To start, I will discuss Dirichlet's proof of infinitude of primes in an arithmetic progression.  This leads up to the study of special values of L-functions and their arithmetic properties.  If time permits, I will try to explain some conjectures and philosophy in this direction.

We study the energy dissipation features of systems comprised of two components one of which is highly lossy and the other lossless. One of the principal results is that all the eigenmodes of any such system split into two distinct classes, high-loss and low-loss,according to their dissipative properties. Interestingly, this splitting is more pronounced the higher the loss of the lossy component. In addition to that, the real frequencies of the high-loss eigenmodes can become very small and even can vanish entirely, which is the case of overdamping. An exhaustive analytical study of the energy, dissipated power, and quality factor for such composite systems is given.

De-mixing problems in spectroscopic imaging often require finding sparse non-negative linear combinations of library functions to match observed data. Due to misalignment and uncertainty in data measurement, the known library functions may not represent the data as well as their proper deformations. To improve data adaptivity, we expand the library to one with a group structure and impose a structured sparsity constraint so that the coefficients for each group should be sparse or even 1-sparse. Since the expanded library is a highly coherent (redundant) dictionary, it is difficult to obtain good solutions using convex methods such as non-negative least squares (NNLS) or L1 norm minimization. We study efficient non-convex penalties such as the ratio/difference of L1 and L2 norms, as sparsity penalties to be added to the objective in NNLS-type models. We show an exact recovery theory of the sparsest solution by minimizing the ratio/difference norms under a uniformity condition. For solving the related unconstrained non-convex models, we develop a scaled gradient projection algorithm that requires solving a sequence of strongly convex quadratic programs.

I will use concrete examples to argue why mathematics is even more important and powerful when computers become more and more powerful and to show what computational mathematics is about.

Which natural numbers occur as the area of a right triangle with three rational sides? This is a very old question and is still unsolved, although partial answers are known (for example, five is the smallest such natural number). In this talk we will discuss this problem and recent progress that has come about through its connections with elliptic curves and other important open questions in number theory.