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.
Bavula slides

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. Bleher -
              slides

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. Crawley-Boevey slides

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. Dean
              slides

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. Esentepe slides

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. Garver-slides

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. Goodearl slides

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. Huang slides

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. Huisgen-Z. slides

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. Jacobsen slides

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. Kaufmann slides

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. Kulkarni slides

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. Luo slides

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. Matherne slides

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. Marko slides

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. Nguyen slides

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. Pevtsova slides

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. Posur slides

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. Prest slides

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. Saorin slides

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. Shah
              slides

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. Slavik slides

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.
Trampel slides

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. Velez slides

Abstract: N/A


Ying Zhou, Tame quivers have finitely many m-maximal green sequences. Zhou slides

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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