Luis Diogo: Monotone Lagrangians in cotangent bundles

11th October, 2016

We show that there is a 1-parameter family of monotone Lagrangian tori in the cotangent bundle of the 3-sphere with the following property: every compact orientable monotone Lagrangian with non-trivial Floer cohomology is not Hamiltonian-displaceable from either the zero-section or one of the tori in the family. The proof involves studying a version of the wrapped Fukaya category of the cotangent bundle which includes monotone Lagrangians. Time permitting, we may also discuss an extension to other cotangent bundles.

Joshua Sabloff: Length and width of Lagrangian cobordisms

11th October, 2016

In this talk, I will discuss two measurements of Lagrangian cobordisms between Legendrian submanifolds in symplectizations: their length and their relative Gromov width. The Gromov width, in particular, is a fundamental global invariant of symplectic manifolds, and a relative version of that width helps understand the geometry of Lagrangian submanifolds of a symplectic manifold. Lower bounds on both the length and the width may be produced by explicit constructions; this talk will concentrate on upper bounds that arise from a filtered version of Legendrian contact homology, a Floer-type invariant.

Kenji Fukaya: Packaging the construction of Kuranishi structure on the moduli space of pseudo-holomorphic curve

4th October, 2016

This is a part of my joint work with Oh-Ohta-Ono and is a part of project to rewrite the whole story of virtual fundamental chain in a way easier to use. In general we can construct virtual fundamental chain on (basically all) the moduli space of pseudo-holomorphic curve. It depends on the choices. In this talk I want to provide a statement to clarify which is the data we need to start with and in which sense the resulting structure is well defined. A purpose of writing such statement is then it can be a black box and can be used without looking the proof. Also it is useful to see some properties of it such as its relation to the (target space) group action or compatibility with forgetful map.

Jonny Evans: Lagrangian cell complexes and Markov numbers

20th September, 2016

Joint work with Ivan Smith. Let p be a positive integer. Take the quotient of a 2-disc by the equivalence relation which identifies two boundary points if the boundary arc connecting them subtends an angle which is an integer multiple of (2π/p). We call the resulting cell complex a 'p-pinwheel'. We will discuss constraints on Lagrangian embeddings of pinwheels. In particular, we will see that a p-pinwheel admits a Lagrangian embedding in ℂP2 if and only if p is a Markov number. Time permitting, I will discuss nondisplaceability results, which are a purely symplectic analogue of the Hacking-Prokhorov classification of Q-Gorenstein degenerations of ℂP2.

Jan Trlifaj: Tree modules, and limits of the approximation theory

19th August, 2016

Classes of modules closed under transfinite extensions often provide for precovers, and hence fit in the machinery of relative homological algebra. However, there are important exceptions: the Whitehead groups, and flat Mittag-Leffler modules over non-perfect rings. The latter class is just the zero dimensional instance (for T = R and n = 0) of non-precovering of the class of all locally T-free modules, where T is any n-tilting module which is not Σ-pure split. The phenomenon occurs even for finite dimensional algebras, when R is hereditary of infinite representation type, and T is the Lukas tilting module. The key tools here are the tree modules, which have recently been generalized in order to solve Auslander's problem on the existence of almost split sequences.

Greg Stevenson: Categories with sufficiently many exact sequences

19th August, 2016

I'll talk around some joint work with Ivo Dell'Ambrogio and Jan Stovicek on the role of Gorenstein module categories in homological algebra. The idea is to reduce understanding universal coefficient theorems to very concrete questions about when a small category is Gorenstein and how one can detect when a representation has finite projective dimension.

Julian Külshammer: Higher Nakayama algebras

19th August, 2016

Nakayama algebras are among the best understood representation-finite algebras. They are defined as those algebras such that each indecomposable projective and each indecomposable injective module admits a unique composition series. An equivalent characterisation is that τjS is simple (or zero) for all j ∈ ℤ and every simple module S. Here, τ denotes the Auslander–Reiten translation. Nakayama algebras can be classified by the sequence of lengths of their indecomposable projective modules, called the Kupisch series.

In this talk, we introduce a higher analogue of a Nakayama algebra for each Kupisch series 𝓁 in the sense of Iyama's higher Auslander–Reiten theory. More precisely, (in type A) the higher Nakayama algebra A𝓁(d) is a quotient of the higher Auslander algebra An(d) of type A, constructed by Iyama and studied extensively by Oppermann and Thomas. In type ̃A, one has to use an infinite version of An(d). The higher Nakayama algebra has a d-cluster-tilting module, i.e. a module M with

add(M) = {N | Exti(M,N) = 0 ∀i = 1, . . . , d−1 } = {N | Exti(N,M) = 0 ∀i = 1, . . . , d−1 }.

There are n simple modules in add(M) and they satisfy that τdjS is simple for all j ∈ ℤ and every simple module S in add(M), where τd = τΩd−1 is Iyama's higher Auslander–Reiten translation.

Igor Burban: Fourier-Mukai transform on Weierstrass cubics and commuting differential operators

19th August, 2016

Any commutative subalgebra A in the algebra of ordinary differential operators admits a natural geometric invariant consisting of an irreducible (possibly singular) projective curve C (called spectral curve) and a semi-stable torsion free sheaf ℱ on it (called spectral sheaf). In the case the rank of A is one (meaning that A contains a pair of differential operators of mutually prime orders), the algebra A can be recovered from its spectral datum (C, ℱ) (Krichever correspondence).

All commutative subalgebras of ordinary differential operators of genus one and rank two were classified in the 80ies by Krichever, Novikov and Gruenbaum. It is a natural problem to describe the spectral sheaves of such algebras. This problem was solved by Previato and Wilson in the case the spectral curve is smooth, their answer was given in terms of Atiyah's classification of vector bundles on an elliptic curve. However, the case of a singular spectral curve remained opened.

In my talk (based on a joint work with Alexander Zheglov: arXiv:1602.08694) I shall explain how the Fourier-Mukai transform allows to describe the spectral sheaf of a genus one commutative subalgebra of ordinary differential operators. As a by-product, I shall also show how the low rank objects of the category of semi-stable sheaves on a cuspidal Weierstrass cubic curve (known to be representation wild) can be classified.

Jiarui Fei: Tensor multiplicity via upper cluster algebras

19th August, 2016

By tensor multiplicity we mean the multiplicities in the tensor product of any two finite-dimensional irreducible representations of a simply connected Lie group. Finding their polyhedral models is a long-standing problem. The problem asks to express the multiplicity as the number of lattice points in some convex polytope.

Accumulating from the works of Gelfand, Berenstein and Zelevinsky since 1970’s, around 1999 Knutson and Tao invented their hive model for the type A cases, which led to the solution of the saturation conjecture. Outside type A, Berenstein and Zelevinsky’s models are still the only known polyhedral models up to now. Those models lose a few nice features of the hive model.

In this talk, I will explain how to use upper cluster algebras, an interesting class of commutative algebras introduced by Berenstein-Fomin-Zelevinsky, to discover new polyhedral models for all Dynkin types. Those new models improve the ones of Berenstein-Zelevinsky's, or in some sense generalize the hive model.

It turns out that the quivers of relevant upper cluster algebras are related to the Auslander-Reiten theory of presentations, which can be viewed as a categorification of these quivers. The upper cluster algebras are graded by triple dominant weights, and the dimension of each graded component counts the corresponding tensor multiplicity.

The proof also invokes another categorification – Derksen-Weyman-Zelevinsky’s quiver-with-potential model for the cluster algebra. The bases of these upper cluster algebras are parametrized by µ-supported g-vectors. The polytopes will be described via stability conditions.

Steven Sam: Noetherian properties in representation theory

18th August, 2016

I’ll explain some recent applications of 'categorical symmetries' in topology, algebraic geometry, and group theory. The general idea is to find an action of a category on the object of interest, prove some niceness property (like finite generation), and then deduce consequences from the general properties of the category.

Shiquan Ruan: Hall polynomials for tame type

18th August, 2016

In this talk we will show that Hall polynomial exists for each triple of decomposition sequences which parameterize isomorphism classes of coherent sheaves of a domestic weighted projective line X over finite fields. These polynomials are then used to define the generic Ringel–Hall algebra of X as well as its Drinfeld double. Combining this construction with a result of Cramer, we show that Hall polynomials exist for tame quivers, which not only refines a result of Hubery, but also confirms a conjecture of Berenstein and Greenstein.

Edward Green: Brauer Configuration Algebras and Multiserial Algebras

18th August, 2016

In joint work with Sibylle Schroll (Univ. of Leicester), we introduce a generalization of Brauer graph algebras which we call Brauer configuration algebras. These will be defined in the talk. Brauer graph algebras are the symmetric special biserial algebras and are currently under active investigation. Defining an algebra KQ/I to be special multiserial if, for each arrow a in the quiver, there is at most one arrow one arrow b such that ab ∉ I and at most one arrow c such that ca ∉ I, we show that KQ/I is a symmetric multiserial algebra if and only if it is a Brauer configuration algebra.

An algebra is called multiserial if the Jacobson radical as a left and as a right module is a Σi Ui of uniserial modules Ui such that the intersection of any two is either (0) or a simple module. We will present a number of results, including the following:
(1) A special multiserial algebra is multiserial.
(2) The trivial extension of an almost gentle algebra by its dual is a Brauer configuration
algebra.
(3) Every symmetric radical cubed zero algebra is a Brauer configuration algebra.
(4) Every special multiserial algebra is the quotient of a Brauer configuration algebra.

We say a module M is multiserial if rad(M) is a sum Σi Ui of uniserial modules Ui such that the intersection of any two is either (0) or a simple module. Although special multiserial algebras are usually of wild representation type, we have the following surprising result which indicates that although wild, the representation theory is worth studying.

Theorem If Λ is a special multiserial algebra and M is a finitely generated Λ-module, then M is a multiserial module.

William Crawley-Boevey: Quiver Grassmannians and orbit closures for representation-finite algebras

18th August, 2016

We show that Auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projective-injective; its endomorphism ring is called the projective quotient algebra. For any representation-finite algebra, we use the projective quotient algebra to construct desingularizations of quiver Grassmannians, orbit closures in representation varieties, and their desingularizations. This generalizes results of Cerulli Irelli, Feigin and Reineke.

Catharina Stroppel: Schur-Weyl dualities in non-semisimple cases

17th August, 2016

'Schur-Weyl duality' is often used to describe a concept in representation theory involving two kinds of symmetry that determine each other. In its original form it goes back to Schur and Weyl (around 1930) and describes an important interplay between the representation theory of the general linear and the symmetric group over the complex numbers. In this talk we will describe some generalizations of this phenomenon with a focus on modern, still open or recently solved questions. In particular we are interested in situations, where the involved algebras are not semisimple. We will indicate the origin of filtrations, homological properties and hidden gradings on the involved algebras and applications to the representation theory of Lie superalgebras.

Steffen Oppermann: d-tilting bundles for Geigle-Lenzing weighted projective spaces

16th August, 2016

This talk is based on joint work with Martin Herschend, Osamu Iyama, and Hiroyuki Minamoto. Classically, the classes of tame (representation infinite, connected) hereditary algebras and Fano Geigle-Lenzing weighted projective lines coincide up to derived equivalence. With the development of Iyama's higher AR-theory, and our work on Geigle-Lenzing projective spaces, it has become natural to ask if there is a higher dimensional analogue of this fact. Here dimension refers to, on the one side the global dimension of the algebra, and on the other side the dimension of the space. Unfortunately, so far a general answer (or general strategy) is elusive. In my talk I will focus on the hypersurface case, and more specifically certain weight sequences within the hypersurface case. For these, I will explain how one may find suitable tilting bundles on the Geigle-Lenzing weighted projective space.

Kyungyong Lee: Positivity for cluster algebras

16th August, 2016

Cluster algebras were first introduced by Fomin and Zelevinsky to design an algebraic framework for understanding total positivity and canonical bases for quantum groups. A cluster algebra is a subring of a rational function field generated by a distinguished set of Laurent polynomials called cluster variables. The Positivity Conjecture, which is now a theorem, asserts that the coefficients in any cluster variable are positive. One proof was given by Schiffler and the speaker, and another proof was obtained by Gross, Hacking, Keel and Kontsevich. We outline the idea of our proof.

Ryan Kinser: K-polynomials of type A quiver orbit closures and lacing diagrams

16th August, 2016

Orbit closures of type A quiver representations are algebraic varieties that arise naturally in several areas of mathematics: for example, in Lusztig's geometric realization of Ringel's work on quantum groups; as generalizations of determinantal varieties in commutative algebra; and in the theory of degeneracy loci of maps of vector bundles.

For equioriented type A quivers, a formula due to Knutson-Miller-Shimozono expresses the equivariant cohomology class of each orbit closure as a sum, over certain 'lacing diagrams', of products of Schubert polynomials. Lacing diagrams were introduced by Abeasis and del Fra in 1982 to visualize direct sum decompositions of type A quiver representations.

In joint work with Allen Knutson and Jenna Rajchgot, we proved a 2004 conjecture of Buch and Rimnyi that generalizes this formula in two ways: to arbitrarily oriented type A quivers, and to equivariant K-classes (a.k.a. K-polynomials), from which equivariant cohomology can be recovered.

The aim of this talk is to explain the combinatorics of (K-theoretic) lacing diagrams and carefully state the formula. Time permitting, I will give some idea of the Gröbner degeneration technique used in the proof.

Srikanth Iyengar: Local Serre duality for modular representations of finite group schemes

16th August, 2016

This talk will be about the representations of a finite group (or a finite group scheme) G defined over a field k of positive characteristic. My plan is to explain the statement and proof of a recent result (obtained in collaboration with Dave Benson, Henning Krause, and Julia Pevtsova) to the effect that the stable module category of finite-dimensional representations of G has local Serre duality.

Eleonore Faber: Non-commutative resolutions of discriminants

16th August, 2016

Let G be a finite subgroup of GLn(K) for a field K whose characteristic does not divide the order of G. The group G acts linearly on the polynomial ring S in n variables over K. When G is generated by reflections, then the discriminant D of the group action of G on S is a hypersurface with a singular locus of codimension 1. In this talk we give a natural construction of a noncommutative resolution of singularities of the coordinate ring of D as a quotient of the skew group ring A = S ∗ G by the idempotent e corresponding to the trivial representation. We will explain how this can be seen in some sense as a McKay correspondence for reflection groups.