Conference Agenda

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Session Overview
Session
MS151, part 1: Cluster algebras and positivity
Time:
Tuesday, 09/Jul/2019:
10:00am - 12:00pm

Location: Unitobler, F-106
30 seats, 54m^2

Presentations
10:00am - 12:00pm

Cluster algebras and positivity

Chair(s): Lisa Lamberti (ETHZ, Switzerland), Khrystyna Serhiyenko (University of California, Berkeley, USA / University of Kentucky, Lexington), Lauren Williams (Harvard, USA)

Cluster algebras are commutative rings whose generators and relations can be defined in a remarkably succinct recursive fashion. Algebras of this kind, introduced by Fomin and Zelevinsky in 2000, are equipped with a powerful combinatorial structure frequently appearing in many mathematical contexts such as Lie theory, triangulations of surfaces, Teichmueller theory and beyond. Coordinate rings of Grassmannians and related invariant rings are well-studied examples of algebras of this type. One important aspect arising from the intrinsic combinatorial structure of cluster algebras is that it uncovers systematic, intriguing and complex positivity properties in these families of rings. For instance, it is expected that for each cluster algebra there is a distinguished basis, such that all elements can be expressed as a "positive" linear combination of basis vectors. Seemingly elementary claims of this type, so far proved only in certain cases, have triggered important developments in research areas at the intersection of geometry, algebra and combinatorics.

In this session, we glimpse at recent developments in this field and discuss open questions.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Toric degenerations of cluster varieties and cluster duality

Konstanze Rietsch
King’s College London

This talk will review some aspects of mirror symmetry for generalised flag varieties G/P and its interaction with the representation theory of G. For a cominuscule homogeneous space G/P there is an expression for the mirror LG model W in terms of coordinates which, by the geometric Satake correspondence in representation theory, are naturally identified with cohomology classes of G/P (joint works with Marsh, Pech, and Williams) leading to some combinatorially very attractive formulas. A relationship between the critical points of W and its tropicalisation, and representation theory (work of Judd) may also be discussed.

 

On mirror symmetry for homogeneous spaces

Lara Bossinger1, Juan Bosco Frías Medina2, Tim Magee2, Alfredo Nájera Chávez2
1Max Planck Institute for Mathematics in the Sciences, 2Instituto de Matematicas UNAM, Mexico

Cluster varieties are a particularly nice class of log Calabi-Yau varieties-- the non-compact analogue of usual Calabi-Yaus. They come in pairs (A,X), with A and X built from dual tori. The punchline of this talk will be that compactified cluster varieties are a natural progression from toric varieties. Essentially all features of toric geometry generalize to this setting in some form, and the objects studied remain simple enough to get a hold of and do calculations.

Compactifications of A and their toric degenerations were studied extensively by Gross, Hacking, Keel, and Kontsevich. These compactifications generalize the polytope construction of toric varieties-- a construction which is recovered in the central fiber of the degeneration. Compactifications of X were introduced by Fock and Goncharov and generalize the fan construction of toric varieties. Recently, Lara Bossinger, Juan Bosco Frías Medina, Alfredo Nájera Chávez, and I introduced the notion of an X-variety with coefficients, expanded upon the notion of compactified X-varieties, and for each torus in the atlas gave a toric degeneration where each fiber is a compactified X-variety with coefficients. We showed that these fibers are stratified, and each stratum is again a compactified X-varieties with coefficients. In the central fiber, we recover the toric variety associated to the fan in question, and we show that strata of the fibers degenerate to toric strata. This talk is based on arXiv:1809.08369.

 

Generalised friezes and the weak Ptolemy map

Ilke Canakci, Peter Jørgensen
Newcastle University, UK

Frieze patterns, introduced by Conway, are infinite arrays of numbers where neighbouring numbers satisfy a local arithmetic rule. Frieze patterns with positive integer values are of a special interest since they are in one-to-one correspondence with triangulations of polygons by Conway--Coxeter. Remarkably, this established a connection to cluster algebras–predating them by 30 years– and to cluster categories. Several generalisations of frieze patterns are known. Joint with Jørgensen, we associated frieze patterns to dissections of polygons where the entries are over a (commutative) ring. Furthermore, we introduced an explicit combinatorial formula for the entries of these friezes by generalising the 'T-path formula' of Schiffler which was introduced to give explicit formulas for cluster variables for cluster algebras of type A.

 

Perfect matching modules for dimer algebras

Ilke Canakci1, Alastair King2, Matthew Pressland3
1Newcastle University, UK, 2University of Bath,UK, 3Universität Stuttgart, D

The theory of dimer models, or bipartite graphs on surfaces, first arose in theoretical physics, and later found diverse applications in geometry and representation theory. Recently, there has been much interest in dimer models on the disk, particularly those arising from Postnikov diagrams, and their relationship to Grassmannian cluster algebras and categories. Perfect matchings of dimer models play a central role in the theory. In joint work with İlke Çanakçı and Alastair King, we provide an algebraic viewpoint on these objects, by defining and studying a module for the dimer algebra for each perfect matching. As an application, we explain the relationship between combinatorial and homological formulae for computing Grassmannian cluster variables.