Applications of commutative algebra to spline theory and string theory

108 pagesIn this thesis, we study two problems: (1) the dimension problem on splines, and (2) Gorenstein Calabi-Yau varieties with regularity 4 and codimension 4. They come from approximation theory and physics, respectively, but can be studied with commutative algebra. Splines play an important role in approximation theory, geometric modeling, and numerical analysis. One key problem in spline theory is to determine the dimension of spline spaces. The Schenck-Stiller "2r +1" conjecture is a conjecture on this problem. We present a counter-example to this conjecture and prove it with the spline complex. We also conjecture a new bound for the first homology of the spline complex. Calabi-Yau varieties, especially Calabi-Yau threefolds, play a central role in string theory. A first example of a Calabi-Yau threefold is a quintic hypersurface in P7. Generalizing this construction, we may consider complete intersection Calabi-Yaus (CICY), or more generally Gorenstein Calabi-Yaus (GoCY). In 2016, Coughlan, Golebiowski, Kapustka and Kapustka found 11 families of Gorenstein Calabi-Yau threefolds in P7 and they ask if it is a complete list. We consider the Artinian reduction and find there are 8 Betti diagrams for these GoCYs. There are another 8 Betti diagrams corresponding to Artinian Gorenstein rings of regularity 4 and codimension 4. We prove they cannot be Betti diagrams of Gorenstein threefolds in P7. Our result can be viewed as a step towards answering the CGKK question. These two topics may seem to be unrelated at first sight. However, Macaulay's inverse systems provide a unifying theme. We discuss some of these topics as future directions of research