Math 412 - Topology with Applications

Highlights of Data Expedition: • Students explored daily observations of local climate data spanning the past 35 years. • Topological Data Analysis, or TDA for short, provides cutting-edge tools for studying the geometry of data in arbitrarily high dimensions. • Using TDA tools, students discovered intrinsic dynamical features of the data and learned how to quantify periodic phenomenon in a time-series. • Since nature invariably produces noisy data which rarely has exact periodicity, students also considered the theoretical basis of almost-periodicity and even invented and tested new mathematical definitions of almost-periodic functions. Summary The dataset we used for this data expedition comes from the Global Historical Climatology Network. “GHCN (Global Historical Climatology Network)-Daily is an integrated database of daily climate summaries from land surface stations across the globe.” Source: https://www.ncdc.noaa.gov/oa/climate/ghcn-daily/ We focused on the daily maximum and minimum temperatures from January 1, 1980 to April 1, 2015 collected from RDU International Airport. Through a guided series of exercises designed to be performed in Matlab, students explore these time-series, initially by direct visualization and basic statistical techniques. Then students are guided through a special sliding-window construction which transforms a time-series into a high-dimensional geometric curve. These high-dimensional curves can be visualized by projecting down to lower dimensions as in the figure below (Figure 1), however, our focus here was to use persistent homology to directly study the high-dimensional embedding. The shape of these curves has meaningful information but how one describes the “shape” of data depends on which scale the data is being considered. However, choosing the appropriate scale is rarely an obvious choice. Persistent homology overcomes this obstacle by allowing us to quantitatively study geometric features of the data across multiple-scales. Through this data expedition, students are introduced to numerically computing persistent homology using the rips collapse algorithm and interpreting the results. In the specific context of sliding-window constructions, 1-dimensional persistent homology can reveal the nature of periodic structure in the original data. I created a special technique to study how these high-dimensional sliding-window curves form loops in order to quantify the periodicity. Students are guided through this construction and learn how to visualize and interpret this information. Climate data is extremely complex (as anyone who has suffered from a bad weather prediction can attest) and numerous variables play a role in determining our daily weather and temperatures. This complexity coupled with imperfections of measuring devices results in very noisy data. This causes the annual seasonal periodicity to be far from exact. To this end, I have students explore existing theoretical notions of almost-periodicity and test it on the data. They find that some existing definitions are also inadequate in this context. Hence I challenged them to invent new mathematics by proposing and testing their own definition. These students rose to the challenge and suggested a number of creative definitions. While autocorrelation and spectral methods based on Fourier analysis are often used to explore periodicity, the construction here provides an alternative paradigm to quantify periodic structure in almost-periodic signals using tools from topological data analysis.

Interpreting Human Genetic Variation through Genomics and In Vivo Models

Improvements in genomic technology, both in the increased speed and reduced cost of sequencing, have expanded the appreciation of the abundance of human genetic variation. However the sheer amount of variation, as well as the varying type and genomic content of variation, poses a challenge in understanding the clinical consequence of a single mutation. This work uses several methodologies to interpret the observed variation in the human genome, and presents novel strategies for the prediction of allele pathogenicity.

Using the zebrafish model system as an in vivo assay of allele function, we identified a novel driver of Bardet-Biedl Syndrome (BBS) in CEP76. A combination of targeted sequencing of 785 cilia-associated genes in a cohort of BBS patients and subsequent in vivo functional assays recapitulating the human phenotype gave strong evidence for the role of CEP76 mutations in the pathology of an affected family. This portion of the work demonstrated the necessity of functional testing in validating disease-associated mutations, and added to the catalogue of known BBS disease genes.

Further study into the role of copy-number variations (CNVs) in a cohort of BBS patients showed the significant contribution of CNVs to disease pathology. Using high-density array comparative genomic hybridization (aCGH) we were able to identify pathogenic CNVs as small as several hundred bp. Dissection of constituent gene and in vivo experiments investigating epistatic interactions between affected genes allowed for an appreciation of several paradigms by which CNVs can contribute to disease. This study revealed that the contribution of CNVs to disease in BBS patients is much higher than previously expected, and demonstrated the necessity of consideration of CNV contribution in future (and retrospective) investigations of human genetic disease.

Finally, we used a combination of comparative genomics and in vivo complementation assays to identify second-site compensatory modification of pathogenic alleles. These pathogenic alleles, which are found compensated in other species (termed compensated pathogenic deviations [CPDs]), represent a significant fraction (from 3 – 10%) of human disease-associated alleles. In silico pathogenicity prediction algorithms, a valuable method of allele prioritization, often misrepresent these alleles as benign, leading to omission of possibly informative variants in studies of human genetic disease. We created a mathematical model that was able to predict CPDs and putative compensatory sites, and functionally showed in vivo that second-site mutation can mitigate the pathogenicity of disease alleles. Additionally, we made publically available an in silico module for the prediction of CPDs and modifier sites.

These studies have advanced the ability to interpret the pathogenicity of multiple types of human variation, as well as made available tools for others to do so as well.

Akrasia and the Aesthetic: Human Agency and the Site of Literature, 1760-1820

This study focuses on a series of foundational stylistic and formal innovations in eighteenth-century and Romantic literature, and argues that they can be cumulatively attributed to the distinct challenges authors faced in representing human action and the will. The study focuses in particular on cases of “acting against better judgment” or “failing to do what one knows one ought to do” – concepts originally theorized as “akrasia” and “weakness of the will” in ancient Greek and Scholastic thought. During the Enlightenment, philosophy increasingly conceives of human minds and bodies like systems and machines, and consequently fails to address such cases except as intractable or incoherent. Yet eighteenth-century and Romantic narratives and poetry consistently engage the paradoxes and ambiguities of action and volition in representations of akrasia. As a result, literature develops representational strategies that distinguish the epistemic capacities of literature as privileged over those of philosophy.

The study begins by centering on narratives of distempered selves from the 1760s. Jean-Jacques Rousseau’s Confessions and Laurence Sterne’s A Sentimental Journey narrate cases of knowingly and weakly acting against better judgment, and in so doing, reveal the limitations of the “philosophy of the passions” that famously informed sentimental literature at the time. These texts find that the interpretive difficulties of action demand a non-systematic and hermeneutic approach to interpreting a self through the genre of narrative. Rousseau’s narrative in particular informs William Godwin’s realist novels of distempered subjects. Departing from his mechanistic philosophy of mind and action, Godwin develops the technique of free indirect discourse in his third novel Fleetwood (1805) as a means of evoking the ironies and self-deceptions in how we talk about willing.

Romantic poetry employs the literary trope of weakness of will primarily through the problem of regretted inaction – a problem which I argue motivates the major poetic innovations of William Wordsworth and John Keats. While Samuel Taylor Coleridge sought to characterize his weakness of will in philosophical writing, Wordsworth turns to poetry with The Prelude (1805), revealing poetry itself to be a self-deceiving and disappointing form of procrastination. More explicitly than Wordsworth, John Keats identifies indolence as the prime symbol and basis of what he calls “negative capability.” In his letters and poems such as “On Seeing the Elgin Marbles” (1817) and “Ode on Indolence” (1819), Keats reveals how the irreducibly contradictory qualities of human agency speak to the particular privilege of “disinterested aesthetics” – a genre fitted for the modern era for its ability to disclose contradictions without seeking to resolve or explain them in terms of component parts.