Abstract: Recently developed methods and a deep articulation of ideas from the Calculus of Variations, Geometric Measure Theory, and nonlinear Partial Differential Equations (PDEs), have been instrumental in the mathematical rigorous understanding of nonlinear phenomena in a plethora of physical and technological applications, ranging from analyzing instabilities in novel advanced materials to the denoising of medical images. This leads to the two talks of this series: Variational Problems in Materials Science and Variational Problems in Imaging Science.
Lecture 1: March 15, 2018. Time and location: TBA
Abstract: Quantum dots are man-made nanocrystals of semiconducting materials. Their formation and assembly patterns play a central role in nanotechnology, and in particular in the optoelectronic properties of semiconductors. Changing the dots’ size and shape gives rise to many applications that permeate our daily lives, such as the new Samsung QLED TV monitor that uses quantum dots to turn “light into perfect color”!
Quantum dots are obtained via the deposition of a crystalline overlayer (epitaxial film) on a crystalline substrate. When the thickness of the film reaches a critical value, the profile of the film becomes corrugated and islands (quantum dots) form. As the creation of quantum dots evolves with time, materials defects appear. Their modeling is of great interest in materials science since material properties, including rigidity and conductivity, can be strongly influenced by the presence of defects such as dislocations.
In this talk we will use methods from the calculus of variations and partial differential equations to model and mathematically analyze the onset of quantum dots, the regularity and evolution of their shapes, and the nucleation and motion of dislocations.
Lecture 2: March 16, 2018. Time and location: TBA
Abstract: The mathematical treatment of image processing is strongly hinged on variational methods, partial differential equations, and machine learning. The bilevel scheme combines the principles of machine learning to adapt the model to a given data, while variational methods provide model-based approaches which are mathematically rigorous, yield stable solutions and error estimates. The combination of both leads to the study of weighted Ambrosio-Tortorelli and Mumford-Shah variational models for image processing.