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oai:www.ideals.illinois.edu:2142/102425 2023-07-11 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Tyson, Jeremy Wu, Jang-Mei Fernandes, Rui Kaufman, Robert Jung, Derek 2019-02-06T19:32:49Z 2019-02-06T19:32:49Z 2018-11-08 2018-12 "For $k,n\ge 1$, the jet space $J^k(\R^n)$ is the set of $k^{th}$-order Taylor polynomials of functions in $C^k(\R^n)$. Warhurst constructs a Carnot group structure on $J^k(\R^n)$ such that the jets of functions in $C^{k+1}(\R^n)$ are horizontal. Like in all Carnot groups, one can define a Carnot-Carath\'eodory metric on $J^k(\R^n)$ by minimizing lengths of horizontal paths. Unfortunately, exact forms or even the regularities of geodesics connecting generic pairs of points are not known for $J^k(\R^n)$. After describing the Carnot group structure of $J^k(\R^n)$, we will prove that there exists a biLipschitz embedding of $\mathbb{S}^n$ into $J^k(\R^n)$ that does not admit a Lipschitz extension to $\mathbb{B}^{n+1}$. This strengthens a result of Rigot and Wenger \cite{RW:LNE} and generalizes a result for $\mathbb{H}^n$ of Dejarnette, Haj{\l}asz, Lukyanenko, and Tyson. We will then consider a problem related to Gromov's conjecture on the H\""older equivalence of Carnot groups. We will prove that for all $m\ge 2$ and $\epsilon>0$, there does not exist an injective, locally $(\frac{1}{2}+\epsilon)$-H\""older mapping $f:\R^m\to J^k(\R)$ that is locally Lipschitz as a mapping into $\R^{k+2}$. This builds on a result of Balogh, Haj{\l}asz, and Wildrick for $\mathbb{H}^n$. We will conclude by proposing analogues of horizontal and vertical projections for $J^k(\R)$. We prove Marstrand-type results for these mappings. This continues efforts of Balogh, Durand-Cartagena, F\""assler, Mattila, and Tyson over the past decade to prove Marstrand-type theorems in a sub-Riemannian setting. We will study the metric structure of $J^k(\R^n)$, focusing primarily on the model filiform jet spaces $J^k(\R)$." Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2019-02-05 without embargo terms The student, Derek Jung, accepted the attached license on 2018-11-06 at 07:31. The student, Derek Jung, submitted this Dissertation for approval on 2018-11-06 at 07:40. This Dissertation was approved for publication on 2018-11-08 at 09:07. DSpace SAF Submission Ingestion Package generated from Vireo submission #13061 on 2019-02-05 at 11:08:54 Made available in DSpace on 2019-02-06T19:32:49Z (GMT). No. of bitstreams: 6 JUNG-DISSERTATION-2018.pdf: 545800 bytes, checksum: 7552089aec29e8d004b6453984c42e8b (MD5) ThesisJung.bib: 41450 bytes, checksum: fc920505d125290eaa05a78b45121911 (MD5) Thesis_jung_Nov41209.bbl: 10027 bytes, checksum: 151cdbe145dd2a020e4a140a8d01900b (MD5) Thesis_jung_Nov41209.tex: 297189 bytes, checksum: d5eb59dcd20df1c79139ec45ef0a1d6e (MD5) LICENSE.txt: 4207 bytes, checksum: 0c96b859403c9fbce7a0c55577afec26 (MD5) PROQUEST_LICENSE.txt: 4553 bytes, checksum: cb50808d7910c57a6b5d4ca2cb4fcb4d (MD5) Previous issue date: 2018-11-08 application/pdf http://hdl.handle.net/2142/102425 en Copyright 2018 Derek Jung Sub-Riemannnian Geometry Jet spaces Carnot groups Geometric Analysis Lipschitz and Holder mappings into jet space Carnot groups text text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.