4 points in the two sphere

Matthew Sean Miller

Research Interests



Edge colored hypergraph
The braid arrangment in four variables





My interest in the topology of subspace arrangements and configuration spaces has led me to study many facets of mathematics including algebraic and geometric topology, combinatorics, algebra, differential geometry, and soon algebraic geometry. The goal of my research is to use topology, particularly rational homotopy theory, and its connections to other parts of mathematics to define and compute invariants of topological manifolds.

Here is a summary of my research aimed at a general mathematical audience.
Here is a more detailed and more technical description of my research.


Current projects and preprints

(in reverse chronological order)
  • Subspace arrangements.
    A major part of my current research focuses on the topology of the complement of subspace arrangements. I am particularly interested in studying the rational homotopy type of complements of arrangements using various combinatorial tools.

    • Formality of the complement of an arrangement. (with Max Wakefield)
      This project is still in its early stages. We are working towards understanding the necessary combinatorial conditions to show that the complement of a complex arrangement is formal.


    • Massey products and k-equal arrangements.
      I have a couple results about the existence and non-existence of (higher order) Massey products in k-equal arrangements that I am currently writing up.


    • Pascal arrangements. (with Max Wakefield)
      In this little paper we describe an infinite family of examples and prove that their intersection lattices are not geometric but their complements are rationally formal. Two of the intersection lattices are illustrated to the right.
      Pascal lattices
      This paper has been submitted and a link to the pdf file will appear here soon.
      Pascal arrangements (coming soon)


    • Edge colored hypergraphic subspace arrangements. (with Max Wakefield)
      We introduce the notion of an edge-colored hypergraphic subspace arrangement and prove some combinatorial and topological results about these arrangements. These results have some consequences for k-equal arrangements. We are in the process of furthering the topological results of this paper.

      This paper has been accepted for publication by Pure and Applied Mathematics Quarterly
      and it has been posted on the arxiv as arxiv:0903.4221.

      Here is a direct link to a pdf file of the paper.


  • Relatives of the square peg problem.
    The "square peg problem" is a hundred year old problem which asks if a simple closed curve in the plane always contains four points that form a square. With Corey Dunn I have started investigating variants of this problem for other manifolds. Our work was inspired by the recent paper posted on the arxiv by Vrecica and Zivaljevic.


  • Configuration spaces of lens spaces.
    I completed my doctoral work in 2007 with Dev Sinha at the University of Oregon. The title of my thesis is The rational homotopy types of configuration spaces of three-dimensional lens spaces. As the title suggests, I build rational models for the configuration spaces of three-dimensional lens spaces. To build these models I compute all possible Massey products using explicit geometric representatives. The diagram on the right shows a schematic picture of how to construct a Massey triple product. I also describe generators for the rational homotopy groups. In the complete version I construct certain computable maps which are homeomorphism invariants of lens spaces. One of my current projects is to classify these rational models up to quasi-isomorphism.

    Here are links to pdf files of this work:
    The complete thesis
    A Massey product
    This shorter version has been accepted for publication in Homology, Homotopy and Applications.


  • Nets in the complex projective plane.
    My first unpublished paper is joint work with Corey Dunn, Max Wakefield and Sebastian Zwicknagl. We prove that the only (4,k) net in CP^2, for k less than 6 is the (4,3) net known as the Hessian. The diagram to the right illustrates the combinatorics of the Hessian. This work stemmed from a problem stated in Yuzvinsky's paper Realization of finite Abelian groups by nets in P^2. After completing this paper we found that most (if not all) of these results were previously known. But it is a nice little computation that uses some group theory and some linear algebra.

    This paper is posted on the arxiv: math.CO/0703142
    The Hessian



Scholarly activities
Here is a partial list of seminars and conferences that I have spoken at (or will speak at soon):
In addition to the conferences I have spoken at, I have attended many conferences and workshops at home and abroad, here is a partial list:
  • Eastern Section Meeting of the AMS, Special Session on Homotopy Theory, October 2009
  • Conference in honor of Bob Stong, University of Virginia, November 2007
  • Pacific Institute of Mathematics Summer Schools:
    • Topics in homotopy theory, August 2005, University of Calgary
    • Workshop on knots and three-manifolds, July 2004, University of British Columbia
  • Workshop on configuration spaces, June 2004, UCL Louvain-la-Neuve, Belgium
  • Cascade Topology Seminar, May 2006, November 2004