Last updated: May 04, 2019,
11:10 EDT
Conference Titles, Abstracts, and Slides
Claire Amiot, A
complete derived invariant for gentle algebras.
Abstract: In this talk I will explain how the geometric model
for the derived category of a gentle algebra permits to
exhibit a complete numerical derived invariant for gentle
algebras. This is joint work with Pierre-Guy Plamondon and
Sibylle Schroll.
Véronique Bazier-Matte, Realization of the
associahedron of cyclic Dynkin quivers.
Abstract: Consider Q be mutation-equivalent to a simply-laced
Dynkin quiver. Let A=kQ/I be the corresponding cluster-tilted
algebra and $\mathscr{A}_Q$ be the associated cluster algebra.
For M any A-module, we give an explicit construction of a
polytope which is the convex hull of the dimension vectors of
submodules of M. If every indecomposable A-module appears as a
direct summand of M, we show that the 1-skeleton of this
polytope realizes the exchange graph of $\mathscr{A}_Q$. This
work builds on papers by Arkani-Hamed, Bai, He and Yan and by
Bazier-Matte, Douville, Thomas, Yıldırım and Mousavand. The
former paper gives this construction for a linearly oriented
A_n quiver, motivated by the problem of scattering amplitudes
in quantum field theory. The latter paper generalizes the
construction to all simply-laced Dynkin types. Joint work with
Guillaume Douville.
Charlie Beil, Nonnoetherian singularities and
their noncommutative blowups.
Abstract: I will describe a new fundamental class of
varieties with nonnoetherian coordinate rings.
Specifically, given an affine variety X and a finite
collection of non-intersecting positive dimensional
subvarieties Y_i of X, a nonnoetherian coordinate ring may
be constructed whose variety coincides with X except that
each Y_i is identified as a distinct positive dimensional
(closed) point. Although it is expected that these varieties
do not admit commutative resolutions, I will explain how
their noncommutative blowups are desingularizations that
share nearly identical primitive spectra. Finally, I will
discuss how this bizarre geometry may be useful in
formulating a quantum theory of gravity, and the new
mathematical questions that arise.
Frauke Bleher, Cup products on curves over finite
fields.
Abstract: This is joint work with Ted Chinburg. Let C be a
smooth projective curve over a finite field k, and let $\ell$
be a prime number different from the characteristic of k. In
this talk I will discuss triple cup products on the first
étale cohomology group of C with coefficients in the constant
sheaf of $\ell$th roots of unity. These cup products are
important for finding explicit descriptions of the $\ell$-adic
completion of the étale fundamental group of C and also for
cryptographic applications. For the latter, one restricts the
cup product to triples of cyclic groups of order $\ell$ inside
the first cohomology group. I will describe an upper and a
lower bound for the number of different non-degenerate
trilinear maps obtained this way. Using work of McCallum and
Sharifi on cup products in Iwasawa theory, I will also present
a formula of the value of the triple cup product on a given
element in its domain. We do not know if this formula can in
general be computed in polynomial time.
Ben Blum-Smith, Cohen-Macaulayness of invariant
rings is determined by inertia groups.
Abstract: If a finite group G acts on a Cohen-Macaulay ring A,
and the order of G is a unit in A, then the invariant ring AG
is Cohen-Macaulay as well, by the Hochster-Eagon theorem. On
the other hand, if the order of G is not a unit in A then the
Cohen-Macaulayness of AG
is a delicate question that has attracted research attention
over the last several decades, with answers in several special
cases but little general theory. In this talk we show that the
statement that AG
is Cohen-Macaulay is equivalent to a statement quantified over
the inertia groups for the action of G on A acting on strict
henselizations of appropriate localizations of A. In a case of
long-standing interest — a permutation group acting on a
polynomial ring — we show how this can be applied to find an
obstruction to Cohen-Macaulayness that allows us to completely
characterize the permutation groups whose invariant ring is
Cohen-Macaulay regardless of the ground field. This is joint
work with Sophie Marques.
Diego Bravo, Igusa-Todorov and triangular matrix
algebras.
Abstract: For an abelian category $\mathcal{A}$, we define the
category PEx($\mathcal{A}$) of pullback diagrams of short
exact sequences in $\mathcal{A}$, as a subcategory of the
functor category Fun($\Delta, \mathcal{A}$) for a fixed
diagram category $\Delta$. For $M\in {\rm Ob}({\rm
PEx}(\mathcal{A}))$ we prove the existence of a short exact
sequence $0 {\to} K {\to} P {\to} M {\to} 0$ of functors, were
the objects are in PEx($\mathcal{A}$) and $P(i) \in {\rm
Proj(\mathcal{A})}$ for any $i \in \Delta$. We show that
PEx($\mathcal{A}$) is closed un direct sums and summands and
we exhibit the shape of the projective objects in
PEx($\mathcal{A}$). As an application, we prove that if
($\mathcal{C}, \mathcal{D}, \mathcal{E}$) is a triple of
syzygy finite classes of objects in mod$\Lambda$ satisfying
special conditions, then $\Lambda$ is an Igusa-Todorov
algebra. As a corollary, we get that if $\Lambda$ is a
triangular matrix algebra of $n$-syzygy finite algebras, then
$\Lambda$ is Igusa-Todorov. We also get as corollary a Theorem
by J. Wei.
Corey Bregman, Kodaira Fibrations with invariant
cohomology.
Abstract: A Kodaira fibration is a complex surface admitting a
holomorphic submersion onto a complex curve, whose fibers have
nonconstant moduli. Let X be a Kodaira fibration with
nontrivial holomorphic invariants in degree one. When the
dimension of the invariants is at most two, we show that X
admits a branched covering over a product of curves inducing
an isomorphism on rational cohomology in degree one.
Michael Brown, Standard
conjecture D for matrix factorizations.
Abstract: In 1968, Grothendieck proposed a family of
conjectures concerning algebraic cycles called the Standard
Conjectures. Grothendieck wrote the following in his paper
Standard Conjectures on Algebraic Cycles: "Alongside the
problem of resolution of singularities, the proof of the
standard conjectures seems to me to be the most urgent task in
algebraic geometry". The conjectures have been proven in some
special cases, but they remain wide open in general. In 2011,
Marcolli-Tabuada realized the Standard Conjectures as special
cases of a more general family of conjectures involving a
"noncommutative version" of algebraic cycles, which they call
the Noncommutative Standard Conjectures. The goal of this talk
is to discuss a proof, joint with Mark Walker, of one of the
Noncommutative Standard Conjectures in a special case which
does not fall under the purview of Grothendieck's original
conjectures: namely, in the setting of matrix factorizations.
Mandy Cheung, Stability conditions on Fukaya-Seidel
category for the A2 quiver.
Abstract: Kontsevich proposed the Homological mirror symmetry
which describes mirror pair (X, Y) as an equivalence of a
triangulated category between the derived category of coherent
sheaves of X and the derived Fukaya category of Y. Thus it is
reasonable to compare the stable objects in both categories.
Together with Fan and Lin, we look at the rational elliptic
surface which the corresponding derived Fukaya-Seidel category
is the derived category of the A2 quiver. While the stable
objects in for the A2 quiver is clear, the special Lagrangian
discs in the Fukaya categories is not well studied. We wish to
match the special Lagrangian with the stable objects in the
derived category of coherent sheaves.
Özgür Esentepe, Orders over Cohen-Macaulay local
rings with positive CM-relative dominant dimension.
Abstract: Over a CM local ring, there is an interesting
class of orders that has been studied intensively due to
their geometric significance, namely non-singular orders. In
this talk I will not talk about non-singular orders.
Instead, I will focus on orders which are not non-singular.
After explaining what this means, I will discuss the notion
of CM-relative dominant dimension of an order and present
some results regarding special tilting modules in the
positive CM-relative dominant dimension case, following
closely the corresponding artinian case. This is joint work
with Graham Leuschke.
Francesca Fedele, Grothendieck groups of
triangulated categories with higher cluster tilting
subcategories.
Abstract: Let C be a suitable triangulated category with a
Serre functor and an n-cluster tilting subcategory T=
add(t), for some integer n>1. In this setup, every
indecomposable in T appears in a so called Auslander-Reiten
(n+2)-angle in T. We show that the Grothendieck group of C
can be expressed as a quotient of the split Grothendieck
group of T by some elements determined by the
Auslander-Reiten (n+2)-angles in T.
Al Garver, Reverse plane partitions via quiver
representations.
Abstract: A reverse plane partition is an order reversing
map from poset to the nonnegative integers. We study reverse
plane partitions defined on the so-called minuscule posets,
which arise in the context of Lie theory. A minuscule poset
is defined by choosing Dynkin diagram and a so-called
minuscule vertex of the Dynkin diagram. We show that there
is a bijection between reverse plane partitions on a
minuscule poset and representations of a Dynkin quiver of
the corresponding type all of whose indecomposable summands
are supported on the minuscule vertex. If there is time, we
will also discuss applications of our work to a cyclic
action on reverse plane partitions known as promotion. This
is joint work with Rebecca Patrias and Hugh Thomas.
Christof Geiss, Indecomposable rigid modules and
Schur roots.
Abstract: In joint work with B. Leclerc and J. Schröer we
constructed for any generalized symmetrizable Cartan matrix
C with symmetrizer D and orientation \Omega over any field F
an 1-Iwanaga-Gorenstein algebra H:=H_F(C,D,\Omega) in terms
of a quiver with relations. After reviewing this
construction, we show
that the rigid indecomposable H-modules of finite projective
dimension are naturally parametrized by the real Schur roots
associated to (C,\Omega).
Moreover, the dimension vectors of the left finite bricks
are given by the real Schur roots associated to
(C^T,\Omega). An important device in our proof is
the construction of a F[[t]]-order which helps us to connect
H with a species and Auslander's characterization of local
regular rings.
Hernán Giraldo, Shapes of the irreducible
morphisms and Auslander-Reiten triangles in the stable
category of modules over repetitive algebras.
Abstract: For the stable category of modules over a
repetitive algebra, we show that the irreducible morphisms
fall into three canonical forms: first, all the component
morphisms are split monomorphisms (smonic case), second, all
of them are split epimorphims (sepic case), and third, there
is exactly one irreducible component (sirreducible case).
Finally, we describe the shape of the Auslander-Reiten
triangles by using the properties of the irreducible
morphisms as above. This is joint work with Yohny
Calderon-Henao and Jose Velez-Marulanda.
Emily Gunawan, Frieze vectors and unitary friezes.
Abstract: We study friezes of type Q as homomorphisms from
the cluster algebra to an arbitrary integral domain. In
particular, we show that every positive integral frieze of
affine Dynkin type A is unitary, which means it is obtained
by specializing each cluster variable in one cluster to the
constant 1. This completes the answer to the question of
unitarity for all positive integral friezes of Dynkin and
affine Dynkin types. For an arbitrary quiver Q, we introduce
a new class of integer vectors which we call frieze vectors.
These frieze vectors are defined as solutions of certain
Diophantine equations given by the cluster variables in the
cluster algebra. We establish a bijection between the
positive unitary frieze vectors and the clusters in the
cluster algebra.
Eric Hanson, The picture space of a gentle
algebra.
Abstract: Given a tau-tilting finite algebra Λ, one can
associate a finite CW complex called the picture space. It
is known that if the 2-simple
minded collections for Λ can be defined using pairwise
compatibility conditions, then the picture
space is a K(π,1), that is the cohomology of the space is
isomorphic to the cohomology of its fundamental group. This
property has been verified when Λ is hereditary or Nakayama.
The focus of this talk is to explain a counterexample
showing that in general, 2-simple minded collections cannot
be defined using pairwise compatibility conditions. More
precisely, we show that if Λ is a gentle algebra whose
quiver contains a vertex of degree at least 3 and no loops
or 2-cycles (for example, if Λ is cluster tilted of type An
for n ≥ 4), then Λ fails to have this property. This is joint work with Kiyoshi
Igusa.
Lutz Hille, Optimal slopes for Dynkin quivers.
Abstract: Stability for quiver
representations is defined by a slope function, a slope is
the quotient of an additive function by a positive additive
function. For quivers without oriented cycles there is
always a trivial slope, so that only simple modules are
stable. Conversely, one may ask for slopes with a maximal
set of stable representations, we call such a slope optimal.
In particular, slopes can be naturally ordered and one is
interested in the change of the set of stable
representations depending on the slope. In this note we show
the existence of an optimal slope for any Dynkin quiver,
where the crucial part is to solve the problem for any
orientation. For Dynkin quivers a slope is optimal,
precisely when each indecomposable representation is stable.
We also show, how this problems depends on the orientation
and how it depends on the positive function in the slope. We
also discuss open problems for tame and wild quivers.
Miodrag Iovanov, On the representation type of
quantum groups.
Abstract: Algebras of finite representation type are a
central theme in the representation theory of finite
dimensional algebras. We will give an overview of various
results and classifications of such algebras which are also
quantum groups (Hopf algebras), as well as of other Hopf
algebras with other desirable properties such as being
monomial, Nakayama, serial, etc. Our attention will be
focused mostly on pointed Hopf algebras (algebras where
simple modules are 1-dimensional), and several recent
results, current on-going research (based on various
collaborations), as well as some open questions will be
discussed.
David Jorgensen, A converse to a construction of
Eisenbud-Shamash.
Abstract: Let (Q, n, k) be a commutative local Noetherian
ring, f1, . . . , fc a Q-regular sequence in n, and R = Q/(f1,
. . . , fc). Given a complex of finitely generated free
R-modules, we give a construction of a complex of finitely
generated free Q-modules having the same homology. A key
application is when the original complex is an R-free
resolution of a finitely generated R-module. In this case our
construction is a sort of converse to a construction of
Eisenbud-Shamash yielding a free resolution of an R-module M
over R given one over Q.
Peter Jørgensen, The index in higher homological
algebra and an application to higher tropical friezes.
Abstract: We will introduce the notion of index in higher
homological algebra and show how it gives rise to higher
tropical friezes as introduced by Oppermann and Thomas.
Yuta Kimura, Finiteness of the number of torsion
classes over Noetherian algebras.
Abstract: Let R be a commutative Noetherian ring.
Classification problems of subcategories in the category of
finitely generated R-modules have been studied by many
mathematicians. For example, Serre subcategories and torsion
classes bijectively correspond to specialization closed
subsets of SpecR, by Gabriel, Stanley-Wang. In this talk, we
study torsion classes of a module finite R-algebra A. The
above results mean that there are infinitely many torsion
classes in modA, in general. Thus, we restrict R to be one
dimensional integral domain, and study when the number of
torsion classes in modA is finite.
Alistair King, Stability conditions for quivers.
Abstract: I will explain this (by now classical) notion and
how it relates to recent developments in cluster theory.
Ellen Kirkman, The Jacobian, reflection
arrangement, and discriminant for reflection Hopf algebras.
Abstract: "Let k be an algebraically closed field of
characteristic zero. When H is a semisimple Hopf algebra that
acts inner faithfully and homogeneously on an Artin-Schelter
algebra A so that A^H is also Artin-Schelter regular, we call
H a reflection Hopf algebra for A; when H=k[G] and A = k[x_1,
... , x_n] then H is a reflection Hopf algebra for A if and
only if G is a reflection group. We give some examples of
reflection Hopf algebras and show that there exist notions of
the Jacobian, reflection arrangement, and discriminant that
extend the definitions used for reflection groups actions on
polynomial algebras to this noncommutative setting.
Sanjeevi Krishnan, Bundle theory for
representations of small categories.
Abstract: Not submitted.
Kyu-Hwan Lee, Geometric description of c-vectors
and real Loesungen of cluster algebras.
Abstract: We propose a combinatorial/geometric model to
describe the c-matrices of an arbitrary skew-symmetrizable
matrix and formulate several conjectures. In particular, we
introduce real Loesungen as an analogue of real roots and
conjecture that c-vectors are real Loesungen given by
non-self-crossing curves on a Riemann surface. We investigate
the rank 3 quivers which are not mutation equivalent to any
acyclic quiver and show that all our conjectures are true for
these quivers. This is a joint work with Kyungyong Lee.
Zongzhu Lin, Representations of quantum groups at pr-th
root of 1 over p-adic fields.
Abstract: click here
Isaías David Marín Gaviria, Algorithms of
differentiation to classify some equipped posets.
Abstract: Equipped posets were introduced by A. G. Zavadskij
and his doctoral student A. V. Zabarilo at the earliest
2000’s. Soon afterwards A. G. Zavadskij classified equipped
posets of finite growth representation type by using his
algorithm of differentiation with respect to a suitable pair
of points and some suitable algorithms DVII-DIX introduced by
him for equipped posets. We recall that algorithms of
differentiation are functors with the main role of diminishing
dimension of objects of the corresponding categories. Since
the classification of equipped posets was obtained without pay
attention to the behavior of morphisms of the categories
involved in the process, the main problem of the theory of the
algorithms of differentiation consists of proving that such
functors induce a categorical equivalence. In this talk, we
describe some categorical properties of the algorithms of
differentiation DVII-DIX and some applications of these
results in the classification of equipped posets.
Frank Marko, Supersymmetry of Littlewood-Richardson
tableaux.
Abstract: The Littlewood-Richardson coefficients
C^{\lambda'}_{\mu'\nu}, counting the number of
Littlewood-Richardson tableaux of skew shape \lambda'/\mu' and
content \nu appear in the description of Hook Schur functions.
They are characters of simple supermodules over general linear
supergroups G=GL(m|n) and the ground field \mathbb{C}. The
representation theory of GL(m|n) enjoys a natural
"supersymmetry", while Littlewood-Richardson tableaux,
originating in the representation theory of symmetric groups,
lack such a "supersymmetry" . The purpose of the talk is to
show how to replace Littlewood-Richardson tableaux T by a pair
(T^+,T^-) of tableaux related by a "supersymmetry".
Agustín Moreno Cañadas,
Matrix problems associated to some Brauer
configuration algebras and categorification of magic squares.
Abstract: Bijections between solutions of the Kronecker
problem and the four subspace problem with indecomposable
projective modules over some Brauer configuration algebras are
obtained by interpreting elements of some integer sequences as
polygons of suitable Brauer configurations. This kind of
configurations are also used to categorify (in the sense of
Ringel and Fahr) magic sums.
Bach Nguyen, Action of Hopf algebras
and noncommutative prime spectra.
Abstract: Let H be a Hopf algebra, and A be any associative
unital algebra. Suppose A is a left H-module algebra. Then the
spectrum of A admits a stratification with its strata indexed
by H-prime ideals of A. Using this stratification to study A
was initiated by K. Goodearl and E. Letzter, then M. Lorenz,
where they considered group actions. In this talk, we will
discuss a generalization of their results to the setting of
cocommutative Hopf algebra, where the H-strata can be
described in term of the prime spectrum of certain commutative
algebra. This is a joint work with M. Lorenz and R. Yammine.
Joe Reid, Indecomposable objects determined by
their index in Higher Homological Algebra.
Abstract: For a positive integer d, the notion of triangulated
categories extends to (d+2) angulated categories. An analogous
extension of cluster tilting subcategories has been introduced
by Oppermann and Thomas, which allows a definition of index in
this higher dimensional setting. We show that for odd d and
under a technical assumption, this uniqueness of index holds
for indecomposable objects in (d+2) angulated
categories. The technical assumption is satisfied for
(d+2)-angulated higher cluster categories of Dynkin type A.
Job Rock, Continuous clusters and continuous
mutation.
Abstract: Igusa and Todorov introduced continuous cluster
categories in 2015. In this talk we'll examine a new model for
a continuous cluster category; the previous model is a
localisation of the new model. We also introduce continuous
clusters and continuous mutation as generalizations of
clusters and mutation. Joint work with Kiyoshi Igusa and
Gordana Todorov.
Philipp Rothmaler, Positive primitive torsion.
Abstract: Martsinkovsky and Russell introduced a torsion
radical—called injective torsion—as the injective
stabilization of the tensor product with the module, evaluated
at the ring. In joint work with Martsinkovsky I proved that
this torsion coincides with what I earlier called elementary
torsion for the case where the vanishing class of modules is
that of flat modules (on the other side) and the pp formulas
in question are those which vanish on that class (or simply,
on the regular module). I will discuss this and a
generalization—now called pp torsion—to arbitrary classes of
modules resp., arbitrary sets of pp formulas.
Dylan Rupel, From cluster agebras to quiver
Grassmannians.
Abstract: A quiver Grassmannian is a variety parametrizing
subrepresentations of a given quiver representation. Reineke
has shown that all projective varieties can be realized as
quiver Grassmannians. In this talk, I will study a class of
smooth projective varieties arising as quiver Grassmannians
for (truncated) preprojective representations of an
n-Kronecker quiver, i.e. a quiver with two vertices and n
parallel arrows between them. The main result I will present
is a recursive construction of cell decompositions for these
quiver Grassmannians motivated by the theory of rank two
cluster algebras. If there is time I will discuss a
combinatorial labeling of the cells by which their dimensions
may conjecturally be directly computed. This is a report on
joint work with Thorsten Weist.
Ralf Schiffler, Frieze varieties: a
characterization of the finite-tame-wild trichotomy.
Abstract: We introduce a new class of algebraic varieties
which we call frieze varieties. Each frieze variety is
determined by an acyclic quiver. The frieze variety is defined
in an elementary recursive way by constructing a set of points
in affine space. From a more conceptual viewpoint, the
coordinates of these points are specializations of cluster
variables in the cluster algebra associated to the quiver. We
give a new characterization of the finite--tame--wild
trichotomy for acyclic quivers in terms of their frieze
varieties. We show that an acyclic quiver is representation
finite, tame, or wild, respectively, if and only if the
dimension of its frieze variety is 0,1, or >1 ,
respectively.
Markus Schmidmeier, Gorenstein projective Z[ε]-modules.
Abstract: We identify the finitely generated Gorenstein
projective modules over the ring Z[ε] = Z[T]/(T2)
as the embeddings of a subgroup in a free abelian group of
finite rank. We classify the indecomposable objects in this
category and note that the Krull-Remak-Schmidt property
(uniqueness of the decomposition factors) fails. However, the
stable category modulo projectives, being equivalent to the
category of finite abelian groups, has the Krull-Remak-Schmidt
property.
Jeanne Scott, Content for the Fibonacci lattice.
Abstract: The Young-Fibonacci lattice is a differential poset
which, like the lattice of integer partitions, is the
branching poset for a tower of algebras (call the Okada
algebras) whose irreducible representations can be described
using complete chains in the lattice. In this talk I'll report
on a current project whose goal is to develop a theory of
Jucys-Murphy elements for the Okada algebras together with a
notion of content for the lattice. I'll explain what should be
expected of this idea of content from the perspective of the
character theory of the Okada algebras and properties of the
Okada-Schur functions.
Emre Sen, Syzygy filtrations of cyclic Nakayama
algebras.
Abstract: We introduce a method "syzygy filtration" to give
building blocks of syzygies appearing in projective
resolutions of indecomposable Λ-modules where Λ is a cyclic
Nakayama algebra. We interpret homological invariants of Λ
including finitistic dimension, phi-dimension, Gorenstein
dimension, dominant dimension and their upper bounds in terms
of this new filtration. We give a simpler proof of theorem
relating Gorenstein dimension to phi-dimension. Moreover we
show that difference of phi-dimension and finitistic dimension
is at most one.
Amit Shah, Auslander-Reiten theory in quasi-abelian
and Krull-Schmidt categories.
Abstract: In the 70s, Auslander and Reiten introduced the
study of what is now known as Auslander-Reiten theory, and
this work has played a large role in understanding the
representation theory of artin algebras. After motivating why
I care about Auslander-Reiten theory in more general contexts,
I will explain why many of the results of Auslander and Reiten
hold for quasi-abelian categories. We will also see that there
are particularly nice consequences when we are dealing with
Krull-Schmidt, quasi-abelian categories.
David Speyer, Between representation theory and
lattice theory.
Abstract: I'll attempt to give an overview of the connections
between the torsion classes of path algebras, preprojective
algebras and related objects, and the lattice theory of
Coxeter groups and their Cambrian lattice quotients.
Asish Srivastava, Partial morphisms in additive
exact categories.
Abstract: We develop a general theory of partial morphisms in
additive exact categories which extends the model theoretic
notion introduced by Ziegler in the particular case of
pure-exact sequences in the category of modules over a ring.
We relate partial morphisms with (co-)phantom morphisms and
injective approximations and study the existence of such
approximations in these exact categories.
Torkil Stai, 0-cocompactness in triangulated
categories.
Abstract: This talk is a report on joint work with Steffen
Oppermann and Chrysostomos Psaroudakis. In a triangulated
category, an object is called compact if any morphism from it
to a direct sum, factors through a finite subsum. We will
start by explaining why compact objects are important. In
particular, compact objects generate so-called t-structures,
and---when there are "enough" of them---they give a celebrated
representability theorem. Unfortunately, "cocompact" objects
do not really appear in the categories we wish to study. In
the last part of the talk, we will introduce a weaker notion
called 0-cocompactness, covering more of our favorite examples
while letting us keep some of the upshots of compact objects.
Xi Tang, Generalized tilting theory in functor
categories.
Abstract: Generalized tilting theory in functor categories was
introduced by Martinez-Villa and Ortiz-Morales in 2013. In
this talk we will further develop this theory in several
directions. First, we extend Miyashita's proof of the
generalized Brenner-Butler Theorem to arbitrary functor
categories of the form Mod (C) with C an annuli variety.
Secondly, we construct a hereditary and complete cotorsion
pair generated by a generalized tilting subcategory T of Mod
(C). Some applications of these two results include an
isomorphism between the Grothendieck groups K0(C)
and K0(T), the existence of a new abelian model
structure on the category of complexes in Mod (C) and a
t-structure on the derived category of Mod (C).
José Velez-Marulanda, Deformations of
Gorenstein-projective modules over Nakayama and triangular
matrix algebras.
Abstract: Let $\mathbf{k}$ be a fixed field of arbitrary
characteristic, and let $\Lambda$ be a finite dimensional
$\mathbf{k}$-algebra. Assume that $V$ is a left
$\Lambda$-module of finite dimension over $\mathbf{k}$. F. M.
Bleher and the author previously proved that $V$ has a
well-defined versal deformation ring $R(\Lambda,V)$ which is a
local complete commutative Noetherian ring with residue field
isomorphic to $\mathbf{k}$. Moreover, $R(\Lambda,V)$ is
universal if the endomorphism ring of $V$ is isomorphic to
$\mathbf{k}$. In this article we prove that if $\Lambda$ is a
basic connected Nakayama algebra without simple modules and
$V$ is a Gorenstein-projective left $\Lambda$-module, then
$R(\Lambda,V)$ is universal. Moreover, we also prove that the
universal deformation rings $R(\Lambda,V)$ and $R(\Lambda,
\Omega V)$ are isomorphic, where $\Omega V$ denotes the first
syzygy of $V$. This result extends the one obtained by F. M.
Bleher and D. J. Wackwitz concerning universal deformation
rings of finitely generated modules over self-injective
Nakayama algebras. In addition, we also prove the
following result concerning versal deformation rings of
finitely generated modules over triangular matrix finite
dimensional algebras. Let $\Sigma=\begin{pmatrix} \Lambda
& B\\0& \Gamma\end{pmatrix}$ be a triangular matrix
finite dimensional Gorenstein $\mathbf{k}$-algebra with
$\Gamma$ of finite global dimension and $B$ projective as a
left $\Lambda$-module. If $\begin{pmatrix}
V\\W\end{pmatrix}_f$ is a finitely generated
Gorenstein-projective left $\Sigma$-module, then the versal
deformation rings $R\left(\Sigma,\begin{pmatrix}
V\\W\end{pmatrix}_f\right)$ and $R(\Lambda,V)$ are isomorphic.
José Vivero, Generalized Igusa-Todorov functions.
Abstract: In this talk I am going to present a new approach to
the Igusa-Todorov functions, which will lead to the definition
of generalized versions of them. I will state some of the
fundamental properties of these functions and their relation
with the original ones. Also we will see how they can be
applied to cast some more light upon the finitistic dimension
conjecture.
Daping Weng, Cluster structures on double
Bott-Samelson cells.
Abstract: Let G be a Kac-Peterson group associated to a
symmetrizable generalized Cartan matrix. Let (b, d) be a pair
of positive braids associated to the root system. We define
the double Bott-Samelson cell associated to G and (b,d) to be
the moduli space of configurations of flags satisfying certain
relative position conditions. We prove that they are affine
varieties and their coordinate rings are upper cluster
algebras. We construct the Donaldson-Thomas transformation on
double Bott-Samelson cells and show that it is a cluster
transformation. In the cases where G is semisimple and the
positive braid (b,d) satisfies a certain condition, we prove a
periodicity result of the Donaldson-Thomas transformation, and
as an application of our periodicity result, we obtain a new
geometric proof of Zamolodchikov's periodicity conjecture in
the cases of D ⨂ An. This is joint work with Linhui
Shen.
Nicholas Williams, Higher analogues of Grassmannian
clusters.
Abstract: Scott described the coordinate ring of the
Grassmannian as a cluster algebra. This description, along
with results from Oh, Postnikov, and Speyer, puts the clusters
in bijection with collections of subsets which satisfy a
certain combinatorial condition. This combinatorial condition
has a natural generalisation but it is not immediate that this
case behaves as the classical case does. However, recent
results of ours go some way towards showing that this
generalised case is reasonably well-behaved. This is based on
joint work-in-progress with Jordan McMahon.
Emine Yildirim, The bounded derived category for
cominuscule posets.
Abstract: Cominuscule posets come from root posets and have
connections to Lie theory and Schubert calculus. We are
interested in whether the bounded derived category of the
incidence algebra of a cominuscule poset is fractionally
Calabi-Yau. In other words, we ask if some non-zero power of
the Serre functor is a shift functor. We answer this question
on the level of the Grothendieck groups. On the Grothendieck
group this functor becomes an endomorphism called the Coxeter
transformation. We show that the Coxeter transformation has
finite order for two of the three infinite families of
cominuscule posets, and for the exceptional cases. Our
motivation comes from a conjecture by Chapoton which states
that the bounded derived category of incidence algebra of root
posets is fractionally Calabi-Yau. Our result can be thought
of as a parabolic analogue of Chapoton’s conjecture.
Mohamed F. Yousif, Modules with the exchange
property.
Abstract: In this talk, we unify old results and provide new
ones on the long standing open question of Crawley and Jónsson
that asks whether the finite exchange property always implies
the full exchange property.
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