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You searched for `+publisher:"Wayne State University" +contributor:("John R. Klein")`

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Wayne State University

1. Qin, Lizhen. Moduli spaces and cw structures arising from morse theory.

Degree: PhD, Mathematics, 2011, Wayne State University

URL: https://digitalcommons.wayne.edu/oa_dissertations/328

In this dissertation, we study the moduli spaces and CW Structures arising from Morse theory.
Suppose M is a smooth manifold and f is a Morse function on it. We consider the negative gradient flow of f. Suppose the flow satisfies transversality. This naturally defines the moduli spaces of flow lines and gives a stratication of M by its unstable manifolds. The gluing of broken flow lines can also be constructed.
We prove that, under certain assumptions, these moduli spaces can be compactified and the compactified spaces are smooth manifolds with corners. Moreover, these compactified manifolds satisfy certain orientation formulas. We also prove that the stratication of M is actually a CW decomposition of M with explicit characteristic maps, which has good properties. Finally, we show that the associativity of gluing of broken flow lines exclusively follows from the compatibility of the manifold structures of the compactified moduli spaces, which establishes the associativity of gluing in certain cases.
In order to obtain the above results, we also prove some results on the dynamical aspects of negative gradient flows, which may be of independent interest.
*Advisors/Committee Members: John R. Klein.*

Subjects/Keywords: Manifolds; Morse Theory; Topology; Mathematics

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Qin, L. (2011). Moduli spaces and cw structures arising from morse theory. (Doctoral Dissertation). Wayne State University. Retrieved from https://digitalcommons.wayne.edu/oa_dissertations/328

Chicago Manual of Style (16^{th} Edition):

Qin, Lizhen. “Moduli spaces and cw structures arising from morse theory.” 2011. Doctoral Dissertation, Wayne State University. Accessed December 12, 2019. https://digitalcommons.wayne.edu/oa_dissertations/328.

MLA Handbook (7^{th} Edition):

Qin, Lizhen. “Moduli spaces and cw structures arising from morse theory.” 2011. Web. 12 Dec 2019.

Vancouver:

Qin L. Moduli spaces and cw structures arising from morse theory. [Internet] [Doctoral dissertation]. Wayne State University; 2011. [cited 2019 Dec 12]. Available from: https://digitalcommons.wayne.edu/oa_dissertations/328.

Council of Science Editors:

Qin L. Moduli spaces and cw structures arising from morse theory. [Doctoral Dissertation]. Wayne State University; 2011. Available from: https://digitalcommons.wayne.edu/oa_dissertations/328

Wayne State University

2. Peter, John Whitson. Stabilization and classification of poincare duality embeddings.

Degree: PhD, Mathematics, 2012, Wayne State University

URL: https://digitalcommons.wayne.edu/oa_dissertations/470

We define a space E(K,X) of Poincare Duality embeddings and show that such spaces admit a highly connected stabilization map.
This serves as a tool for classifying Poincare Duality embeddings in terms of the homotopy types of their complements. In
particular, a Poincare embedding gives rise to a fiberwise duality
map in the category of retractive spaces over X. We use this construction to obtain a highly connected classification map with target a moduli space of unstable complements for Poincare embeddings. As consequences, we obtain stabilization and classication results for
smooth embeddings.
*Advisors/Committee Members: John R. Klein.*

Subjects/Keywords: Mathematics

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Peter, J. W. (2012). Stabilization and classification of poincare duality embeddings. (Doctoral Dissertation). Wayne State University. Retrieved from https://digitalcommons.wayne.edu/oa_dissertations/470

Chicago Manual of Style (16^{th} Edition):

Peter, John Whitson. “Stabilization and classification of poincare duality embeddings.” 2012. Doctoral Dissertation, Wayne State University. Accessed December 12, 2019. https://digitalcommons.wayne.edu/oa_dissertations/470.

MLA Handbook (7^{th} Edition):

Peter, John Whitson. “Stabilization and classification of poincare duality embeddings.” 2012. Web. 12 Dec 2019.

Vancouver:

Peter JW. Stabilization and classification of poincare duality embeddings. [Internet] [Doctoral dissertation]. Wayne State University; 2012. [cited 2019 Dec 12]. Available from: https://digitalcommons.wayne.edu/oa_dissertations/470.

Council of Science Editors:

Peter JW. Stabilization and classification of poincare duality embeddings. [Doctoral Dissertation]. Wayne State University; 2012. Available from: https://digitalcommons.wayne.edu/oa_dissertations/470

Wayne State University

3. Catanzaro, Michael Joseph. A Topological Study Of Stochastic Dynamics On CW Complexes.

Degree: PhD, Mathematics, 2016, Wayne State University

URL: https://digitalcommons.wayne.edu/oa_dissertations/1433

In this dissertation, we consider stochastic motion of subcomplexes of a CW complex, and explore the implications on the underlying space. The random process on the complex is motivated from Ito diffusions on smooth manifolds and Langevin processes in physics. We associate a Kolmogorov equation to this process, whose solutions can be interpretted in terms of generalizations of electrical, as well as stochastic, current to higher dimensions. These currents also serve a key function in relating the random process to the topology of the complex. We show the average current generated by such a process can be written in a physically familiar form, consisting of the solution to Kirchhoffâ€™s network problem and the Boltzmann distribution, suitably generalized to arbitrary dimensions. We analyze these two components in detail, and discover they reveal an unexpected amount of information about the topology of the CW complex. The main result is a quantization result for the average current in the low temperature, adiabatic limit. As an application, we express the Reidemeister torsion of the complex, a topological invariant, in terms of these quantities.
*Advisors/Committee Members: Vladimir Y. Chernyak, John R. Klein.*

Subjects/Keywords: Mathematics

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Catanzaro, M. J. (2016). A Topological Study Of Stochastic Dynamics On CW Complexes. (Doctoral Dissertation). Wayne State University. Retrieved from https://digitalcommons.wayne.edu/oa_dissertations/1433

Chicago Manual of Style (16^{th} Edition):

Catanzaro, Michael Joseph. “A Topological Study Of Stochastic Dynamics On CW Complexes.” 2016. Doctoral Dissertation, Wayne State University. Accessed December 12, 2019. https://digitalcommons.wayne.edu/oa_dissertations/1433.

MLA Handbook (7^{th} Edition):

Catanzaro, Michael Joseph. “A Topological Study Of Stochastic Dynamics On CW Complexes.” 2016. Web. 12 Dec 2019.

Vancouver:

Catanzaro MJ. A Topological Study Of Stochastic Dynamics On CW Complexes. [Internet] [Doctoral dissertation]. Wayne State University; 2016. [cited 2019 Dec 12]. Available from: https://digitalcommons.wayne.edu/oa_dissertations/1433.

Council of Science Editors:

Catanzaro MJ. A Topological Study Of Stochastic Dynamics On CW Complexes. [Doctoral Dissertation]. Wayne State University; 2016. Available from: https://digitalcommons.wayne.edu/oa_dissertations/1433