Projects

Papers

  • The Ihara Zeta Function of the Infinite Grid (arXiv preprint)
    Abstract: The infinite grid is the Cayley graph of ZxZ with the usual generators. In this paper, the Ihara zeta function for the infinite grid is computed using elliptic integrals and theta functions. The zeta function of the grid extends to an analytic, multivalued function which satisfies a functional equation. The set of singularities in the domain is finite. The grid zeta function is the first computed example which is non-elementary, and which takes infinitely many values at each point of the domain.
    Related Mathematica files: ComplexUtilities.m, GridZeta.m, GridZeta.nb.
    November 3, 2013 AMS Talk.
  • Zeta Functions of Graphs With Z Actions, Journal of Combinatorial Theory, Series B, Volume 99 # 1, Jan 2009, p48-61. (Online version) | (Preprint version)
    Abstract: Suppose Y is a regular covering of a graph X with covering transformation group G = Z. This paper gives an explicit formula for the L2 zeta function of Y and computes examples. When G = Z, the L2 zeta function is an algebraic function. As a consequence it extends to a meromorphic function on a Riemann surface. The meromorphic extension provides a setting to generalize known properties of zeta functions of regular graphs, such as the location of singularities and the functional equation.
  • With David Letscher, Optimal Strategies for Sports Betting Pools, Operations Research 55 # 6, Nov-Dec 2007, pp 1163-1177.
    pools-or.pdf
    Also see the Optimal Strategies for Sports Betting Pools page for more information, including reports on other seasons results.
    Abstract: Every fall, millions of Americans enter betting pools to pick the winners of each weekend's football games. In the spring, NCAA tournament basketball pools are even more popular. In both cases, teams which are popularly perceived as ``favorites'' gain a disproportionate share of entries. In large pools there can be a significant advantage to picking upsets that differentiate your picks from the crowd.

    In this paper we present a model of betting pools that incorporates opponent behavior. We use the model to derive strategies that maximize the expected return on a bet in both football and tournament style pools. These strategies significantly outperform strategies based on maximizing score or number of correct picks--often by orders of magnitude.

  • With Kevin Whyte, Growth of Betti numbers, Topology, volume 42, #5, p1125-1142 (2003)
    cw02_growth.dvi
    Abstract: Suppose X is any finite complex with vanishing L2 Betti number. We prove upper bounds on the Betti numbers for regular coverings of X, sublinear in the order of covering. The bounds are sensitive to the Novikov-Shubin invariants of X, and are improved in the presence of a spectral gap.

  • With Shahriar Mokhtari-Sharghi, Convergence of zeta functions of graphs, Proceedings of the AMS, volume 130, #7, p1881-1886 (2002)
    cm00.pdf
    Abstract: The L2-zeta function of an infinite graph Y (defined previously in a ball around zero) has an analytic extension. For a tower of finite graphs covered by Y, the normalized zeta functions of the finite graphs converge to the L2-zeta function of Y.

  • With Shahriar Mokhtari-Sharghi, Zeta Functions Of Discrete Groups Acting On Trees, Journal of Algebra 237, p591-620 (2001)
    cm99_zeta.pdf
    Abstract: This paper generalizes Bass' work on zeta functions for uniform tree lattices. Using the theory of von Neumann algebras, machinery is developed to define the zeta function of a discrete group of automorphisms of a bounded degree tree. The main theorems relate the zeta function to determinants of operators defined on edges or vertices of the tree. A zeta function associated to a non-uniform tree lattice with appropriate Hilbert representation is defined. Zeta functions are defined for infinite graphs with a cocompact or finite covolume group action.
  • Residual Amenability and the Approximation of L2-invariants, Michigan Math Journal 46(2), 1999.
    res_amenb.dvi
    Abstract: We generalize Luck's Theorem to show that the L2-Betti numbers of a residually amenable covering space are the limit of the L2-Betti numbers of a sequence of amenable covering spaces. We show that any residually amenable covering space of a finite simplicial complex is of determinant class, and that the L2 torsion is a homotopy invariant for such spaces. We give examples of residually amenable groups, including the Baumslag-Solitar groups.
  • Residual Amenability and the Approximation of L2-invariants, PhD Thesis, U. of Chicago. June 1998.
    clair_thesis.dvi