Reference number : GC-2017-Q1-02202, amendment number : , agreement type : G, recipient legal name : Denham, Graham (The University of Western Ontario), recipient country : CA, recipient province : nan, recipient city : Ontario, recipient postal code : nan, program name english : Discovery Grants Program - Individual, program name french : Programme de subventions à la découverte - individuelles, program purpose english : My research involves interactions between combinatorics and techniques from areas of algebra, topology, and geometry. 
 
Combinatorics is a branch of mathematics which sits at the foundation of computer science and operations research. It involves the study of discrete structures as well as of counting problems. Many interesting objects in other areas of mathematics are fundamentally combinatorial in nature, so methods of combinatorics appear naturally in understanding families of "test" objects such as toric varieties and hyperplane arrangements. These can be used as accessible examples to help identify and understand general mathematical phenomena. 
 
In a different direction, the well-developed techniques of algebra and geometry can sometimes answer combinatorial questions and offer deep reasons for observed or conjectured behaviour of objects in discrete mathematics. For example, if the answer to a counting problem is a number which in itself shows no extra structure, one might find that this number is, in fact, some evaluation of a polynomial. Perhaps the polynomial has coefficients which themselves may be interpreted as volumes of certain polyhedra, or they count the number of high-dimensional "holes" in a geometric object which could be built from the original problem. In this way, one may obtain a better understanding of a combinatorial object, by viewing it as a shadow of something with much more structure and, one hopes, properties which are better known or at least more easily analyzed., program purpose french : My research involves interactions between combinatorics and techniques from areas of algebra, topology, and geometry. 
 
Combinatorics is a branch of mathematics which sits at the foundation of computer science and operations research. It involves the study of discrete structures as well as of counting problems. Many interesting objects in other areas of mathematics are fundamentally combinatorial in nature, so methods of combinatorics appear naturally in understanding families of "test" objects such as toric varieties and hyperplane arrangements. These can be used as accessible examples to help identify and understand general mathematical phenomena. 
 
In a different direction, the well-developed techniques of algebra and geometry can sometimes answer combinatorial questions and offer deep reasons for observed or conjectured behaviour of objects in discrete mathematics. For example, if the answer to a counting problem is a number which in itself shows no extra structure, one might find that this number is, in fact, some evaluation of a polynomial. Perhaps the polynomial has coefficients which themselves may be interpreted as volumes of certain polyhedra, or they count the number of high-dimensional "holes" in a geometric object which could be built from the original problem. In this way, one may obtain a better understanding of a combinatorial object, by viewing it as a shadow of something with much more structure and, one hopes, properties which are better known or at least more easily analyzed., agreement title english : Arrangements of Hyperplanes and Combinatorial Constructions in Topology, agreement title french : Arrangements of Hyperplanes and Combinatorial Constructions in Topology, agreement number : RGPIN, agreement value : 120000.0, agreement start date : 2017-05-10, agreement end date : nan, description english : nan, description french : nan, expected results english : nan, expected results french : nan, additional information english : Grant or Award spanning more than one fiscal year. (2017-2018 to 2022-2023), additional information french : Subvention ou bourse octroyée s'appliquant à plus d'un exercice financier. (2017-2018 à 2022-2023), owner organization : nserc-crsng, owner organization title : Natural Sciences and Engineering Research Council of Canada | Conseil de recherches en sciences naturelles et en génie du Canada