Dr. William Grilliette received his doctor of philosophy from the University of Nebraska – Lincoln on 7 May 2011. His research spans several different areas of mathematics. His main interest is in functional analysis, and operator algebras in specific. He has constructed a modified presentation theory for C*-algebras and proven a corresponding Tietze transformation theorem, analogous to the classical results of Heinrich Tietze for group theory.
In adapting the classical results for group theory to functional analysis, a key tool is the abstraction of category theory. Dr. Grilliette's philosophy regards category theory as an abstract Rosetta Stone, allowing ideas from one area of mathematics to be transferred or mimicked in another. In this way, collaboration can be ignited across starkly different branches of mathematics.
For example, Dr. Grilliette has studied directed multigraphs, or “quivers”, using categorical methods and properties. He has constructed the injective envelope and projective cover of a quiver, analogous to the injective envelope and projective cover of a module. He also collaborated in showing a connection between the blow-up of a quiver and abstract injectivity.
Currently, Dr. Grilliette is investigating the following: generation of matricially normed spaces, constructions and applications of weighted sets, and the category theory of hypergraphs.
- Grilliette, Will. Scaled-free objects II. Ann. Funct. Anal. 6 (2015), no. 3, 216-261.
- Grilliette, Will; Seacrest, Deborah E.; Seacrest, Tyler. On blow-ups and injectivity of quivers. Electron. J. Combin. 20 (2013), no. 2, Paper 40, 16 pp.
- Grilliette, Will. Injective envelopes and projective covers of quivers. Electron. J. Combin. 19 (2012), no. 2, Paper 39, 15 pp.
- Grilliette, Will. Scaled-free objects. New York J. Math. 18 (2012), 275–289.
- Grilliette, Will. Presentations and Tietze transformations of C*-algebras. New York J. Math. 18 (2012), 121–137.