Decoupling and lattice point counting

Abstract:  For many Diophantine equations or systems, the number of solutions within a box of side length N can grow like a power of N. Obtaining a nontrivial upper bound for the exponent is crucial for various problems. Recently, an analytic method called ``decoupling'' has been successful in finding such upper bounds that are often sharp. The highlight of this method is that it exploits the curvature of the manifold containing the underlying lattice points. I will introduce decoupling and examples of its applications in lattice point counting. I will also talk about an important limitation of the decoupling method, and, in a problem where decoupling faces challenges, show how lattice point counting results can help make progress in the reverse direction to solve harmonic analytic questions. Part of this talk is based on joint works with Shaoming Guo, Alex Iosevich, Changkeun Oh and Pavel Zorin-Kranich.