# Browsing by Subject "Topology"

###### Results Per Page

###### Sort Options

Item Open Access Analytic Torsion, the Eta Invariant, and Closed Differential Forms on Spaces of Metrics(2016) Andreae, PhillipThe 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.

Item Embargo DGA maps Induced by Decomposable Fillings with Z-coefficients(2023) Mohanakumar, ChinduTo 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.

Item Open Access Diagrammatics in Categorification and Compositionality(2019) Vagner, DmitryIn 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.

Item Embargo Integrative PTEN Enhancer Discovery Reveals a New Model of Enhancer Organization(2024) Cerda-Smith, Christian GonzaloEnhancers 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.

Item Open Access Math 412 - Topology with Applications(2016-06-24) Ghadyali, Hamza; Bendich, Paul LHighlights 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.Item Embargo Postmitotic Dynamics in Chromatin Modification and Regulatory Topology Underlie Cerebellar Granule Maturation(2023) Ramesh, VijyendraNeurons 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.

Item Open Access The Spherical Manifold Realization Problem(2020-05-09) Davis, A. BlytheThe Lickorish-Wallace theorem states that any closed, orientable, connected 3-manifold can be obtained by integral Dehn surgery on a link in S^3. The spherical manifold realization problem asks which spherical manifolds (i.e., those with finite fundamental groups) can be obtained through integral surgery on a knot in S^3. The problem has previously been solved by Greene and Ballinger et al. for lens space and prism manifolds, respectively. In this project, we determine which of the remaining three types of spherical manifolds (tetrahedral, octahedral, and icosahedral) can be obtained by positive integral surgery on a knot in S^3. We follow methods inspired by those presented by Greene.