We present an efficient high-order discretization scheme for the evaluation of Laplace and Helmholtz layer potentials on smooth surfaces in three dimensions. The scheme is panel based and applies an analytical surface to line integral conversion on each panel to evaluate single layer, double layer, and their normal derivatives accurately. A new basis approximation scheme tailed for these layer potential kernels is proposed. Both nearly singular and singular cases are supported via a unified recursive framework. The Laplace part of the scheme is joint work with Shravan Veerapaneni, and the Helmholtz part is joint work with Shidong Jiang.

In this talk we will discuss how two objects of great interest to both mathematicians and physicists are connected.
On one hand, amplituhedra are the image under a linear map of the positive part of the Grassmannian – where all the Pluckers are nonnegative. Introduced by physicists to encode the probability of certain particle interactions – scattering amplitudes – in Quantum Field Theory, they are semialgebraic sets which generalize polytopes inside the Grassmannian.
On the other hand, cluster algebras are a remarkable class of commutative rings with very nice combinatorics introduced by Fomin and Zelevinsky motivated by the study of total positivity. Many nice algebraic varieties are known to have a cluster algebra structure, including the Grassmannian. They also emerged in physics in the context of scattering amplitudes, where they contributed to both conceptual and computational advances.
We will show how Amplituhedra possesses surprisingly rich cluster structures and how they relate to their geometry and combinatorics.