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Graphical Supercharacters

Luis Garcia (2014); Student Collaborator(s): Jacob Brumbaugh-Smith (2013); Andrew Turner (2014 HMC); Madeleine Bulkow (2014 SCR); Additional Collaborator(s): Matthew Michal*; Mentor(s): Stephan Garcia
*Claremont Graduate University

Abstract: The theory of supercharacters, which generalizes classical character theory, was recently introduced by P. Diaconis and I.M. Isaacs. We study supercharacter theories on (Z=nZ)d induced by the actions of certain matrix groups, demonstrating that a variety of exponential sums of interest in number theory (e.g., Gauss, Ramanujan, and Kloosterman sums) arise in this manner. We also develop the super-Fourier transform, a generalization of the discrete Fourier transform, in which supercharacters play the role of the Fourier exponential basis. For this transform, we provide a corresponding uncertainty principle and compute the associated constants in several cases.
Funding Provided by: Pomona College SURP (LG); Pomona College Mathematics Department (JBS)

Canonical Correlation applied to genes from RNASeq data

Isabelle Ambler (2013); Student Collaborator(s): Jacob Coleman (2013); Mentor(s): Johanna Hardin

Abstract: There has been recent interest in studying natural variation in human gene expression with respect to phenotypic characteristics to find groups of genes with similar function. Previous studies have proposed applying canonical correlation analysis (CCA) to high-throughput data to find maximum correlations between linear combinations of two multidimensional data sets. However, high-throughput data, like RNA-Seq data, is characteristically noisy and, without modification, CCA is susceptible to distortions from these outliers. We demonstrate that robust CCA is a promising approach to computing reliable canonical correlations.

Funding Provided by: Fletcher Jones Foundation (IA); Paul K. Richter and Evalyn E. Cook Richter Memorial Funds (JC)

Estimating Heritability

Karl Kumbier (2013); Student Collaborator(s): Cody Moore (2013); Mentor(s): Johanna Hardin

Abstract: Recent advances in gene sequencing technology have spurred an increase of Genome Wide Association Studies (GWAS), which attempt to find relationships between expressed phenotypes and different loci in the genome. One of the parameters of interest in these studies is heritability (h2), the proportion of phenotypic variance for a given trait accounted for by additive genetic effects. Finding h2 requires an estimate of the relatedness among a sample of individuals; the pairwise estimates make up the genetic relationship matrix (GRM). Our study aims to show that the measurement and sampling errors introduced by estimating the GRM bias estimates of heritability downward. Understanding both how much noise is introduced by using a sample GRM and how much this noise biases estimates of h2 allows future GWAS to correct for sampling and measurement errors.
Funding Provided by: Paul K. Richter and Evalyn E. Cook Richter Memorial Funds (KK); Pomona College Mathematics Department (CM)