WILLIAM DANIEL GILLAM


(Picture)



Office: Kassar House 114
Phone: (401)-863-7957
Address:
Department of Mathematics
Brown University
Box 1917
Providence, RI 02912
Email:
wgillam@math.brown.edu

MATHEMATICAL ACTIVITY


Academic Appointments:
NSF Postdoctoral Research Fellowship (2008-2012)
  • Hosting Institution: Brown University
  • Sponsoring Scientist: Dan Abramovich
    Tamarkin Assistant Professor of Mathematics, Brown University (July 2008-2012)
  • On leave 2008-2009 and 2010-2011 academic years under NSF support

    Education:
    Mathematics Ph.D. Columbia University 2008 (Advisor: M. Thaddeus)
    Mathematics M.Phil. Columbia University 2008
    Mathematics M.A. Columbia University 2004
    Mathematics M.A. Wesleyan University 2003 (Advisor: W. W. Comfort)
    Mathematics B.A. Wesleyan University 2002

    Research Interests:
    Algebraic Geometry
    Curve counting/Gromov-Witten Theory
    Orbifolds and the Crepant Resolution Conjecture
    Knot Theory, topology
    Knot Homology Theories (Khovanov, Heegaard-Floer)
    Homological Algebra and Sheaf Theory
    Logarithmic geometry
    Differential spaces

    Teaching:
    Math 0100: Calculus II (Fall 2011)
    Math 1610: Probability (Fall 2009)
    Math 0520: Linear Algebra (Spring 2010)
    Math 1040: Fundamental Problems of Geometry (Spring 2012)
    Math 0540: Honors Linear Algebra (Spring 2012)


    PhD Thesis:

    Hyperelliptic Gromov-Witten Theory.
    [Download .PDF]

    Publications and Preprints:

    Cleanliness of geodesics in hyperbolic 3-manifolds.
    (with C. Adams, A. Colestock, J. Fowler, and E. Katerman)
    Pacific J. Math. 213 (2004), no. 2, 201-211.
    [Download .PS] [Download .PDF]

    Cusp size bounds from singular surfaces in hyperbolic 3-manifolds.
    (with C. Adams, A. Colestock, J. Fowler, and E. Katerman)
    Trans. Amer. Math. Soc. 358 (2006) 727-741.

    The embeddability ordering of topological spaces.
    Wesleyan University Master's Thesis. Advisor: W. W. Comfort

    The embeddability ordering of topological spaces.
    (with W. W. Comfort)
    Top. App. 153 (2006) 2192-2198.
    [Download .PS] [Download .PDF]

    Embeddability properties of countable metric spaces.
    Top. App. 148 (2005) 63-82.
    [Download .PS] [Download .PDF]

    Computability of a topological poset.
    Top. App. 153 (2006) 1132-1140.
    [Download .PS] [Download .PDF]
    Some calculations for this paper were carried out by computer.
    This C++ program should compile on most systems without difficulty.
    [Download .CPP]

    Knot homology of (3,m) torus knots.
    We calculate Khovanov's homology of the (3,m) torus knots (with integral coefficients) from first principles,
    with a little help from Rasmussen's Lemma.
    [Download .PDF]

    A sheaf-theoretic interpretation of Khovanov's knot homology.
    We show that the "cube of resolutions" used in Khovanov's categorification of the Jones polynomial
    may be viewed as a sheaf on the cube {0,1}^n, viewing the latter as a finite topological space with the order
    topology. Khovanov's homology groups are interpreted as cohomology of this sheaf with support at the closed point.
    The basic long exact sequences in knot homology are interpreted as cohomology long exact sequences associated
    to short exact sequences of sheaves.
    [Download .PDF]

    Computations of Heegaard Floer knot homology.
    (with John A. Baldwin)
    [Download .PDF]
    Also see my related website where you can find a few computer programs.

    The Crepant Resolution Conjecture for an involution of 3d flags.
    Let F be the manifold of flags in a complex three dimensional vector space equipped with a nondegenerate symmetric bilinear form.
    We prove the crepant resolution conjecture of Bryan-Graber for the involution of F taking a flag to its orthogonal complement.
    [Download .PDF]

    Maximal subbundles, Quot schemes, and curve counting
    Last updated: May 15, 2009
    Discusses stable pairs/Donaldson-Thomas residue invariants, virtual intersection theory of the Quot scheme,
    maximal subbundle counts, enumerative significance of residue invariants, etc.
    [Download .PDF]

    Deformation of quotients on a product
    Last updated: August 21, 2009
    Deformation/obstruction theory of quotients of a sheaf on a product pulled back from one of the factors.
    [Download .PDF]

    Oriented real blowup
    Last updated: October 2011
    Discusses the oriented real blowup of an analytic space along a Cartier divisor and how this is related to the Kato-Nakayama spaces of log geometry. Recently added a section on symplectic geometry of oriented real blowups.
    [Download .PDF]

    A shortened version of this appeared as the "Rounding" section in:

    Logarithmic geometry and moduli
    (in the Handbook of Moduli)
    (with Dan Abramovich, Qile Chen, Yuhao Huang, Martin Olsson, Matt Satriano, Shenghao Sun)
    Last updated: June 2010
    [Download .PDF]

    Virtual localization for stable pairs
    Last updated: March 2010
    We explain how to view torus fixed stable pairs in a rank two bundle over a smooth curve as a closed
    subscheme of a product of Quot schemes of symmetric products of the bundle, and we explain the perfect
    obstruction theory on this space inherited as the torus fixed part of the perfect obstruction theory on
    moduli of stable pairs.
    [Download .PDF]

    Relative quotients
    Last updated: September 2010
    We discuss the relative Quot scheme of a vector bundle on a curve, with the goal of proving a
    degeneration formula using log cohomology to recapture vanishing cycles in a degeneration. We also
    interpret the condition for a quotient to be ``well-behaved" along singular loci of expanded degenerations
    as a condition of flatness over one of Olsson's stacks of log structures.
    [Download .PDF]

    On Kapranov's description of moduli of genus zero curves as a Chow quotient
    (with N. Giansiracusa)
    Last updated: December 2010
    We give a simple proof of Kapranov's theorem, which says that the Chow and Hilbert quotients of (P^1)^n by SL_2
    are isomorphic to the usual Deligne-Mumford moduli space of genus zero curves.
    [Download .PDF]

    The evaluation space of logarithmic stable maps
    (with D. Abramovich, Q. Chen, and S. Marcus)
    Last updated: October 2011
    We describe the stacks of log points and log nodes in a log scheme X, where the evaluation maps from the space of log stable maps to X naturally land.
    [Download .PDF]

    Localization of ringed spaces
    Last updated: February 2011
    We introduce the notion of a prime system on a ringed space, and the localization of a ringed space at
    a prime system. We show how this can be used to construct inverse limits of locally ringed spaces and a
    very general Spec functor.
    [Download .PDF]

    Logarithmic stacks and minimality
    Last updated: February 2011
    Given a category fibered in groupoids over schemes with a log structure, one produces (functorially) a category fibered
    in groupoids over log schemes. We describe the essential image of this functor in terms of a categorical notion
    of minimal objects. The result is actual a general statement about groupoid fibrations over fibered categories.
    We discuss several examples from the literature from this point of view, including the log curves of F. Kato
    and the log points from the paper with Abramovich et. al. above.
    [Download .PDF]


    On differential and analytic spaces
    Last updated: September 2011
    We give a cnoncise treatment of the category of differential spaces and construct some new functors to it.
    [Download .PDF]


    Complexification
    Last updated: October 2011
    We study the right adjoint to the inclusion from complexes in an abelian category A into functors from the integers to A and its derived functors.
    [Download .PDF]



    Exposition and Course Notes:

    Algebraic Geometry Exercises
    (compiled with Joe Ross and Matt Deland)
    Last updated: Feb. 16, 2010
    List of problems in algebraic geometry/sheaf theory/algebra of varying difficulty.
    Special sections on: gluing along closed subschemes, cotangent complex, quot schemes
    [Download .PDF]

    Topics in Probability
    Last updated: Dec., 2009
    Notes from the probability class I taught. Pretty rough, but amounts to a fairly complete text for a difficult
    one semester probability class. 77 Pages.
    [Download .PDF]

    Sheaf Theory
    Last updated: Feb., 2010
    Notes on "classical" sheaf theory. 64 Pages.
    [Download .PDF]

    Log Geometry
    Last updated: January 13, 2009
    [Download .PDF]

    Motivic Integration
    Last updated: January 18, 2009
    [Download .PDF]

    On the de Rham cohomology of algebraic varieties
    Discussion of Grothendieck's paper of the same title
    Last updated: Feb. 14, 2012
    [Download .PDF]


    Epistolary:

    Letter to J. Wise.
    August 2, 2008.
    Some discussion of hyperelliptic Hodge integrals.
    [Download .PDF]

    Letter to M. Deland.
    April 9, 2009.
    Our computation of the basic enumerative invariants of the cubic surface by classical and Gromov-Witten techniques.
    Reasonable introductory level example of computation by GW theory.
    [Download .PDF]

    Letter to D. Abramovich et. al.
    October 15, 2010.
    We given a categorical definition of "minimal" objects in a category fibered in groupoids over log schemes and use it
    to explain when such a groupoid fibration "comes from" a groupoid fibration over schemes with log structure.
    This contains some significant errors and is superceded by the "Logarithmic stacks and minimality" article above.
    [Download .PDF]

    Letter to D. Abramovich et. al.
    October 25, 2010.
    We prove the representability of the moduli space of standard log points in a fine log scheme by an algebraic space. This
    moduli space is closely related to the target space for evaluation maps in log Gromov-Witten theory; this connection
    should be explained in future work. This will ultimately be superceded by the "Evaluation space" article above.
    [Download .PDF]

    Letter to Joe Ross and Jarod Alper
    January, 2011.
    We introduce the localization of a ringed space, which ultimately became part of one of the papers above.
    We prove several descent theorems for schemes, ringed spaces, and locally ringed spaces.
    [Download .PDF]

    Letter to D. Abramovich et. al.
    January, 2011.
    A continuation of the Oct. 25, 2010 letter: We prove that the stack of log points in a fine log scheme is algebraic.
    We discuss the evaluation maps of log Gromov-Witten theory in terms of Kato-Nakayama spaces.
    [Download .PDF]


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    W. D. Gillam