I will continue discussing rank 2 parabolic bundles on P^1 in the case of 4 ramification points. I will introduce modifications of vector bundles and Hecke correspondences. These correspondences are crucial for the Langlands program.
Training Machine Learning (ML) models is like finding the quickest path down a winding mountain—too slow, and you never reach the bottom; too fast, and you might veer off course. One way to speed up learning without losing control is momentum, a technique that helps the training algorithm adjust the update direction intelligently. Momentum-based methods, such as Nesterov acceleration, are widely used in ML training, but they are traditionally studied under ideal conditions—when the learning landscape is convex and the gradients are reliable.
I will continue discussing rank 2 parabolic bundles on P^1 in the case of 4 ramification points.