Last updated: May 18,
2018, 09:26 EDT
Conference Titles, Abstracts, and Slides
Karin Baur, Friezes and Grassmannian cluster
categories.
Abstract: In this talk, I will explain how to obtain
SL_k-friezes using Pluecker coordinates by taking certain
extension-closed subcategories of the Grassmannian cluster
categories C(k,n). These are cluster structures associated to
the Grassmannians of k-spaces in n-space. Many of these
friezes arise from specializing a cluster-tilting object to 1.
We use Iyama-Yoshino reduction to reduce friezes to smaller
rank. This is joint work with E. Faber, S. Gratz, G. Todorov
and K. Serhiyenko.
Vladimir Bavula, Localizable and weakly left
localizable rings.
Abstract: Two new classes of rings are introduced - the class
of left localizable and the class of weakly left localizable
rings. Characterizations of them are given.
Charlie Beil, Homotopy dimer algebras on hyperbolic
surfaces.
Abstract: I will report on work in progress with Karin Baur on
a class of quiver algebras called homotopy algebras. The
quiver of a homotopy algebra embeds into a compact surface,
and the relations are induced by path homotopy. I will discuss
certain algebraic and homological properties of homotopy
algebras in the case where the surface is hyperbolic, and
describe how these properties interact with the topology of
the surface.
Frauke Bleher, Top exterior powers in Iwasawa
theory.
Abstract: This talk is about joint work with Ted Chinburg,
Ralph Greenberg, Mahesh Kakde, Romyar Sharifi and Martin
Taylor about higher codimension Iwasawa theory. Iwasawa theory
produces from Galois theory interesting finitely generated
torsion modules M for a Noetherian integral domain R. If all
the localizations of M at codimension one primes of R are
trivial, one says M is pseudo-null. Put differently, the first
Chern class of M is zero. In this talk I will focus on a
purely module theoretic result that relates the maximal
pseudo-null submodules of quotient modules of the top exterior
power of certain M to the maximal pseudo-null submodules of
quotient modules of R. I will then indicate how this leads to
a result on second Chern classes of Iwasawa modules.
İlke Çanakçı, Perfect matchings and the
Grassmannian cluster algebra.
Abstract: The homogeneous coordinate ring of the Grassmannian
admits a cluster algebra structure by Scott. A
categorification of Grassmannian cluster algebras was
introduced by Jensen-King-Su, and so-called dimer algebras
were given as endomorphism algebras of cluster-tilting objects
in this category by Baur-King-Marsh. Certain cluster
variables, namely the twisted Plücker coordinates, in
Grassmannian cluster algebras have been expressed by
Marsh-Scott as dimer partition functions, i.e. as sum over
perfect matchings. The twisted Pluecker coordinates can also
be given by the Caldero-Chapoton cluster character (CC)
formula. We relate the Marsh-Scott formula to the CC formula
by defining perfect matching modules over dimer algebras
associated to perfect matchings appearing in the Marsh-Scott
formula. This is joint work with Alastair King and Matt
Pressland.
William Crawley-Boevey, Representations of equipped
graphs.
Abstract: This is an abstract for one of the expository talks.
An equipped graph can be thought of as a generalization of a
quiver, in which one is allowed not just arrows, but also
edges with two heads or two tails. Gelfand and Ponomarev
introduced representations of equipped graphs by vector spaces
and linear relations, and, generalizing Gabriel's Theorem,
they showed that the equipped graphs of finite representation
type are exactly those whose underlying graph is a Dynkin
diagram. For example, by taking all edges to be two headed or
two tailed, one can study the representation theory of a graph
without needing to choose an orientation. I will discuss an
analogue of Kac's Theorem for equipped graphs, which I
published in 2013, and give illustrations of the
Auslander-Reiten theory for equipped graphs.
Samuel Dean, Auslander's formula and the
MacPherson-Vilonen construction.
Abstract: I will discuss a localisation commonly referred to
as "Auslander's formula", which provides an exact left adjoint
to the Yoneda embedding. I will recall its construction and
explain what it detects. I will show that it can be extended
to a recollement, and give necessary and sufficient conditions
for this recollement to be pre-hereditary. In that case, I
will show that it is also an instance of the
MacPherson-Vilonen construction.
Özgür Esentepe, Annihilation of cohomology over
curve singularities.
Abstract: For a commutative Gorenstein ring, we study the
cohomology annihilator ideal which consists of all ring
elements that annihilate all Ext-modules Ext^n(M,N) for any
finitely generated R-modules M, N and sufficiently large n.
This ideal is closely related to the singularities or our
commutative ring. Indeed, an element is in this ideal if and
only if annihilates (the "stable" endomorphism ring of) every
object in the singularity category. We will discuss this
relation and show that, under reasonable assumptions, in
dimension 1 the cohomology annihilator ideal is equal to the
conductor ideal. If time permits, we will also make
connections with noncommutative resolutions.
Mboyo Esole, Minimal models, hyperplane
arrangements, and gauge theories.
Abstract: I will review new developments in string geometry
focussing on methods of hyperplane arrangements inspired by
gauge theory to study the structure of the cone of movable
curves of crepant resolutions of elliptic fibrations. As a
bonus, I will also review a new pushforward formula in
intersection theory that relies on an old formula of
Jacobi. I will explain how this pushforward formula is used to
compute Euler characteristics of varieties defined by a
sequence of blowups with centers that are regular imbeddings.
Sergio Estrada, The homotopy category of
projectives over a non affine scheme via very flats.
Abstract: The category of quasi-coherent sheaves on a
non-affine scheme is well known not to have enough
projectives. Neeman and Murfet have remedied this lack
by defining the derived category of flats as a suitable
replacement of the homotopy category of projectives. This is
so because a celebrated result by Neeman states that, in the
affine case, the two categories are equivalent. But many
concrete schemes satisfy the so-called resolution property,
i.e. they have enough locally frees (so, in particular, enough
infinite dimensional vector bundles in the sense of Drinfeld)
which constitute a good replacement of the flats. In the talk
we will show that, for such schemes, the derived category of
flats is still triangulated equivalent to the derived category
of vector bundles. The equivalence is indirect and strongly
uses the class of very flat sheaves as recently defined by
Positselski. The talk is based on a joint paper with Alexander
Slávik.
Al Garver, Biclosed sets in representation theory.
Abstract: It was recently proved by Demonet, Iyama, Reading,
Reiten, and Thomas that the lattice of torsion classes of a
finite dimensional representation finite algebra is
congruence-uniform. We study another class of
congruence-uniform lattices called biclosed sets. Every such
lattice is defined by the choice of a gentle algebra. For
gentle algebras with the property that every indecomposable
module is a brick, we show that the corresponding lattice of
biclosed sets is isomorphic to a lattice of certain
subcategories of the module category of an analogue of the
preprojective algebra, which we call torsion shadows. If time
permits, we present a similar description of the shard
intersection order of these lattices. This is joint work with
Thomas McConville and Kaveh Mousavand.
Ken Goodearl, Poisson clusters and unique
factorization.
Abstract: Many cluster algebras are known to be UFDs, with
clusters consisting of prime elements. Moreover, large classes
of algebras that are known or conjectured to be cluster
algebras have compatible Poisson structures. (This is, in
fact, natural and expected for semiclassical limits of quantum
algebras.) We will discuss the reverse direction -- how
combinations of unique factorization and Poisson structures
lead to cluster structures, without any a priori cluster
combinatorics. A large class of Poisson UFDs thus
automatically become cluster algebras. This is joint work with
Milen Yakimov.
Sira Gratz, A class of A^1-homotopy phantoms.
Abstract: In joint work with Greg Stevenson we show that in
many cases, A^1-homotopy invariants of the (ungraded)
singularity category of a finitely generated algebra admitting
a suitable grading can be computed relatively easily. Rather
than discuss this in depth, we will take the opportunity of
this talk to advertise a result that follows directly from
computations by Tabuada: Cluster categories of Dynkin type
A_2n are "A^1-homotopy phantoms". Discussing this example will
illustrate some of the methods used for the computation of
A^1-homotopy invariants for singularity categories.
Amihay Hanany, Coulomb branches and symplectic
singularities.
Abstract: This talk deals with construction of symplectic
singularities using techniques from super symmetric gauge
theories - the so called Coulomb branch. There are quiver
operations that give relations between different moduli spaces
which will be presented and discussed in detail.
Lutz Hille, Spherical and exceptional
modules for the Auslander algebra of k[T]/T^nv.
Abstract: (joint with David Ploog) In this talk we classify
spherical and exceptional modules over the Auslander algebra
of the truncated polynomial ring k[T]/T^n. Using the
classification of full exceptional sequences we can construct
tilting modules for this algebra, parametrized by pairs of
permutations of n elements. We compare this result with the
recent classifcation of all tilting modules by Geuenich.
An Huang, Hasse-Witt matrices and period integrals.
Abstract: I shall explain a program to relate the arithmetic
of Calabi-Yau hypersurfaces in toric varieties or flag
varieties, to their period integrals at the large complex
structure limit in mirror symmetry. In particular, this leads
to a proof of a recent conjecture of Vlasenko regarding higher
Hasse-Witt matrices. It is a joint work with Bong Lian,
Shing-Tung Yau and Cheng-long Yu.
Birge Huisgen-Zimmermann, Truncated path algebras,
a geometric and homological stepping stone.
Abstract: Given any basic finite dimensional algebra \Lambda
over an algebraically closed field, there is a unique
truncated path algebra \Lambda_{{trunc}} that shares quiver
and Loewy length with \Lambda. Clearly, the category
\Lambda-mod embeds into \Lambda_{{trunc}}-mod, and, for any
dimension vector d of the quiver of \Lambda, the
parametrizing variety Rep_{d}(\Lambda) embeds into
Rep_{d}(\Lambda_{trunc}) as a closed subvariety. Based
on these simple observations, the strategy of moving back and
forth between the two algebras provides an effective approach
to exploring \Lambda-mod. This motivates the program of
developing the representation theory of truncated path
algebras to a level matching that attained for hereditary
algebras, the latter being the ``simplest" algebras that
provide a similar foundation for the algebras with acyclic
Gabriel quivers.
Karin Jacobsen, d-Abelian quotients of
(d+2)-angulated categories.
Abstract: Let C be a suitable (d+2)-angulated category for an
integer d⩾1. If T is a cluster tilting object in the sense of
Oppermann-Thomas and I=add T is the ideal of morphisms
factoring through an object of add T, then we show that C/I is
d-abelian. We actually show that if Γ=End T is the
endomorphism algebra of T, then C/I is equivalent to a
d-cluster tilting subcategory of mod Γ in the sense of Iyama;
this implies that C/I is d-abelian. Moreover, we show that Γ
is a d-Gorenstein algebra. More general conditions which imply
that C/I is d-abelian will also be determined. This is joint
work with Peter Jørgensen.
Ryo Kanda, Normal extensions of Artin-Schelter
regular algebras and flat families of Calabi-Yau central
extensions.
Abstract: This is a joint work with Alex Chirvasitu and S.
Paul Smith. We introduce a new method to construct
4-dimensional Artin-Schelter regular algebras as normal
extensions of 3-dimensional ones. When this is applied to a
3-Calabi-Yau algebra, we obtain 4-Calabi-Yau algebras that
form a flat family over a projective space. Our method is a
rich source of new 4-dimensional regular algebras. Some of the
4-dimensional regular algebras discovered by
Lu-Palmieri-Wu-Zhang also arise as outputs of our construction
and our result gives a new proof of regularity for those
algebras.
Ralph Kaufmann, Feynman categories in geometry and
physics.
Abstract: We will discuss Feynman categories. These are
abstract categories designed to handle higher operations, such
as those coming from correlation functions. Concrete examples,
which we will discuss, are built on graphs. The
mathematical/categorical treatment allows us to use standard
constructions, such as push-forward and pull-back. Pairing
this with a transformation that takes combinatorial input and
outputs topological spaces in terms of cubical complexes, we
discuss moduli spaces and the complexes arising in Cutkosky
rules and Outer space.
Mark Kleiner, Preprojective quiver of a
Coxeter group.
Abstract: Certain results on representations of quivers have
analogs in the structure theory of general Coxeter
groups. A fixed Coxeter element c turns the Coxeter
graph into an acyclic quiver, the c-quiver. A positive root is
c-preprojective if a positive power of c takes it to a
negative root. A Coxeter group is finite if and only if
every positive root is c-preprojective. The graded
c-preprojective quiver is an enlargement of the
c-quiver. The construction is analogous to, but
different from, that of the graded preprojective algebra of a
general quiver. The c-preprojective roots are explicitly
described in terms of the graded paths in the c-preprojective
quiver.
Maitreyee Kulkarni, A combinatorial Fourier
transform for quiver representation varieties in type A.
Abstract: For a given dimension vector d, we consider the
space of representations of the linearly-oriented type A
quiver. A product of general linear groups acts on this
space, and the orbits are isomorphism classes of
representations with dimension vector d. In this
setting, we introduce a combinatorial algorithm to describe
the Fourier--Sato transform; this algorithm matches up orbits
for the type A quiver with orbits for its reversed quiver in
an interesting way. Last year at the Auslander
Conference, we learned from Thomas Brüstle and Lutz Hille of
another map between these collections of orbits (called the
Knight--Zelevinsky multisegment duality). We now know
that our combinatorial Fourier--Sato algorithm and its inverse
both give the same map as multisegment duality. The only
proof we know that these three algorithms are the same is
purely geometric. This is joint work with Pramod Achar
and Jacob Matherne.
Xiahua Luo, Gorenstein projective modules for the
working algebraist.
Abstract: In my talk, I'll give a brief survey on the
construction of Gorenstein-projective modules. These modules
was firstly introduced by M. Auslander in the name of
G-dimension zero modules in 1967. Enochs and Jenda used the
notion of Gorenstein-projective modules in 1995. It turns out
that these modules play a very important role in relative
homological algebra, Tate cohomological theory, poset
representation theory, invariant subspace of linear operator
and so on. However, until around ten years ago, the
construction of these modules was far from being known well.
From then, our workshop in Shanghai began to study this
problem. Frist, the explicit construction over upper
triangular matrix algebras was given by P. Zhang, Z. W. Li and
B. L. Xiong etc. Then the Gorenstein-projective quiver
representations were described via separated monic
representations in joint work with P. Zhang. In this talk,
I'll give more details about the construction of these modules
over tensor products of finite dimensional algebras. This is a
joint work with W. Hu, B.L. Xiong and G.D. Zhou.
Ivan Martino, Finite groups generated in real
codimension two.
Abstract: In this talk, I introduce the new notions of finite
linear groups generated by elements that fix subspace of
codimension one or two. This leads to a generalization of
finite reflection groups.
Jacob Matherne, Singular Hodge theory of matroids.
Abstract: To any matroid, I will associate a certain ring
that, when the matroid is realizable, is the cohomology ring
of a certain variety called the semi-wonderful model. I
will show how the Hodge theory of this ring can conjecturally
be used to establish the "top-heavy conjecture" of Dowling and
Wilson from 1974, as well as the non-negativity of the
Kazhdan-Lusztig polynomials of Elias, Proudfoot, and
Wakefield. This is joint work with Tom Braden, June Huh,
Nick Proudfoot, and Botong Wang.
Jordan McMahon, Fabric idempotent ideals and
homological dimensions.
Abstract: For a finite-dimensional algebra A, and an A-module
M, it is interesting to analyse which terms in the projective
resolution of M are generated by a particular projective
A-module. This question was related to the homological
properties of idempotent ideals by Auslander-Platzeck-Todorov.
We introduce the notion of a fabric idempotent in order to
illustrate this theory for classes of algebras arising from
higher Auslander-Reiten theory. Time permitting, we use
similar techniques to describe singularity categories for
higher Nakayama algebras, generalising a result of Chen-Ye.
Frantisek Marko, Supersymmetric elements in
positive characteristic.
Abstract: We start with supersymmetric polynomials over a
field of positive characteristic. Then we explain analogous
concepts for the distribution algebra of a torus of a general
linear supergroup and for the divided powers algebra.
Van Nguyen, Finite generation of the
cohomology rings of some pointed Hopf algebras.
Abstract: Over a field of prime characteristic p > 2, we
prove that the cohomology rings of some pointed Hopf algebras
of dimension p^3 are finitely generated. These are Hopf
algebras arising in the ongoing classification of finite
dimensional pointed Hopf algebras in positive characteristic.
They include bosonizations of Nichols algebras of Jordan type
in a general setting. Our proofs are based on an algebra
filtration and a lemma of Friedlander and Suslin, drawing on
both twisted tensor product resolutions and Anick resolutions
to locate the needed permanent cocycles in May spectral
sequences. In this talk, I will describe the two resolutions
of interest and explain our main finite generation results.
Julia Pevtsova, Super elementary subgroups.
Abstract: For a finite group G, classical theorems of Quillen
and Chouinard tell us how to detect whether a cohomology class
in mod p cohomology is nilpotent or
whether a module is projective: one has to restrict to
elementary abelian subgroups of G. For other finite
dimensional algebras with interesting cohomology rings
families of subalgebras which detect nilpotent and
projectivity take on a more sophisticated shape if they can be
found at all. I will review the classical theory
starting with Quillen and describe what plays the role of an
elementary abelian subgroup for finite group schemes and
finite supergroup schemes.
Sebastian Posur, A constructive approach to Freyd
categories.
Abstract: In this talk we demonstrate that important parts of
category theory such as the theory of Freyd categories are
inherently algorithmic and in fact implementable in computer
algebra systems. Freyd categories are a universal way of
equipping given additive categories with cokernels. Their
constructive nature yields unified data structures for
explicit computations within the category of finitely
presented modules over computable rings, finitely presented
functors over computable abelian categories, and free abelian
categories associated to path algebras.
Matthew Pressland, A tilting viewpoint on higher
Auslander algebras.
Abstract: To any finite-dimensional algebra, one may associate
two sequences, of 'canonical' tilting and cotilting modules
respectively, the length of which depends on the dominant
dimension of the algebra. Typically these sequences do not
intersect---indeed, they do so if and only if each is the
other read backwards, and this property characterises
(minimal) Auslander--Gorenstein algebras, in the sense of
Iyama--Solberg. If time allows, I will also explain some other
special properties and applications of the modules appearing
in these sequences.
Mike Prest, Tensor products on free abelian
categories and Nori motives.
Abstract: This is joint work with Luca Barbieri-Viale and
Annette Huber, see arXiv:1803.00809, motivated by asking
whether the tensor product structure on Nori motives can be
obtained through the algebraic construction of these via free
abelian categories. There is indeed a general
construction which includes lifting a tensor product on
R-modules to the associated functor category on finitely
presented modules. In model-theoretic terms, this lifts
a tensor product on R-modules to one on pp-pairs.
Manuel Saorín, t-Structures in the base of a
derivator for which the heart is a Grothendieck category.
Abstract: We give a natural definition of t-structure on a
strong stable derivator and show that it is completely
determined by its restriction to the base of the derivator.
With this at hand, we tackle the problem of giving sufficient
conditions on a t-structure on that base so that the
corresponding heart is an AB5 abelian category or/and has a
generator. For the AB5 condition we introduce the concept of
homotopically smashing t-structure, which is shown to lie
between the classical concepts of compactly generated and
smashing t-structure. We will show that any homotopically
smashing t-structure has a heart which is AB5. As for the
property of having a generator, we show that it is always the
case when the triangulated category is the homotopy category
of a combinatorial stable model category and the objects of
the co-aisle satisfies certain homotopical smallness
condition with respect to a large enough regular cardinal. As
a consequence, it will follow that in the homotopy category of
any combinatorial stable model category each compactly
generated t-structure has a heart which is a Grothendieck
category.
Ralf Schiffler, Cluster algebras and Jones
polynomials.
Abstract: This talk is on a very concrete connection between
cluster algebras and knot theory. A special class of knots,
the 2-bridge knots (or links), are parametrized by continued
fractions. On the other hand, we can associate to every
continued fraction a so-called snake graph and then define a
cluster variable whose Laurent expansion is given as a sum
over all perfect matchings of the snake graph. We thus obtain
a cluster variable for every 2-bridge knot. Knot invariants
is one of the main branches in knot theory and the Jones
polynomial is an important knot invariant. It is a Laurent
polynomial in one variable t. We show that up to normalization
by the leading term, the Jones polynomial of a 2-bridge
knot (or link) is equal to the specialization of the
associated cluster variable obtained by setting all initial
cluster variables to 1 and specializing the initial principal
coefficients of the cluster algebra as follows: y_1 =
t^{−2} and y_i =−t^{−1}, for all i>1. As a
consequence we obtain a direct formula for the Jones
polynomial of a 2-bridge link as the numerator of a continued
fraction of Laurent polynomials in q=−t^{−1}. This is
joint work with Kyungyong Lee.
Sibylle Schroll, A geometric model of the derived
category of a gentle algebra.
Abstract: On joint work with Sebastian Opper and Pierre-Guy
Plamondon. In this talk we will associate to every finite
dimensional gentle algebra A an oriented surface with marked
points in the boundary S_A. This surface gives a geometric
model of bounded derived category D^b(A) of A. Furthermore, it
follows from work of Haiden-Katzarkov-Kontsevich and
Lekili-Polishchuk that if A is homologically smooth then
D^b(A) is equivalent to the partially wrapped Fukaya category
of S_A.
Amit Shah, Quasi-abelian hearts of twin cotorsion
pairs on triangulated categories.
Abstract: Nakaoka defines a twin cotorsion pair on a
triangulated category to be a pair of cotorsion pairs ((S, T),
(U, V)) satisfying S is contained in U. A certain subfactor
category H (called the heart) is associated to such a pair,
and it is shown to be semi-abelian. We prove that, under a
mild assumption on the twin cotorsion pair, H is a
quasi-abelian category. In particular, for a cluster category
C (with shift [1]) and rigid object r in C, Nakaoka calculates
that H = C/X, with ((S, T), (U, V)) = ((add r[1], X), (X,
Y[-1])), where X = Ker(Hom(r,−)) and Y = {a in C | Hom(X,a) =
0}. Buan & Marsh establish that C/X is integral and that a
canonical localisation of it is equivalent to mod(End r)^op.
An application of our result shows that C/X is in fact
quasi-abelian, and hence there are many aspects of
Auslander-Reiten theory that hold in C/X.
Alexander Slávik, Flat modules over noetherian
rings with countable spectrum.
Abstract: N/A
Louis-Philippe Thibault,
Graded bimodule Calabi-Yau
algebras.
Abstract: Higher preprojective algebras were defined as part
of Iyama's higher Auslander-Reiten theory. They are endowed
with a grading that gives them a structure of bimodule
Calabi-Yau algebra of Gorenstein parameter 1. We explain that
this grading does not exist in two different settings, namely
when considering tensor products of Koszul preprojective
algebras as well as some skew-group algebras of finite
subgroups of SL(n,k) acting on polynomial rings. The
latter setting contrasts with the classical case n=2, in which
every skew-group algebra is Morita equivalent to a
preprojective algebra. These classes of bimodule Calabi-Yau
algebras are however endowed with a natural grading that gives
them a Gorenstein parameter higher than 1. We thus conclude by
considering questions regarding these algebras, whose
properties seem to be less known.
Kurt Trampel, Noncommutative discriminants of
quantum cluster algebras.
Abstract: The notion of a discriminant has found many uses in
studying noncommutative algebras over the past five years.
However, directly computing the discriminant can be difficult.
We present a theorem describing the discriminant for certain
subalgebras of quantum cluster algebras at a root of unity.
Hipolito Treffinger, On c-vectors of finite
dimensional algebras.
Abstract: In the theory of cluster algebras, g-vectors and
c-vectors play a fundamental role. With the introduction of
tau-tilting theory by Adachi, Iyama and Reiten, as many other
objects arising from the cluster setting, c-vectors and
g-vectors started to have a natural formulation in the
language of representation theory of finite dimensional
algebras. In this talk we will show that the set c-vectors of
an algebra is always the set of dimension vectors of certain
bricks in its module category. As an application we will show
how the c-vectors determine the wall and chamber structure of
the algebra.
José Vélez-Marulanda, Derived tame Nakayama
algebras.
Abstract: N/A
Ying Zhou, Tame quivers have finitely many
m-maximal green sequences.
Abstract: Keller introduced the concept of maximal green
sequences. Brustle-Dupont-Perotin proved that tame quivers
have finitely many maximal green sequences. We have
generalized the result to m-maximal green sequences.
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