Browsing by Subject "Topology"

The central idea of this dissertation is to interpret certain invariants constructed from Laplace spectral data on a compact Riemannian manifold as regularized integrals of closed differential forms on the space of Riemannian metrics, or more generally on a space of metrics on a vector bundle. We apply this idea to both the Ray-Singer analytic torsion

and the eta invariant, explaining their dependence on the metric used to define them with a Stokes' theorem argument. We also introduce analytic multi-torsion, a generalization of analytic torsion, in the context of certain manifolds with local product structure; we prove that it is metric independent in a suitable sense.

To every Legendrian link in R3, we can assign a differential graded algebra (DGA) called the Chekanov-Eliashberg DGA. An exact Lagrangian cobordism between two Legendrian links induces a DGA map between the corresponding Chekanov-Eliashberg DGAs, and this association is functorial. This DGA map was written down explicity for exact, decomposable Lagrangian fillings as Z_2-count of certain pseudoholomorphic disks by Ekholm, Honda, and K ́alm ́an, and this was combinatorially upgraded to an integral count by Casals and Ng. However, this upgrade only assigned an automorphism class of DGA maps. We approach the same problem of integral lifts by a different strategy, first done for the differential in the Chekanov-Eliashberg DGA by Ekholm, Etnyre, and Sullivan. Here, we find the precise DGA maps for all exact, decomposable Lagrangian cobordisms through this more analytic method.

In the present work, I explore the theme of diagrammatics and their capacity to shed insight on two trends—categorification and compositionality—in and around contemporary category theory. The work begins with an introduction of these meta- phenomena in the context of elementary sets and maps. Towards generalizing their study to more complicated domains, we provide a self-contained treatment—from a pedagogically novel perspective that introduces almost all notion via diagrammatic language—of the categorical machinery with which we may express the broader no- tions that found the sequel. The work then branches into two seemingly unrelated disciplines: dynamical systems and knot theory. In particular, the former research defines what it means to compose dynamical systems in a manner analogous to how one composes simple maps. The latter work concerns the categorification of the slN link invariant. In particular, we use a virtual filtration to give a more diagrammatic reconstruction of Khovanov-Rozansky homology via a smooth TQFT. Finally, the work culminates in a manifesto on the philosophical role of category theory.

Enhancers possess both structural elements mediating promoter looping and functional elements mediating gene expression. Traditional models of enhancer-mediated gene regulation imply genomic overlap or immediate adjacency of these elements. We test this model by combining densely-tiled CRISPRa screening with nucleosome-resolution Region Capture Micro-C topology analysis. Using this integrated approach, we comprehensively define the cis-regulatory landscape for the tumor suppressor PTEN, identifying and validating 10 distinct enhancers and defining their 3D spatial organization. Unexpectedly, we identify several long-range functional enhancers whose promoter proximity is facilitated by chromatin loop anchors several kilobases away, and demonstrate that accounting for this spatial separation improves the computational prediction of validated enhancers. Thus, we propose a new model of enhancer organization incorporating spatial separation of essential functional and structural components.

Neurons are remarkably long-lived cells that are born early on in development and maintained over the lifespan of an organism. Their birth is followed by their iterative maturation into functional neurons that can participate in CNS circuitry. This involves the temporally regulated programs of gene transcription enabling them to migrate, generate axons, dendrites, synapses, and spatial connections with other cells within their niche. This is mediated at least in part by dynamics in chromatin biology, as mutations in chromatin regulators are strongly implicated in the advent of neurodevelopmental and neuropsychiatric disorders. The mechanisms by which chromatin dynamics orchestrate neuronal maturation remain poorly understood. We find that the postnatal maturation of cerebellar granule neurons (CGNs) requires dynamic changes in the genomic distribution of histone H3 lysine 27 trimethylation (H3K27me3), demonstrating a causal function for this chromatin modification in gene regulation beyond its canonical role in cell fate specification. The developmental loss of H3K27me3 at promoters of genes that turn on as CGNs mature is facilitated by the lysine demethylase, and ASD-risk gene, Kdm6b, through its catalytic activity. Interestingly, inhibition of the H3K27 methyltransferase EZH2 in newborn CGNs not only blocks the repression of progenitor genes but also impairs the induction of mature CGN genes, showing the importance of bidirectional H3K27me3 regulation across the genome. We also find dynamics in regulatory chromatin topology to facilitate the interaction between cerebellar enhancers and their cognate genes during cerebellar maturation, that appears to be poised by H3K27me3. These data demonstrate that dynamics at the level of chromatin primary, secondary, and tertiary structures in developing postmitotic neurons regulate the temporal coordination of gene expression programs that underlie functional neuronal maturation.