Kaehler
potential.

Dec 192017
4237MIT
David Smyth, Tufts
Intersection numbers on logcanonical models of moduli spaces of curves
abstract±
The WittenKontsevich theorem concerns the recursive structure of top intersections of psiclasses on the moduli space of `stable' pointed curves. These intersection numbers depend on a definition of stability, with different stability conditions leading to different compactifications. In genus one, there is a natural sequence of mstability conditions, where varying the parameter m gives all logcanonical models appearing in the HassettKeel program. In this talk, I’ll explain how to compute the top intersections of psiclasses on the moduli space of mstable genus one curves, for arbitrary m.
This seminar is organized by Joe Harris (Harvard), Philip Engel (Harvard), Brooke Ullery (Harvard), Davesh Maulik (MIT), Georg Oberdieck (MIT), Dhruv Ranganathan (MIT). 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.