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Undergraduate Research in Mathematics

Student Research at Pomona

Learn more about student research in mathematics.

Pomona is a community of daring minds. It is a place for students who are venturesome by choice, who have talent, passion, and independence of spirit, and who are prepared to dream big and work hard in order to make a difference in the world.

One way that Pomona College provides opportunities for students to excel is through research opportunities. Conducting research as an undergraduate not only gives students an advantage when applying for fellowships or graduate school; it also gives them a chance to tackle real-world problems and to find out what it’s like to be treated as colleagues by their professors, many of whom are the leading experts in their fields.

Recent Student Research

A robust extension of Sparse Canonical Correlation Analysis for the analysis of genomic data

Joseph Replogle (2013); Student Collaborator(s): Jake Coleman (2013); Mentor(s): Johanna Hardin

Abstract: Medical genomics seeks to explain complex phenotypes based on variations in genetic, epigenetic, and environmental elements. To this end, high-throughput genomic and molecular biology technologies generate vast biological datasets that provide for examination of many variables simultaneously. In order to illuminate the mechanisms and pathways underlying human traits and unveil novel therapeutic avenues, creative statistical techniques must help integrate these diverse genetic datasets. Canonical Correlation Analysis (CCA), a statistical method that maximizes the correlation between linear combinations of sets of variables, and particularly Sparse Canonical Correlation Analysis (SCCA), which performs CCA on a small subset of variables extracted using a penalty function, are fruitful techniques for analysis of the complex relationships found in genomic data. Here we extend SCCA using Spearman Rank Correlation to make the method more robust to outliers. We use a combination of simulated and real data to show that our method outperforms previously proposed SCCA methods in the presence of the noisy data commonly found in biology. Additionally, we propose a permutation test for assessing the significance of multiple canonical variates. We hope that our robust SCCA will allow biologists to better characterize the associations between genetic datasets in order to improve understanding of human disease.
Funding Provided by: Howard Hughes Medical Institute

Averages in the Period 2 Region of the Logistic Map

Maricela Cruz (2014); Mentor(s): Johanna Hardin; Ami Radunskaya

Abstract: The logistic map is a nonlinear difference equation well studied in literature, used to model reproduction and starvation in certain populations. Here we study the distributional characteristics of the stochastic logistic map giving evidence that the map has a stable distribution over the period 2 region. We use simulations in R to support the claim that regardless of the initial distribution of x (for example, x = population size), the logistic map iterates to a unique stable distribution. That is, after 10,000 iterations of the logistic map we arrive at a unique stable distribution of x. We also examine the relationship between the mean of the stochastic logistic equation and the mean of the deterministic logistic equation for period 2. Our initial results show that the relationship between the two averages in period 2 is opposite of that in period one. In the period 2 case the mean of the stochastic logistic equation is greater than the mean of the deterministic logistic equation. We investigate this relationship as the parameter, λ, changes.
Funding Provided by: Linares Family SURP