Applications of commutative algebra

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My research lies in the intersection of three areas of mathematics: algebra, geometry, and combinatorics. The overlap of algebra and geometry is a rich area of mathematics that dates back to at least the first half of the 17th century. The mathematician/philosopher Rene Descartes was among the first to demonstrate the power of associating geometric objects with algebraic equations and vice versa. Describing a line or a parabola with an equation is an example of this technique that many Canadians will remember from their mathematics courses in high school. In a similar way, there are ways to connect algebra with problems in combinatorics (counting problems and discrete structures), allowing one to study the same problem through multiple lenses. While many new tools and techniques have been developed since Descartes' time, the theme of studying problems from many different angles has remained a constant.**The research of the proposal is interested in exploring and uncovering new connections between algebra, geometry, and combinatorics, and in particular, applying the tools and techniques of commutative algebra to questions in other areas of mathematics. A brief summary of three of my proposed projects are given below. **One facet of my proposed research focuses on the algebraic structures attached to a graph, in order to build a bridge between graph theory and algebra. A graph is a collection of nodes with lines joining some of the nodes together. When we colour a graph, we want to assign a colour to each node subject to the rule that nodes that are joined together by a line must receive a different colour. One would then like to know what is the least number of colours you need to colour the graph. Colouring problems can be related to questions about scheduling and even solving Sudoku problems. Over the last couple of years, I have been interested in studying how this colouring information can be encoded algebraically. My current research proposal is interested in finding graphs with interesting colouring properties in order to build unique and interesting algebraic structures.**Another project of my proposal also hopes to strengthen this bridge between graph theory and algebra. I am interested in objects called toric ideals which can be constructed from a graph. Toric ideals have rich structure, and can be studied geometrical, algebraically, or even combinatorially. Toric ideals are of great interest because they have applications to areas such as biology and statistics. I am interested in determining how the graph theory information gets encoded into an algebraic object called a minimal free resolution. **A third project of my research proposal is to use geometry and algebra to study problems about matrices. Matrices, which are rectangular arrays of numbers, are commonly introduced as a tool to study systems of equations. I am currently interested in problems of the following type: what properties about the matrix can we deduce if we are only given some partial information about the matrix, e.g. the location of the non-zero entries? The origin of this type of question can be traced back to work of P. Samuelson, the Nobel Prize winner in Economics. More recently, problems of this type have appeared in the context of modelling diseases. My proposed research plans to study problems of this type through the lenses of algebraic geometry and commutative algebra, thus introducing new tools and techniques to the area.**These projects, which are part of my ongoing research programme to understand homological and algebraic invariants, will hopefully provide a theoretical basis for future applications.