I will explain how one may associate some moduli spaces of meromorphic differential equations to certain (special) finite graphs. In a sense (that will be explained) this extends some work of Okamoto (computing symmetry groups of Painleve equations) in the case when the moduli spaces are two-dimensional, and work of Crawley-Boevey and others in the case when the differential equations only have regular singularities (and the graph is star-shaped).
We review how quiver theories inevitably arise within string theory as world-volume theory of branes in various spacetime dimensions. We discuss intimate relations between the gauge theory, encoded by the quiver representations, the algebraic geometry of affine Calabi-Yau varieties, and auxiliary combinatorics involving dimers, tilings and plethystics.
Somos-k sequences are a family of sequences defined by quadratic recurrence relations. They arise in number theory (Morgan Ward's elliptic divisibility sequences), combinatorics (Fomin & Zelevinsky's cluster algebras) and discrete integrable systems (discrete Hirota equations, Quispel-Roberts-Thompson maps). For k=3,4,5,6,7 it is known that the Somos-k recurrence exhibits the Laurent phenomenon, but for higher values of k it does not. Here it is explained how the Laurent property is connected with the integrability of the recurrences, in a suitable sense, and two methods for constructing invariants for them are presented; one method is based on the existence of higher order relations, and the other method is a by-product of the construction of Lax pairs.
I will describe a surprising relationship between Dynkin preprojective algebras and minimal rational CFTs, how it can be `explained' in terms of functors on Temperley-Lieb categories and how it can be generalised.
Joint work with Allan Fordy (Leeds)
We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which have higher periodicity.
The periodicity means that sequences given by recurrence relations arise in a natural way from the associated cluster algebras. We present a number of interesting, new families of nonlinear recurrences, necessarily with the Laurent property, of both the real line and the plane, containing integrable maps as special cases. In particular, we show that some of these recurrences can be linearised and, with certain initial conditions, give integer sequences which contain all solutions of some particular Pell equations.
Hypergeometric systems (HGS) in the sense of Gelfand, Kapranov, Zelevinsky (GKZ) are certain systems of PDE's on an N-dimensional complex linear space on which a k-dimensional torus acts. The Laurent polynomials in the title have N monomial terms in k variables and give a family of functions on that torus. Integrals involving these Laurent polynomials are functions of the coefficients which satisfy the GKZ HGS. The singular locus of the HGS coincides which the locus in the space of coefficients for which the Laurent polynomial has a critical point in the torus. The torus acts on the space of coefficients and the Chow quotient is a model for the orbit space. The first part of the talk will give a brief introduction to the aforementioned topics.
The second part of the talk will report on a dimer model (i.e. quiver with superpotential) which is naturally associated with the above in case k=N-2.
The final part of the talk will present the theorem which gives equations for the singular locus of the HGS and for the Chow quotient as the determinant of a suitable version of the Kasteleyn matrix of the dimer model.
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