We advocate the use of cluster algebras and their y-variables in the study of hyperbolic 3-manifolds. We study hyperbolic structures on the mapping tori of pseudo-Anosov mapping classes of punctured surfaces, and show that cluster y-variables naturally give the solutions of the edge-gluing conditions of ideal tetrahedr…
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New solutions to 3D integrability equations using quantum cluster algebras.
problem Constructing solutions to the tetrahedron and 3D reflection equations.
method Extending quantum cluster algebra approach to Fock-Goncharov quivers and investigating cluster transformations.
result Explicit formulas for matrix elements of solutions derived for typical representations.
Aggregated Hold-Outmath.ST
Aggregated hold-out (Agghoo) is a method which averages learning rules selected by hold-out (that is, cross-validation with a single split). We provide the first theoretical guarantees on Agghoo, ensuring that it can be used safely: Agghoo performs at worst like the hold-out when the risk is convex. The same holds true…
Develops quantum cluster algebra approach to solve tetrahedron equation.
problem Investigates a three-dimensional generalization of the Yang-Baxter equation.
method Quantum cluster algebra approach with realization of quantum Y-variables in terms of q-Weyl algebras.
result Obtains a solution with three spectral parameters and reproduces Sergeev's R matrix.
Researchers classify and characterize Bäcklund transformations for hyperbolic Monge-Ampère systems.
problem Classifying and characterizing Bäcklund transformations for hyperbolic Monge-Ampère systems.
method Completely determined a subclass of Bäcklund transformations for Type A systems and formulated conditions for Type B systems.
result Parametrized Bäcklund transformations for Type A systems and provided an invariant condition for Type B systems.
Unified 3D R-matrices from quantum cluster algebra.
problem Constructing new solutions to the tetrahedron equation.
method Symmetric butterfly quiver, quantum cluster algebra, quantum dilogarithms, q-Weyl algebra.
result Unified 3D R-matrices from various sources.
Learning theory has largely focused on two main learning scenarios. The first is the classical statistical setting where instances are drawn i.i.d. from a fixed distribution and the second scenario is the online learning, completely adversarial scenario where adversary at every time step picks the worst instance to pro…