• The problem with deconvolving structural signals (in preparation).
    Through a combinatorial analysis, we find that the standard prediction methods for structural distributions of RNA sequences fail to identify scenarios where multiple structures are present in a distribution. RNA research has exploded in the past couple decades, as biologists realized that RNA sequences play a diverse set of biological roles, and thus cannot easily be understood through DNA analysis alone. The structure of an RNA sequence encodes valuable information about its function. Biologists derived several computational models to predict the single structure of an RNA sequence from its list of nucleotides alone. However, evidence has arisen indicating that some RNA sequences can naturally fold into more than one structure, contrary to previous assumptions. Research with collaborator Christine Heitsch identified problems current models encounter in predicting multimodal distributions. We conclude that the standard methods of identifying distributions are often unable to identify that more than one structure is present when incorporating data modelled from a weighted mixture of two structures. Even when both structures can be identified, the correct weighting of the structures cannot.
  • Multivariate generating functions: asymptotics and examples (in preparation). Talk available online.
    Using singularity analysis, we found asymptotic expressions for the coefficients of generating functions with algebraic singularities. Such generating functions appear in many contexts: the Catalan numbers, many probabilistic generating functions, and functions encoding the outputs of context free grammars are all algebraic. These generating functions provide a challenge because the branch cuts in algebraic functions prevented the use of previous techniques that relied on residues. Through the Cauchy integral formula and explicit contour defomrations, we derive an asymptotic formula for the coefficients of the generating functions. The results are applied to a stochastic context-free grammar (SCFG) that predicts RNA secondary structures. Here, we find asymptotic formulae for the probability that a structure was produced with specific ratios of structural features.
  • Sublinear variance in Euclidean first passage percolation (in preparation).
    We prove that in Euclidean first passage percolation, the passage time of fluid over a distance of length n has a variance at most Cn/log(n) for some constant, C. First passage percolation (FPP) has gathered attention in the world of probability as a simple model for natural phenomena. Originally proposed as a model for fluid flow and biological growth, FPP can be adapted for many applied settings, such as modelling the growth of cancer in biomedicine, deposition of materials in physics, and auction theory in economics. Despite the simplicity of the model, many of its properties are notoriously difficult to analyze. For example, the limiting asymptotic shape of how far fluid has spread after time n is still unknown as n approaches infinity, although such a limit is known to exist. In work with Megan Bernstein and Michael Damron, we look at a related model called Euclidean first passage percolation, which has the advantage of rotational symmetry, and thus a clear limiting asymptotic shape. However, this rotational symmetry comes with the price: analyzing other aspects of the distance is technical. To show sublinear variance, we rely on the Falik-Samorodnitsky inequality and log-Sobolev inequalities to reduce the problem to bounds on sums of derivatives. These sums can be handled using results on greedy lattice animals.
  • Asymptotics of Bivariate Generating Functions with Algebraic Singularities, 2018. Journal of Combinatorial Theory, Series A. 153: 1-30.
    In this paper, we use the multivariate analytic techniques of Pemantle and Wilson to derive asymptotic formulae for the coefficients of a broad class of multivariate generating functions with algebraic singularities. Then, we apply these results to a generating function encoding information about the stationary distributions of a graph coloring algorithm studied by Butler, Chung, Cummings, and Graham (2015). Historically, Flajolet and Odlyzko (1990) analyzed the coefficients of a class of univariate generating functions with algebraic singularities. These results have been extended to classes of multivariate generating functions by Gao and Richmond (1992) and Hwang (1996, 1998), in both cases by immediately reducing the multivariate case to the univariate case. Pemantle and Wilson (2013) outlined new multivariate analytic techniques and used them to analyze the coefficients of rational generating functions. These multivariate techniques are used here to analyze functions with algebraic singularities.
  • Quantum random walk on the integer lattice: Examples and phenomena   (with A. Bressler, R. Pemantle, and M. PetkovÅ¡ek), 2010. In: Algorithmic Probability and Combinatorics, AMS Contemporary Mathematics series, vol. 520, pp. 221-240.
    We apply results of Baryshnikov, Bressler, Pemantle, and collaborators, to compute limiting probability profiles for various quantum walks in one and two dimensions. Using analytical machinery, we obtain some features of the limit distribution that are not evident in an empirical intensity plot of the time 10,000 distribution. Some conjectures are stated, and computational techniques are discussed as well.