The tauopathies are a group of neurodegenerative diseases characterized by abnormal tau neuropathology and include frontotemporal dementia (FTD), progressive supranuclear palsy (PSP), corticobasal degeneration (CBD), and Alzheimer's disease (AD), collectively with a heavy burden on the lives of patients and on public health. Uncertainty of the underlying disease mechanisms hinders the development of effective treatments and diagnostics. While most cases of each disease are sporadic, a strong genetic contribution has been identified and provides an opening for the discovery of clues to their pathophysiology.

I led a series of genetic screens to systematically search for genetic contributors to the tauopathies. A genome-wide association study of PSP was performed to pinpoint common polymorphisms that contribute to disease risk. Aside from supporting associations at loci that had already been reported near MAPT, STX6, MOBP, and EIF2AK3, novel associations were discovered at genome-wide significance (near RUNX2 and SLCO1A2) and suggestive significance (near DUSP10, SP1, ASAP1, and WDR63). Furthermore, we identified genetic correlations between PSP, Parkinson's disease, and amyotrophic lateral sclerosis, indicating that the role of tau may extend to more neurodegenerative diseases than currently appreciated. Leveraging the coverage of the genotyping array, copy number variants were also called in a subset of this disease cohort. A previously unreported duplication spanning part of the tau gene was found in two PSP patients, expanding the scope of mutations found in the disease.

While the SNP genotyping array in the previous study provided genome-wide coverage of common polymorphisms, a newly designed exome array specifically typed rare exonic variants found in large population cohorts. We performed one of the first exome array studies aimed at uncovering causal genetic risk factors in AD, FTD, and PSP. We identified a contribution of exonic variants in the ABCA7 gene to AD, presaging a body of current literature focusing on protein-disrupting mutations of this gene. Additionally, we provided support for the candidate genes PAXIP1 and DYSF in AD. Whole genome sequencing studies outlined a contribution of rare protein-disrupting mutations in predicted damaging genes. The MAPT A152T variant was also confirmed as a risk factor in tauopathies, with our work expanding the phenotypic spectrum.

We further established the downstream molecular effects of tauopathy risk alleles. The H1 haplotype of chromosome 17q21.31, a major risk factor for PSP and CBD that is also polymorphic in the general population, had a large influence on DNA methylation in the region and mRNA expression of nearby genes. This methylation pattern appeared to mediate some of the risk conferred from the H1 haplotype. We also identified another methylation signature on the promoter of the gene encoding IL-1β that was correlated with aging and methylation, with mechanistic support from mouse models of sirtuin 1 function. Finally, we linked rs242557 - a risk allele within MAPT - with plasma tau concentration, a potential biomarker of clinical utility.

Future work will focus on improving statistical power by increasing the size of sample cohorts and integrating additional layers of genomic data, including those from public epigenomics and mRNA expression studies. I outline a path for the translation of the results from completed and future work to enabling precision medicine for patients suffering from tauopathies.

The dissertation presents a collection of interrelated works in the philosophy of mathematics. They are roughly unified by their focus on descriptive set theory, which is investigated through the lens of mathematical practice. Chapters 2 and 3 examine the roles that confluence plays in mathematical practice, such as providing justification for the Church-Turing Thesis. An extensive survey of the technical literature will attest to the ubiquity of justification by confluence, and it will be shown to serve a wide variety of justificational purposes that are largely orthogonal to each other. Chapter 4 presents a series of theorems and proofs that involve increasingly substantial use of metamathematical methods. Reflecting on the question of whether the metamathematical elements can be translated away without loss of insight, it attempts to shed light on our practical taxonomy of proofs by their methodology, as well as on the specific question of whether a proof can be said to make substantial use of metamathematical methods. Chapter 5 traces the pre-history of the theory of Borel equivalence relations, with the specific aim of identifying the early ancestors to this theory prior to its sudden emergence in the 1990s.

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