41 citations · 153 across the 15 of their papers we have counts for
15 papers
Monoidal categorification of genus zero skein algebras
Dylan G. L. Allegretti, Hyun Kyu Kim, Peng Shan
We prove a conjecture of the first and third named authors relating the Kauffman bracket skein algebra of a genus zero surface with boundary to a quantized -theoretic Coulomb br…
Dualities of -theoretic Coulomb branches from a once-punctured torus
Dylan G. L. Allegretti, Peng Shan
We consider the quantized -character variety of a once-punctured torus. We show that this quantized algebra has three -invariant subalgebras that are i…
Skein algebras and quantized Coulomb branches
Dylan G. L. Allegretti, Peng Shan
To a compact oriented surface of genus at most one with boundary, we associate a quantized -theoretic Coulomb branch in the sense of Braverman, Finkelberg, and Nakajima. In the…
Stability conditions and Teichmüller space
Dylan G. L. Allegretti
We consider a 3-Calabi-Yau triangulated category associated to an ideal triangulation of a marked bordered surface. Using the theory of harmonic maps between Riemann surfaces, we c…
On the wall-crossing formula for quadratic differentials
Dylan G. L. Allegretti
We prove an analytic version of the Kontsevich-Soibelman wall-crossing formula describing how the number of finite-length trajectories of a quadratic differential jumps as the diff…
Stability conditions, cluster varieties, and Riemann-Hilbert problems from surfaces
Dylan G. L. Allegretti
We consider two interesting spaces associated to a quiver with potential: a space of stability conditions and a cluster variety. In the case where the quiver with potential arises…
Categorified canonical bases and framed BPS states
Dylan G. L. Allegretti
We consider a cluster variety associated to a triangulated surface without punctures. The algebra of regular functions on this cluster variety possesses a canonical vector space ba…
Voros symbols as cluster coordinates
Dylan G. L. Allegretti
We show that the Borel sums of the Voros symbols considered in the theory of exact WKB analysis arise naturally as Fock-Goncharov coordinates of framed -local sy…
The monodromy of meromorphic projective structures
Dylan G. L. Allegretti, Tom Bridgeland
We study projective structures on a surface having poles of prescribed orders. We obtain a monodromy map from a complex manifold parameterising such structures to the stack of fram…
Stability conditions and cluster varieties from quivers of type
Dylan G. L. Allegretti
We describe the relationship between two spaces associated to a quiver with potential. The first is a complex manifold parametrizing Bridgeland stability conditions on a triangulat…
The geometry of cluster varieties from surfaces
Dylan G. L. Allegretti
Cluster varieties are geometric objects that have recently found applications in several areas of mathematics and mathematical physics. This thesis studies the geometry of a large…
On the asymptotic behavior of Jacobi polynomials with first varying parameter
Oleg Szehr, Rachid Zarouf
We investigate the large behavior of Jacobi polynomials with varying parameters for and . This is a well-studied topic…
A duality map for quantum cluster varieties from surfaces
Dylan G. L. Allegretti, Hyun Kyu Kim
We define a canonical map from a certain space of laminations on a punctured surface into the quantized algebra of functions on a cluster variety. We show that this map satisfies a…
The cluster symplectic double and moduli spaces of local systems
Dylan G. L. Allegretti
We prove a conjecture of Fock and Goncharov which provides a birational equivalence of a cluster variety called the cluster symplectic double and a certain moduli space of local sy…
Laminations from the symplectic double
Dylan G. L. Allegretti
Let be a compact oriented surface with boundary together with finitely many marked points on the boundary, and let be the same surface equipped with the opposite orie…