Seminar

Stability of some Fractional function spaces

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.

A new solvability condition for $L^p$ boundary value problems

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.

Accelerated Optimization in Machine Learning

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.