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Abstracts are listed in the alphabetical order of the speakers' surnames.

Abstracts for the Flash Talks are at the bottom of the page.

Title: Models of algebraic group actions

 

Abstract: Given a rational action of an algebraic group on a variety X, Weil's regularization theorem asserts that there is a variety Y equipped with a regular G-action and a G-equivariant birational map to X. One would like to have additional information on the model Y: can it be taken smooth and projective?

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The talk will present an affirmative answer to this question in characteristic 0; in positive characteristic, the model can be taken projective and normal. The main tools are two analogues of induction of representations, in the setting of varieties with group action.

Giovanni Cerulli Irelli(Watch the recorded Talk)

Title: Algebras with self-duality
 

Abstract: Symmetric quivers and more generally algebras with a self-duality have been introduced by Derksen and Weyman in 2002 with the aim to get actions of groups like the orthogonal and the symplectic group on quiver representations. More precisely these groups act on the subset of so-called symmetric representations of the representation varieties which is stable by the action of the usual structure groups. The orbits for the "small" groups are described by a theorem of Magyar-Weyman-Zelevinsky which states that they are the intersection of the orbit by the big group with the subset of symmetric representations. The main open problem in this theory is to describe the orbit closures.

 

In a joint work with M. Boos we give another proof of the theorem of Magyar-Weyman-Zelevinsky and we describe the orbit closures for symmetric quivers of finite type. The problem of describing the orbit closures for this action come from the study of 2-nilpotent Borel orbits for the classical groups and with the connection with linear degenerations of flag varieties of type B,C and D.

Calin Chindris(Watch the recorded Talk)

Title: Sigma-critical quiver representations and applications to the Paulsen Problem in Frame Theory

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Abstract: Sigma-critical representations are quiver representations that satisfy certain matrix equations. They arise naturally in the context of the Kempf-Ness theorem on closed orbits in Invariant Theory.

 

After introducing all the relevant concepts, I will first describe a result that gives necessary and sufficient conditions for the orbit of a quiver representation to contain a sigma-critical representation. I will then explain how this result can be used to solve the Paulsen Problem for matrix frames. This is based on joint work with Jasim Ismaeel.

William Crawley-Boevey(Watch the recorded Talk)

Title: Semilinear clannish algebras

 

Abstract: Over 30 years ago, the speaker introduced clannish algebras, which generalize string algebras, and gave a classification of their indecomposable modules. It is similar to the classification for string algebras, but the classes of strings and bands each divide into so-called asymmetric and symmetric subclasses. This solved a problem raised by Gelfand at the ICM in 1970.

 

The present talk is about ongoing work with Raphael Bennett-Tennenhaus, generalizing the classification for clannish algebras to allow representations which involve maps which are semilinear with respect to automorphisms of the underlying field or division ring (and also to allow the so-called special loops to be bound by quadratic polynomials which are irreducible).

 

An analogous generalization for string algebras has been considered by Ringel in unpublished work, and has applications to questions in arithmetic such as the existence of F-crystals.

 

The setup of semilinear clannish algebras can be used to classify modules for rings arising from surfaces with orbifold points of order 2, unsplit Dedekind-like rings, and other examples.

Title: Ranks of matrix pencils and orbit degenerations

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Abstract: Consider the conjugation action of the general linear group on m-tuples of n x n matrices. In Operator Theory,  Curto and Herrero  conjecture in 1985 that  an m-tuple A lies in the closure of the orbit of B if and only if rk f(A) is no larger than rk f(B)  for every noncommutative polynomial f in m variables.  Hadwin and Larson showed in 2003 that this is false and conjectured that  the Curto-Herrero conjecture is true when we allow f to be matrices with noncommutative polynomials as entries. I will discuss recent results with Igor Klep, Visu Makam and Jurij . We showed that the Hadwin-Larson conjecture is false in general. But the conjecture is true if we assume  equality rk f(A)=rk f(B) for all matrices with noncommutative polynomial entries. In fact, it suffices to have equality for all f that are matrices with linear entries. Among other things, we use a theorem of Auslander and an example of Carlson from the representation theory of finite dimensional algebras.

Title: Homomorphisms between representations of skewed-gentle algebras

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Abstract: Skewed gentle algebras are a special class of clannish algebras. Over a field of characteristic different from 2, they are Morita equivalent to skew group algebras of certain gentle algebras with an involution. These algebras appear for example as Jacobian algebras associated to certain triangulations of punctured surfaces with nonempty boundary.

 

Our motivation is to describe all decorated tau-reduced irreducible component of the module varieties of those algebras in terms of laminations. To this end, we need a convenient description of the homomorphisms between the indecomposable representations. Such a description was found for the string case about 20 years ago, whilst the homomorphisms involving certain asymmetric bands leads to an annoying combinatorial problem. We can fix this now for skewed-gentle algebras over an algebraically closed field which is not of characteristic 2.

 

This is part of joint project with D. Labardini-Fragoso and J. Wilson.

Martina Lanini(Watch the recorded Talk)

Title: Simple minded collections arising from perverse sheaves

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Abstract: Simple minded collections appeared for the first time about twenty years ago, in Rickard's work on equivalences of derived equivalences for symmetric algebras. They are. families of objects in a triangulated category satisfying certain vanishing and generating conditions.

 

In this talk I will report on an ongoing joint project with Alessio Cipriani, in which we investigate simple minded collections arising from perverse sheaves on a given stratified space. It is a natural, and in general open question, to ask how simple minded collections are related to each other. We show that any pair of perverse simple minded collections differ by a sequence of appropriate (HRS-)tilts.

Title: Symmetric quivers and symmetric varieties

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Abstract: Since the 1980s, mathematicians have found connections between orbit closures in type A quiver representation varieties and Schubert varieties in type A flag varieties. For example, singularity types appearing in type A quiver orbit closures coincide with those appearing in Schubert varieties in type A flag varieties; combinatorics of type A quiver orbit closure containment is governed by Bruhat order on the symmetric group; and classes of type A quiver orbit closures in equivariant cohomology and K-theory (as well as classes of associated degeneracy loci) can be expressed in terms of formulas involving Schubert polynomials, Grothendieck polynomials, and other objects from Schubert calculus.

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After recalling some of this story, I will discuss the related setting of Derksen-Weyman's symmetric quivers and their representation varieties. I will show how one can adapt results from the ordinary type A setting to unify aspects of the equivariant geometry of type A symmetric quiver representation varieties with Borel orbit closures in corresponding symmetric varieties G/K (G = general linear group, K = orthogonal or symplectic group). This is joint work with Ryan Kinser and Martina Lanini.

Markus Reineke(Watch the recorded Talk)

Title: Dimension expanders and quiver representations

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Abstract: We interpret the notion of dimension expanders of Wigderson and Lubotzky-Zelmanov in terms of stable quiver representation, and use this to prove existence results and to speculate on potential generalizations.

Title: Algebro-geometric tau-tilting theory

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Abstract: Starting from an algebra of finite representation type, I will describe an affine algebraic variety the combinatorics of whose totally non-negative part recovers the structure of the tau-tilting fan. If the algebra is the path algebra of a Dynkin quiver, there is a connection to the cluster X-varieties of Fock and Goncharov. The motivation for this work comes from the physics of scattering amplitudes, and I will say something about that connection. This is part of a joint project with Nima Arkani-Hamed, Hadleigh Frost, Pierre-Guy Plamondon, and Giulio Salvatori.

Title: On the structure of Gorenstein Ideals of codimension 4

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Abstract: I will discuss recent results on the structure of Gorenstein ideals of codimension 4 with n generators. The main point is that explicit classification of such ideals is possible for n< 8, while for n> 9 it will be much more difficult. The main idea is to relate the classification problem to the Kac-Moody Lie algebras of type E_n.

Grzegorz Zwara(Watch the recorded Talk & Slides)

Title: On tangent spaces to orbit closures for quiver representations

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Abstract: Let Q be a finite quiver. The representations of Q with a given dimension vector d form an affine spaces equipped with an action of a product of general linear groups. The orbits under this action correspond to the isomorphism classes of representations, and the Zariski closures of the orbits form a rich class of affine varieties which includes many classical determinantal varieties.

 

During the talk we discuss results and methods concerning the (Zariski) tangent spaces to these orbit closures, especially when Q is a Dynkin quiver. We also pose some open problems.

------------Flash Talks
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Title: Hochschild cohomology and the fundamental group(s) of an algebra

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Abstract: I will try to (very quickly) explain how one can classify maximal tori in the first Hochschild cohomology Lie algebra using the fundamental groups of bound quivers, and (hopefully) give a few examples. This can be used to get derived (and stable) invariants out of these fundamental groups. As one consequence, we prove that there are only finitely many monomial algebras within any derived equivalence class of finite dimensional algebras. This is all joint work with Lleonard Rubio y Degrassi.

Title: Frobenius Splitting of K-orbit Closures

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Abstract: Let K be a symmetric subgroup of a complex reductive algebraic group $G$, defined as the fixed points of some holomorphic involution of $G$, and $B$ some fixed Borel subgroup of $G$. It is interesting to ask which results about Schubert varieties (which are $B$-orbit closures in $G/B$) can be adapted to the study of $K$-orbit closures in $G/B$. For example, it is known that Schubert varieties can be Frobenius split, arising as compatibly split subvarieties of a certain Frobenius splitting of $G/B$. In this setting, the connection between line bundles on $G/B$ and the representations of $G$ plays an important role.

 In my talk and extended abstract, I will highlight what the corresponding results are for $K$-orbit closures of $G/B$, based on joint work with Jenna Rajchgot. Building on the approach from Brion and Kumar for Schubert varieties, we can use a natural generalization of a Bott-Samelson variety (called a Barbasch-Evens-Magyar variety) to construct Frobenius splittings for $K$-orbit closures. I will briefly illustrate some of these key ideas via a concrete example during the flash talk and elaborate on the details in the extended abstract.

Title: Torus actions on quotients of affine spaces

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Abstract: We study a torus action on a GIT quotient of a complex vector space by a reductive complex algebraic group which acts linearly. We show that, under the assumption that $G$ acts freely on the stable locus, the components of the torus fixed point locus of the GIT quotient are again GIT quotients of linear subspaces by Levi subgroups. The talk is based on joint work with Ana-Maria Brecan.

Andras Lorincz(Watch the recorded Talk & Extended abstract​)

Title: On the collapsing of homogeneous bundles in arbitrary characteristic

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Abstract: In this talk, I present results on the geometry of equivariant, proper maps from homogeneous bundles over flag varieties, called collapsing maps. Kempf showed that, provided the bundle is completely reducible, the image of a collapsing has rational singularities in characteristic zero. We extend this result to positive characteristics showing that for the analogous bundles the image is strongly F-regular if its coordinate ring has a good filtration, and we give several criteria for the existence of the latter. Furthermore, we generalize the results to Schubert varieties, and develop a technique for finding defining equations. These results give a uniform, characteristic-free approach for the study of the geometry of a number of remarkable varieties, including representation varieties of radical square zero algebras (e.g. varieties of complexes).

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