Thesis (Ph.D.)--Mathematics, Georgia Institute of Technology, 2008. / Committee Chair: Prof. Jean Bellissard; Committee Member: Prof. Claude Schochet; Committee Member: Prof. Michael Loss; Committee Member: Prof. Stavros Garoufalidis; Committee Member: Prof. Thang Le.

Combinatorial techniques have changed the paradigm of materials research by allowing a faster data acquisition in complex problems with multidimensional parameter space. The focus of this thesis is to demonstrate biomaterials design and characterization via preparation of two dimensional combinatorial libraries with chemically-distinct structured patterns. These are prepared from blends of biodegradable polymers using thickness and temperature gradient techniques. The desired pattern in the library is chemically-distinct cell adhesive versus non-adhesive micro domains that improve library performance compared to previous implementations that had modest chemical differences. Improving adhesive contrast should minimize the competing effects of chemistry versus physical structure. To accomplish this, a method of blending and crosslinking cell adhesive poly(季aprolactone) (PCL) with cell non-adhesive poly(ethylene glycol) (PEG) was developed. We examine the interaction between MC3T3-E1 osteoblast cells and PCL-PEG libraries of thousands of distinct chemistries, microstructures, and roughnesses. These results show that cells grown on such patterned biomaterial are sensitive to the physical distribution and phases of the PCL and PEG domains. We conclude that the cells adhered and spread on PCL regions mixed with PEG-crosslinked non-crystalline phases. Tentatively, we attribute this behavior to enhanced physical, as well as chemical, contrast between crystalline PCL and non-crystalline PEG.

We study the K-theory and cohomology of spaces of aperiodic and repetitive tilings with finite local complexity. Given such a tiling, we build a spectral sequence converging to its K-theory and define a new cohomology (PV cohomology) that appears naturally in the second page of this spectral sequence. This spectral sequence can be seen as a generalization of the Leray-Serre spectral sequence and the PV cohomology generalizes the cohomology of the base space of a Serre fibration with local coefficients in the K-theory of its fiber. We prove that the PV cohomology of such a tiling is isomorphic to the Cech cohomology of its hull. We give examples of explicit calculations of PV cohomology for a class of 1-dimensional tilings (obtained by cut-and-projection of a 2-dimensional lattice). We also study the groupoid of the transversal of the hull of such tilings and show that they can be recovered: 1) from inverse limit of simpler groupoids (which are quotients of free categories generated by finite graphs), and 2) from an inverse semi group that arises from PV cohomology. The underslying Delone set of punctures of such tilings modelizes the atomics positions in an aperiodic solid at zero temperature. We also present a study of (classical and harmonic) vibrational waves of low energy on such solids (acoustic phonons). We establish that the energy functional (the "matrix of spring constants" which describes the vibrations of the atoms around their equilibrium positions) behaves like a Laplacian at low energy.