Seminar

I will finish my series of talks about parabolic bundles.

Fractional Sobolev spaces, which generalize the concept of classical Sobolev spaces, have been a central tool in harmonic analysis, variation problems, and PDE for several decades. One of the main issues of these spaces is that they are unstable as the fractional parameter s goes to 1. In this talk, I will share some of the new results that I've obtained regarding stabilizing these spaces, and how these results compare to the classical Bourgain-Brezis- Mironescu formula.

I will continue discussing Rank 2 parabolic bundles on P^1 . Time permits, I will discuss the morphism from rank two connections with 4 regular singular points to our moduli stack of bundles.

Abstract: We are discussing the elliptic operator $L:=\mathrm{div}(A\nabla\cdot)$ and wonder which types of matrices $A$ yield solvability of $L^p$ boundary value problems. It is well-known that the DKP or Carleson condition implies solvability for the Dirichlet and the regularity boundary value problem. Equally, if the domain is the upper half space, independence of the transversal direction $t$ gives solvability of these boundary value problems.

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.