**An application of logarithmic geometry to moduli of curves of genus greater than one.**

This is my PhD thesis, to be submitted in less than two weeks to the University of Colorado, Boulder. I'm pretty sure that I'm not permitted to put it online, so here's the abstract and a link to a version of my defense talk. Part of the results appear in the paper "Contractions of subcurves of log smooth curves"; other results will appear in a paper to be finished later this summer.

DefenseAbstract: We use log geometric data to construct contractions of families of curves of arbitrary genus. We use log blowups and this contraction construction to resolve the rational map between the moduli space of stable curves and some alternate semistable compactifications of the moduli space of curves. In at least two cases, the target space is new.

**Contractions of subcurves of log smooth curves**, ArXiv Preprint at ArXiv:1908.09633

Abstract: Let C be a nodal curve, and let E be a union of semistable subcurves of C. We consider the problem of contracting the connected components of E to singularities in a way that preserves the genus of C and makes sense in families. In order to do this, we introduce the notion of mesa curve, a nodal curve with a logarithmic structure and a nice subcurve. We then show that such a contraction exists for families of mesa curves. Resulting singularities include the elliptic Gorenstein singularities.

**Rank Drops of Recurrence Matrices**in

*Electronic Journal of Linear Algebra*, 2015.

Link

Abstract: A recurrence matrix is a matrix whose terms are sequential members of a linear homogeneous recurrence sequence of order k and whose dimensions are both greater than or equal to k. In this paper, the ranks of recurrence matrices are determined. In particular, it is shown that the rank of such a matrix differs from the previously found upper bound of k in only two situations: When (a_j) satisfies a recurrence relation of order less than k, and when the nth powers of distinct eigenvalues of (a_j) coincide.

**Asymptotic equivalence of group actions on surfaces and Riemann-Hurwitz solutions**in

*Archiv der Mathematik,*with Aaron Wootton. 2014.

Link

Abstract: The topological data of a group action on a compact Riemann surface can be encoded using a tuple (h; m_1, ..., m_s) called its signature. There are two number theoretic conditions on a tuple necessary for it to be a signature: the Riemann-Hurwitz formula is satisfied and each m_i equals the order of a non-trivial group element. We show on the genus spectrum of a group that asymptotically, satisfaction of these conditions is in fact sufficient. We also describe the order of growth for the number of tuples which satisfy these conditions but are not signatures in the case of cyclic groups.

**Interactive dynamic simulations with co-located maglev haptic and 3d graphic display**in

*ACHI 2013,*with Peter Berkelman and Muneaki Miyasaka. 2013.

Link

**Interactive Rigid-Body Dynamics and Deformable Surface Simulations with Co-Located Maglev Haptic and 3D Graphic Display**in

*International Journal On Advances in Intelligent Sytems,*with Peter Berkelman and Muneaki Miyasaka. 2013.

Link

**Co-located haptic and 3D graphic interface for medical simulations**in

*Studies in health technology and informatics,*with Peter Berkelman and Muneaki Miyasaka. 2012.

Link