--- _id: '14445' abstract: - lang: eng text: "We prove the following quantitative Borsuk–Ulam-type result (an equivariant analogue of Gromov’s Topological Overlap Theorem): Let X be a free ℤ/2-complex of dimension d with coboundary expansion at least ηk in dimension 0 ≤ k < d. Then for every equivariant map F: X →ℤ/2 ℝd, the fraction of d-simplices σ of X with 0 ∈ F (σ) is at least 2−d Π d−1k=0ηk.\r\n\r\nAs an application, we show that for every sufficiently thick d-dimensional spherical building Y and every map f: Y → ℝ2d, we have f(σ) ∩ f(τ) ≠ ∅ for a constant fraction μd > 0 of pairs {σ, τ} of d-simplices of Y. In particular, such complexes are non-embeddable into ℝ2d, which proves a conjecture of Tancer and Vorwerk for sufficiently thick spherical buildings.\r\n\r\nWe complement these results by upper bounds on the coboundary expansion of two families of simplicial complexes; this indicates some limitations to the bounds one can obtain by straighforward applications of the quantitative Borsuk–Ulam theorem. Specifically, we prove\r\n\r\n• an upper bound of (d + 1)/2d on the normalized (d − 1)-th coboundary expansion constant of complete (d + 1)-partite d-dimensional complexes (under a mild divisibility assumption on the sizes of the parts); and\r\n\r\n• an upper bound of (d + 1)/2d + ε on the normalized (d − 1)-th coboundary expansion of the d-dimensional spherical building associated with GLd+2(Fq) for any ε > 0 and sufficiently large q. This disproves, in a rather strong sense, a conjecture of Lubotzky, Meshulam and Mozes." article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Uli full_name: Wagner, Uli id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 - first_name: Pascal full_name: Wild, Pascal id: 4C20D868-F248-11E8-B48F-1D18A9856A87 last_name: Wild citation: ama: Wagner U, Wild P. Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes. Israel Journal of Mathematics. 2023;256(2):675-717. doi:10.1007/s11856-023-2521-9 apa: Wagner, U., & Wild, P. (2023). Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes. Israel Journal of Mathematics. Springer Nature. https://doi.org/10.1007/s11856-023-2521-9 chicago: Wagner, Uli, and Pascal Wild. “Coboundary Expansion, Equivariant Overlap, and Crossing Numbers of Simplicial Complexes.” Israel Journal of Mathematics. Springer Nature, 2023. https://doi.org/10.1007/s11856-023-2521-9. ieee: U. Wagner and P. Wild, “Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes,” Israel Journal of Mathematics, vol. 256, no. 2. Springer Nature, pp. 675–717, 2023. ista: Wagner U, Wild P. 2023. Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes. Israel Journal of Mathematics. 256(2), 675–717. mla: Wagner, Uli, and Pascal Wild. “Coboundary Expansion, Equivariant Overlap, and Crossing Numbers of Simplicial Complexes.” Israel Journal of Mathematics, vol. 256, no. 2, Springer Nature, 2023, pp. 675–717, doi:10.1007/s11856-023-2521-9. short: U. Wagner, P. Wild, Israel Journal of Mathematics 256 (2023) 675–717. date_created: 2023-10-22T22:01:14Z date_published: 2023-09-01T00:00:00Z date_updated: 2023-12-13T13:09:07Z day: '01' ddc: - '510' department: - _id: UlWa doi: 10.1007/s11856-023-2521-9 external_id: isi: - '001081646400010' file: - access_level: open_access checksum: fbb05619fe4b650f341cc730425dd9c3 content_type: application/pdf creator: dernst date_created: 2023-10-31T11:20:31Z date_updated: 2023-10-31T11:20:31Z file_id: '14475' file_name: 2023_IsraelJourMath_Wagner.pdf file_size: 623787 relation: main_file success: 1 file_date_updated: 2023-10-31T11:20:31Z has_accepted_license: '1' intvolume: ' 256' isi: 1 issue: '2' language: - iso: eng license: https://creativecommons.org/licenses/by/4.0/ month: '09' oa: 1 oa_version: Published Version page: 675-717 publication: Israel Journal of Mathematics publication_identifier: eissn: - 1565-8511 issn: - 0021-2172 publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 256 year: '2023' ... --- _id: '11777' abstract: - lang: eng text: "In this dissertation we study coboundary expansion of simplicial complex with a view of giving geometric applications.\r\nOur main novel tool is an equivariant version of Gromov's celebrated Topological Overlap Theorem. The equivariant topological overlap theorem leads to various geometric applications including a quantitative non-embeddability result for sufficiently thick buildings (which partially resolves a conjecture of Tancer and Vorwerk) and an improved lower bound on the pair-crossing number of (bounded degree) expander graphs. Additionally, we will give new proofs for several known lower bounds for geometric problems such as the number of Tverberg partitions or the crossing number of complete bipartite graphs.\r\nFor the aforementioned applications one is naturally lead to study expansion properties of joins of simplicial complexes. In the presence of a special certificate for expansion (as it is the case, e.g., for spherical buildings), the join of two expanders is an expander. On the flip-side, we report quite some evidence that coboundary expansion exhibits very non-product-like behaviour under taking joins. For instance, we exhibit infinite families of graphs $(G_n)_{n\\in \\mathbb{N}}$ and $(H_n)_{n\\in\\mathbb{N}}$ whose join $G_n*H_n$ has expansion of lower order than the product of the expansion constant of the graphs. Moreover, we show an upper bound of $(d+1)/2^d$ on the normalized coboundary expansion constants for the complete multipartite complex $[n]^{*(d+1)}$ (under a mild divisibility condition on $n$).\r\nVia the probabilistic method the latter result extends to an upper bound of $(d+1)/2^d+\\varepsilon$ on the coboundary expansion constant of the spherical building associated with $\\mathrm{PGL}_{d+2}(\\mathbb{F}_q)$ for any $\\varepsilon>0$ and sufficiently large $q=q(\\varepsilon)$. This disproves a conjecture of Lubotzky, Meshulam and Mozes -- in a rather strong sense.\r\nBy improving on existing lower bounds we make further progress towards closing the gap between the known lower and upper bounds on the coboundary expansion constants of $[n]^{*(d+1)}$. The best improvements we achieve using computer-aided proofs and flag algebras. The exact value even for the complete $3$-partite $2$-dimensional complex $[n]^{*3}$ remains unknown but we are happy to conjecture a precise value for every $n$. %Moreover, we show that a previously shown lower bound on the expansion constant of the spherical building associated with $\\mathrm{PGL}_{2}(\\mathbb{F}_q)$ is not tight.\r\nIn a loosely structured, last chapter of this thesis we collect further smaller observations related to expansion. We point out a link between discrete Morse theory and a technique for showing coboundary expansion, elaborate a bit on the hardness of computing coboundary expansion constants, propose a new criterion for coboundary expansion (in a very dense setting) and give one way of making the folklore result that expansion of links is a necessary condition for a simplicial complex to be an expander precise." alternative_title: - ISTA Thesis article_processing_charge: No author: - first_name: Pascal full_name: Wild, Pascal id: 4C20D868-F248-11E8-B48F-1D18A9856A87 last_name: Wild citation: ama: Wild P. High-dimensional expansion and crossing numbers of simplicial complexes. 2022. doi:10.15479/at:ista:11777 apa: Wild, P. (2022). High-dimensional expansion and crossing numbers of simplicial complexes. Institute of Science and Technology. https://doi.org/10.15479/at:ista:11777 chicago: Wild, Pascal. “High-Dimensional Expansion and Crossing Numbers of Simplicial Complexes.” Institute of Science and Technology, 2022. https://doi.org/10.15479/at:ista:11777. ieee: P. Wild, “High-dimensional expansion and crossing numbers of simplicial complexes,” Institute of Science and Technology, 2022. ista: Wild P. 2022. High-dimensional expansion and crossing numbers of simplicial complexes. Institute of Science and Technology. mla: Wild, Pascal. High-Dimensional Expansion and Crossing Numbers of Simplicial Complexes. Institute of Science and Technology, 2022, doi:10.15479/at:ista:11777. short: P. Wild, High-Dimensional Expansion and Crossing Numbers of Simplicial Complexes, Institute of Science and Technology, 2022. date_created: 2022-08-10T15:51:19Z date_published: 2022-08-11T00:00:00Z date_updated: 2023-06-22T09:56:36Z day: '11' ddc: - '500' - '516' - '514' degree_awarded: PhD department: - _id: GradSch - _id: UlWa doi: 10.15479/at:ista:11777 ec_funded: 1 file: - access_level: open_access checksum: f5f3af1fb7c8a24b71ddc88ad7f7c5b4 content_type: text/x-python creator: pwild date_created: 2022-08-10T15:34:04Z date_updated: 2022-08-10T15:34:04Z description: Code for computer-assisted proofs in Section 8.4.7 in Thesis file_id: '11780' file_name: flags.py file_size: 16828 relation: supplementary_material - access_level: open_access checksum: 1f7c12dfe3bdaa9b147e4fbc3d34e3d5 content_type: text/x-c++src creator: pwild date_created: 2022-08-10T15:34:10Z date_updated: 2022-08-10T15:34:10Z description: Code for proof of Lemma 8.20 in Thesis file_id: '11781' file_name: lowerbound.cpp file_size: 12226 relation: supplementary_material - access_level: open_access checksum: 4cf81455c49e5dec3b9b2e3980137eeb content_type: text/x-python creator: pwild date_created: 2022-08-10T15:34:17Z date_updated: 2022-08-10T15:34:17Z description: Code for proof of Proposition 7.9 in Thesis file_id: '11782' file_name: upperbound.py file_size: 3240 relation: supplementary_material - access_level: open_access checksum: 4e96575b10cbe4e0d0db2045b2847774 content_type: application/pdf creator: pwild date_created: 2022-08-11T16:08:33Z date_updated: 2022-08-11T16:08:33Z file_id: '11809' file_name: finalthesisPascalWildPDFA.pdf file_size: 5086282 relation: main_file title: High-Dimensional Expansion and Crossing Numbers of Simplicial Complexes - access_level: closed checksum: 92d94842a1fb6dca5808448137573b2e content_type: application/zip creator: pwild date_created: 2022-08-11T16:09:19Z date_updated: 2022-08-11T16:09:19Z file_id: '11810' file_name: ThesisSubmission.zip file_size: 18150068 relation: source_file file_date_updated: 2022-08-11T16:09:19Z has_accepted_license: '1' language: - iso: eng month: '08' oa: 1 oa_version: Published Version page: '170' project: - _id: 2564DBCA-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '665385' name: International IST Doctoral Program publication_identifier: isbn: - 978-3-99078-021-3 issn: - 2663-337X publication_status: published publisher: Institute of Science and Technology status: public supervisor: - first_name: Uli full_name: Wagner, Uli id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 title: High-dimensional expansion and crossing numbers of simplicial complexes type: dissertation user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9 year: '2022' ...