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Curriculum Vita: [pdf]
A wealth of mathematical challenges lies beneath the surface of materials science, biology and other engineering and scientific fields, that present a formidable quest for an inquisite mind. We might not solve all the puzzles, but exploring the mathematical laws of nature still makes for a fascinating journey. Below are some of the areas we are currently exploring.
RESEARCH AREAS
- Mathematical aspects of materials science and engineering
This part of the work is concerned with the internal structure of polycrystalline materials, such as metals, ceramics or semiconductors. The mesoscale, or microstructure, of such materials consists of aggregates of small single-crystal grains joined together at interfaces – grain boundaries – threaded with extended displacements of the lattice structure – such as dislocations – and seeded with point-like defects – such as vacancies and impurities. While the property of these elements is determined by microscopic properties, the interplay between these elements determines macroscopic behavior. For example, atomic forces determine the equilibrium shapes of small grains while the size of the grains partially determines the hardness of a material. In addition, hardness depends both on the dislocations restraining the relative motion of grains under stress and on defects impeding the dislocations. An interface separating grains in this network can be compared to a soap film separating two bubbles of air in soap froth, though it can be extremely more complex. Similar to soap bubbles, grains can meet either at triple junctions or at four-point junctions shared by four grains at a time. The existence of broken bonds in the atoms forming the grain boundary when compared to their arrangement in the interior of the crystal gives rise to the interfacial energy. The energetics and connectivity of this network of interfaces plays a role in many material properties and across many scales of use. Many interesting questions remain about the evolution of materials microstructure, which is the focus of current work.
- Voronoi tesselations: theory and applications
Centroidal Voronoi tessellations (CVTs) are special Voronoi tessellations of a bounded geometric domain such that the generating points of the tessellations are also the centroids (mass centers) of the corresponding Voronoi regions with respect to a given density function. Centroidal Voronoi Tessellations may also be defined in more abstract and more general settings. Figure below gives some examples of such tessellations. Due to the natural optimization properties enjoyed by CVTs, they have many applications in diverse fields such as data compression, optimal quadratures, vector quantization, image analysis, clustering, resource distribution, sensor networks, cellular biology, territorial behavior of animals, mesh generations and numerical solution of PDEs.
- Other directions:
Recently I got interested in biological, chemical and medical applications and started several fruitful collaborations in those areas. My earlier work included the analysis of multidimensional birth-death processes in industrial applications and development of efficient pricing schemes in next generation telecommunication networks, linear algebra with the focus on solving ill-conditioned systems and mathematical models in biology.
More details to come.