Executive Summary : | Homogeneous ideals in polynomial rings that come with underlying graphs or simplicial complexes is an active topic of research in commutative algebra. Hochster proved formulas for Betti numbers of square free monomial ideals in terms of reduced homology of simplicial complexes. Many ideals associate with graphs: edge ideals, path ideals, edge ideals of weighted oriented graphs, binomial edge ideals. Froberg characterised edge ideals with linear resolutions in terms of cochordal graphs. Properties of symbolic powers of ideals is an important topic and naturally this has been studied for ideals that come with combinatorics. The nth symbolic power of an ideal I (Notation: I^(n)) is defined as the intersection of the localisations of I^n at the associated primes of I. The following questions are studied by many and we'd like to study them: Q1. For which positive integers a, b one has I^(a) contained in I^b for various classes of ideals? In particular when is symbolic power same as ordinary power? Q2: Find various classes of ideals whose symbolic powers have linear resolutions. Last part of Q1 is related to the famous packing problem. For edge ideals I^(3) is contained in I^2 and edge ideals for which the ordinary and symbolic powers are same to be the edge ideals of bipartite graphs. Explicit description of generators of symbolic powers in terms of the underlying graphs are used in the proof of both. This motivates us to study the following: Q3. For ideals with underlying combinatorial structures how to describe the symbolic powers combinatorially? Hochster and Huneke used characteristic p methods to symbolic powers for and Frobenius power of an ideal is central to their work. We believe that Frobenius powers of monomial ideals will be extremely useful in the study of symbolic powers and would like to explore this in this project. Graphs have lots of application to network medicine. Graph invariants of biological networks summarises the biology well and useful in finding drug targets . Recently we showed that minimum size of a vertex cover and matching numbers of immunological networks summarises the biology well and very useful to find drug targets. These are connected to height and Castelnuovo-Mumford regularity of edge ideals. One may ask how does other invariants of edge ideals summarise biological information. Also it seems worth exploring that instead of just the edges, which summarises interaction between two molecules, more complex combinatorial structures influences the biology. It is known that the triangles in a graph are generators of second symbolic powers of edge ideals along with products of edges(that are not divisible by some triangle). These motivates the following question: Q4: How does various homological invariants of edge ideal, its second power and second symbolic powers capture the biology of various immunological networks and is there a way to use them in drug target detection? |