Georgia Institute of Technology College of Sciences

Sharp trace inequalities play a major role in the world of Mathematics. Not only do they give a connection between 'boundary values' of the trace and 'interior values' of the function, but also the truest form of it, many times with a complete classification of when equality is attained. The result presented here, motivated by such inequality proved by Jose' Escobar, is a new trace inequality, connecting between the fractional laplacian of a function and its restriction to the intersection of the hyperplanes x_(n)=0, x_(n-1)=0, ..., x_(n-j+1)=0 where 1<=j<=n. We will show that the inequality is sharp and discussed the natural space for it, along with the functions who attain equality in it. The above result is based on a joint work with Prof. Michael Loss.

Fourier series provide a way of writing almost any signal as a superposition of pure tones, or musical notes.  But this representation is not local, and does not reflect the way that music is actually generated by instruments playing individual notes at different times.  We will discuss time-frequency representations, which are a type of local Fourier representation of signals.  This gives us a mathematical model for representing music.  While the model is crude for music, it is in fact apowerful mathematical representation that has appeared widely throughout mathematics (e.g., partial differential equations), physics (e.g., quantum mechanics), and engineering (e.g., time-varying filtering).  We ask one very basic question: are the notes in this representation linearly independent?  This seemingly trivial question leads to surprising mathematical difficulties.

By means of examples, I will illustrate the connection between orthogonal hypergeometric polynomials which satisfy interesting spectral and self-dual properties and representations of Lie algebras.

This talk is an introduction to using variational approaches for image reconstruction and segmentation. This talk will start with Total Variation minimization (TV) model and discuss Mumford-Shah and Chan-Vese model for image segmentation. I will mainly focus on multiphase segmentation and its extensions.