Overview:
Noncommutative projective geometry first emerged as a distinct discipline
during the late `80s and early `90s, as part of the work of Artin, Schelter,
Stafford, Tate, Van den Bergh, Zhang, and others, who imported techniques
from algebraic geometry to understand the Sklyanin algebras. From these beginnings,
a general theory eventually emerged in which powerful techniques from classical
algebraic geometry were modified to give insight into the class of noncommutative
graded algebras. The subject has since grown rapidly and links with commutative
algebra, algebraic geometry, representation theory, and other disciplines
have become apparent. This workshop will give experts in these fields the
opportunity to exchange ideas and will allow postdocs and graduate students
to interact with established researchers and develop their research programs.
The workshop will comprise two types of talks: expository and specialized.
The expository talks consist of three 50 minutes introductory lectures:
- Matthew Ballard (University of South Carolina) on connections
between geometric algebra and mirror symmetry
- Graham Leuschke (Syracuse University) on connections between
noncommutative geometry and commutative algebra via noncommutative
resolutions
- Toby Stafford (University of Manchester) on a general introduction
to and survey of noncommutative algebraic geometry along with
a review of open problems
The remaining speaking slots will be distributed among both the junior
and senior participants. In order to facilitate communication among the
participants and to foster a sense of inclusion among the junior participants,
we propose that on the first afternoon of the workshop postdocs who are
not giving lectures and each graduate student give short presentations (roughly
five minutes) on their research. All other talks will be 50 minutes in length.
***
Schedule:
| Wednesday July 8, 2015. |
| Time |
Speaker |
Talk Title |
| 9:10-10:00am |
Matthew Ballard
(University of South Carolina) |
|
| 10:10-11:00am |
Toby Stafford
(University of Manchester) |
An Introduction to Noncommutative
Algebraic Geometry 2 : Slides
|
| 11:00-11:30am |
Coffee Break
|
| 11:30-12:20pm |
Graham Leuschke
(Syracuse University) |
|
| 12:20-2:30pm |
Lunch
|
| 2:30-3:20pm |
Free Afternoon
|
| 3:20-4:00pm |
| 4:00-4:50pm |
| 5:00pm |
***
Abstracts:
Matthew Ballard (University of South Carolina)
Mirror symmetry through exceptional collections
Perhaps the most successful approach for establishing Homological Mirror
Symmetry is to identify "nice" generators on each side and make
sure their endomorphisms match up. This leads to a few natural questions:
How does one find these nice generators on each side? Why do they match
up? On the A-side, one often gets almost them for free from the data. On
the B-side, well, I would roughly say from Birational Geometry. By the end
of the lecture series, I hope you will feel satisfied with this answer.
As for the last question, that is something to think about.
The three talks will roughly proceed as follows:
Lecture 1: What is mirror symmetry? An introduction to the categories involved
in mirror symmetry for Fano toric varieties.
Lecture 2: Mirror symmetry in some examples via exceptional collections.
Lecture 3: Finding exceptional collections.
Karin Baur (University of Graz)
Dimer models and categories with Grassmannian structure (coauthors:
Alastair King, Robert Marsh) We associate a dimer algebra A to a Postnikov
diagram D (in a disk) corresponding to a cluster of minors in the cluster
structure of the Grassmannian Gr(k, n). We show that A is isomorphic to
the endomorphism algebra of a corresponding Cohen-Macaulay module T over
the algebra B used to categorify the cluster structure of Gr(k, n) by Jensen-King-Su.
It follows that B can be realised as the boundary algebra of A, that is,
the subalgebra eAe for an idempotent e corresponding to the boundary of
the disk. The construction and proof uses an interpretation of the diagram
D as a dimer model with boundary. We also discuss the general surface case,
in particular computing boundary algebras associated to the annulus.
Pieter Belmans (Departement Wiskunde-Informatica)
Derived categories of noncommutative quadrics and Hilbert schemes of
points
Noncommutative deformations of quadric surfaces have been classified by
Van den Bergh using regular cubic Artin--Schelter $\mathbb{Z}$-algebras
of dimension three. The derived category of a noncommutative quadric surface
can be described using an exceptional sequence of four objects, as in the
commutative case. Orlov has constructed a general procedure that embeds
every triangulated category with a full exceptional collection into the
derived category of a smooth projective variety (see arXiv:1402.7364), but
the construction does not have an interesting interpretation as there are
many choices involved that influence the outcome in non-interesting ways.
In a more recent preprint (arXiv:1503.03174) he constructs an embedding
of the derived category of a noncommutative plane into the derived category
of a variety related to the Hilbert scheme of two points on the commutative
projective plane. We study the more involved case of a noncommutative quadric,
by exhibiting an admissible embedding into the derived category of a variety
related to the Hilbert scheme of two points on the commutative quadric surface.
Joint work with Theo Raedschelders and Michel van den Bergh.
Will Donovan (Kavli IPMU, University of Tokyo)
Contraction algebras and braiding of flops
I explain a construction, joint with M. Wemyss, which associates noncommutative
algebras to suitable rational curves on complex 3-folds. These algebras
control the deformation theory of such curves, and may be used to construct
new actions of braid-type groups on the derived category of coherent sheaves
on the 3-fold. I will give examples, and indicate how these groups arise
topologically from certain simplicial hyperplane arrangements.
Lutz Hille (Universität Münster)
Weighted projective spaces, crepant resolutions and tilting (joint
with R. Buchweitz)
We consider three types of 'weighted projective spaces', the classical one,
the one in the sense of Baer, Geigle and Lenzing, and the toric stacks.
The classical one is defined by a quotient of a C^*-action defined by a
sequence of weights (q_0,...,q_n). If this sequence is reduced, then the
associated toric variety isFano, precisely when each q_i devides the sum
q := \sum q_j. Moreover, let Y be a crepant resolution of a Fano X. Associated
to the weights, there is also a weighted projective space P in the sense
of Baer. This is also a toric stack in the sense of Borisov, Hu and Kawamata.
The latter admits a full, strongly exceptional sequence of line bundles
O(i) for i=0,...,q-1. We construct a full, strongly exceptional sequence
of line bundles on any crepant resolution Y of X, so that the endomorphism
algebra coincides with the one on P. Consequently, Y and P have equivalent
derived categories of coherent sheaves. We close with some examples and
some applications.
Osamu Iyama (Nagoya University)
Partial preprojective algebras and cDV singularities
We discuss tilting theory for partial preprojective algebras, which are
subrings eAe for preprojective algebras A and idempotents e. We classify
certain tilting modules over a partial preprojective algebra in terms of
the Coxeter group and Weyl chambers. We apply this result to study cDV singularities
R. Although R does not necessarily have non-commutative crepant resolutions,
it always has a maximal modifying (=MM) module M. Using correspondence between
MM R-modules and tilting End_R(M)-modules, we classify MM R-modules. This
refines Auslander-McKay correspondence due to Wemyss.
Graham Leuschke (Syracuse University)
Non-commutative desingularizations and Cohen-Macaulay representations
These lectures will describe the work done over the last several years,
by many sets of authors, on non-commutative analogues of resolutions of
singularities. I will start by discussing the McKay Correspondence, a key
motivating example, and then consider several desirable features of any
potential definition of the phrase "non-commutative desingularization",
including Van den Bergh’s definition of non-commutative crepant resolutions.
Throughout I will emphasize the role played by maximal Cohen-Macaulay modules.
Izuru Mori (Shizuoka University)
Stable categories of graded maximal Cohen-Macaulay modules over noncommutative
quotient singularities (coauthors: Kenta Ueyama)
Let S be a noetherian AS-regular Koszul algebra and G is a finite group
acting on S such that SG is an AS-Gorenstein isolated singularity. In this
talk, we will show that the stable category of graded maximal Cohen-Macaulay
modules over SG has a tilting object. This is a noncommutative generalization
of the result due to Iyama and Takahashi with more conceptual proof. The
keys to prove this result are Buchweitz equivalence, Orlov embedding, and
Yamaura tilting.
Dennis Presotto (University of Hasselt)
Homological properties of a certain noncommutative Del Pezzo surface
(coauthors: Louis de Thanhoffer e Volcsey)
Recently, de Thanhoff er de Volcsey and Van den Bergh showed that Grothendieck
groups of "noncommutative Del Pezzo surfaces" with an exceptional
sequence of length 4 are isomorphic to one of three types, the third one
not coming from a commutative Del Pezzo surface.
This proposed talk covers joint work of de Thanhoffer de Volcsey and myself
which led to an explicit construction of this noncommutative surface.
In arXiv:1503.03992 we adapt the theory of noncommutative P^1-bundles as
appearing in the work of Van den Bergh, Nyman and Mori, culminating in the
construction of the desired noncommutative surface.
Theo Raedschelders (Vrije Universiteit
Brussel)
Universal coacting Hopf algebras, Schur-Weyl duality, and derived Tannaka-Krein
(joint work with Michel Van den Bergh)
For any Koszul Artin-Schelter regular algebra A, we consider the universal
Hopf algebra aut(A) coacting on A, introduced by Manin. To study the representations
(i.e. fd comodules) of this Hopf algebra, we use the Tannaka-Krein formalism
and construct a sufficiently combinatorial rigid monoidal category U, equipped
with a functor M to vector spaces that 'knows enough' about the representations
of aut(A). Using this pair (U,M) we show that aut(A) is quasi-hereditary
as coalgebra and we deduce some nice consequences.
Alice Rizzardo (University of Edinburgh)
An example of a non-Fourier-Mukai functor between derived categories
of coherent sheaves (Coauthors: Michel Van den Bergh)
Orlov’s famous representability theorem asserts that any fully faithful
exact functor between the bounded derived categories of coherent sheaves
on smooth projective varieties is a Fourier-Mukai functor. We will show
that this result is false without the fully faithfulness hypothesis. This
is joint work with Michel Van den Bergh.
S. Paul Smith (University of Washington)
Exotic elliptic algebras (joint work with Alex Chirvasitu)
The algebras of the title are new Artin-Schelter regular algebras of global
and GK dimension 4 or higher. They are noetherian and have many of the homological
properties of the polynomial ring with its standard grading. They are constructed
from Sklyanin algebras of dimension 4 or higher by a cocycle twist or descent-like
procedure that is well-enough behaved that one can transfer (some) properties
from the Sklyanin algebras to the new algebras. Like the Sklyanin algebras
their graded representation theory is controlled by an elliptic curve and
a translation automorphism. We focus on the 4-dimensional case and show
the new algebras exhibit several new features (20 point modules, for example)
that ``test'' our understanding of AS regular algebras. Their line modules
are particularly interesting, and are parametrized by 7 curves in an appropriate
Grassmanian, three of which are elliptic curves, and 4 of which are plane
conics.They also provide new non-commutative analogues of quadric surfaces.
(arXiv: 1502.01744)
Spela Spenko (University of Ljubljana)
Non-commutative resolutions of quotient singularities
This is joint work with Michel Van den Bergh. We generalize standard results
about non-commutative resolutions of quotient singularities for nite groups
to arbitrary reductive groups. We show in particular that quotient singularities
for reductive groups always have non-commutative resolutions in an appropriate
sense.
We discuss a number of examples, both new and old, that can be treated using
our methods; twisted non-commutative crepant resolutions exist in previously
unknown cases for determinantal varieties of symmetric and skew-symmetric
matrices.
Toby Stafford (University of Manchester)
Noncommutative Projective Algebraic Geometry
We will survey noncommutative algebraic geometry, with a particular emphasis
on the classification and structure of noncommutative surfaces. Time permitting
we will also discuss how the module theory, and related moduli spaces, of
these surfaces, are related to other areas.
Louis de Thanhoffer de Volcsey (University
of Toronto)
numerical classification of exceptional collections of length 4 (coauthors:
Michel Van den Bergh)
In an upcoming paper, we considered the problem of classifying exceptional
sequences on the Grothendieck group of Del Pezzo surfaces. We showed that
these groups in fact satisfy a number of interesting conditions (all of
which involve the Serre automorphism). If we axiomatically impose these
conditions on a free abelian group with a bilinear form, we can define a
number of notions which coincide with important invariants in the geometric
case such as the Picard group, the canonical sheaf, the Riemann-Roch formula,
etcetera. In this proposed talk, we will explain the details of this idea
and prove as an application that any such group with an exceptional sequence
of length 4 must be isomorphic to the Grothendieck group of either P1×P1,
F1 or a more exotic third ‘noncommutative surface'
Chelsea Walton (Temple University)
Quantum groups for non-Noetherian AS regular algebras of dimension 2
(Coauthors: Xingting Wang)
We investigate homological and ring-theoretic properties of universal quantum
linear groups that coact on Artin-Schelter regular algebras A of global
dimension 2, especially with central homological codeterminant. We also
establish conditions when Hopf quotients of these quantum groups, that also
coact on A, are cocommutative.
James Zhang (University of Washington)
The Tits Alternative
In 1972, J. Tits proved the following dichotomy: every subgroup of the
linear automorphism group of a finite dimensional vector space is either
virtually solvable or contains a free subgroup of rank two. Automorphism
groups of deformations of a polynomial ring have been studied extensively
by many mathematicians, for example, J. Alev, J. Dixmier, M. Kontsevich,
T. Lenagan, M. Yakimov and others. In this talk, we will explain why the
discriminant controls some global structures of a family of automorphism
groups. By using the discriminant, a new version of the Tits alternative
can be proved.
***
Participants:
| Last Name |
First Name |
Institution |
| Azimi |
Sepinoud |
Abo Akademi University |
| Ballard |
Matthew |
University of South Carolina |
| Baur |
Karin |
University of Graz |
| Beil |
Charlie |
University of Bristol |
| Beil |
Jason |
University of Waterloo |
| Belmans |
Pieter |
Universiteit Antwerpen |
| Briggs |
Benjamin |
University of Toronto |
| Broomhead |
Nathan |
|
| Bush Hipwood |
(Luke) Dominic |
University of Manchester |
| Chan |
Kenneth |
University of Washington |
| Chirvasitu |
Alexandru |
University of Washington |
| Crawford |
Simon |
University of Edinburgh |
| De Laet |
Kevin |
University of Antwerp Belgium |
| de Thanhoffer |
Louis |
UHasselt |
| Donovan |
Will |
Kavli IPMU |
| Ebrahim |
Ebrahim |
UCSB |
| Elle |
Susan |
UCSD |
| Esentepe |
Özgür |
University of Toronto |
| Faber |
Eleonore |
University of Vienna |
| Gaddis |
Jason |
Wake Forest University |
| Hille |
Lutz |
Universität Münster |
| Hossain |
Ehsaan |
University of Waterloo |
| Im |
Jeffrey |
University of Toronto |
| Ingalls |
Colin |
University of New Brunswick |
| Iyama |
Osamu |
Nagoya University |
| Keeler |
Dennis |
Miami University |
| Kirkman |
Ellen |
Wake Forest University |
| Madill |
Blake |
University of Waterloo |
| Mialebama Bouesso |
Andre Saint Eudes |
AIMS-SA |
| Mori |
Izuru |
Shizuoka University |
| Mousavidehshikh |
Ali |
University of Toronto |
| Mukhtar |
Muzammil |
GC University |
| Nafari |
Manizheh |
|
| Nanayakkara |
Basil |
Brock university |
| Nasr |
Amir |
University of New Brunswick |
| Nguyen |
Van |
Northeastern University |
| Nolan |
Brendan |
University of Kent |
| Omale |
Kooje |
Benue State University |
| Oppermann |
Steffen |
NTNU |
| Presotto |
Dennis |
UHasselt |
| Purohit |
Dr Rakeshwar Purohit |
university college of science MLSU UDAIPUR |
| Quddus |
Safdar |
National Institute of Science Education and Research,
Deaprtment of Atomic Energy |
| Raedschelders |
Theo |
Vrije Universiteit Brussel |
| Rayan |
Steven |
University of Toronto |
| Rizzardo |
Alice |
University of Edinburgh |
| Rogalski |
Daniel |
UCSD |
| Rosen |
Julian |
University of Waterloo |
| Shipman |
Ian |
University of Michigan |
| Sierra |
Susan |
University of Edinburgh |
| Smith |
Paul |
University of Washington |
| Spenko |
Spela |
University of Ljubljana |
| Stangle |
Joshua |
Syracuse University |
| Stevenson |
Greg |
Bielefeld University |
| Thibault |
Louis-Philippe |
University of Toronto |
| van Roosmalen |
Adam-Christiaan |
Charles University |
| Veerapen |
Padmini |
Tennessee Tech University |
| Walton |
Chelsea |
Temple University |
| Wicks |
Elizabeth |
University of Washington |
| Won |
Robert |
University of California, San Diego |
| Wu |
Quanshui |
Fudan University |
| Yazdani |
Fereshteh |
University of New Brunswick |
| Yee |
Daniel |
University of Wisconsin Milwaukee |
| Zhang |
James |
University of Washington |
| Zhang |
Pu |
Shanghai Jiao Tong University |
2015 Participants at the Fields Institute -
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