[{"publication_status":"published","publication_identifier":{"issn":["0021-2172"],"eissn":["1565-8511"]},"file_date_updated":"2023-10-31T11:20:31Z","has_accepted_license":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"intvolume":"       256","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."}],"volume":256,"article_type":"original","date_created":"2023-10-22T22:01:14Z","author":[{"first_name":"Uli","orcid":"0000-0002-1494-0568","full_name":"Wagner, Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","last_name":"Wagner"},{"id":"4C20D868-F248-11E8-B48F-1D18A9856A87","full_name":"Wild, Pascal","last_name":"Wild","first_name":"Pascal"}],"day":"01","scopus_import":"1","title":"Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes","oa_version":"Published Version","issue":"2","citation":{"chicago":"Wagner, Uli, and Pascal Wild. “Coboundary Expansion, Equivariant Overlap, and Crossing Numbers of Simplicial Complexes.” <i>Israel Journal of Mathematics</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s11856-023-2521-9\">https://doi.org/10.1007/s11856-023-2521-9</a>.","ista":"Wagner U, Wild P. 2023. Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes. Israel Journal of Mathematics. 256(2), 675–717.","apa":"Wagner, U., &#38; Wild, P. (2023). Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes. <i>Israel Journal of Mathematics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s11856-023-2521-9\">https://doi.org/10.1007/s11856-023-2521-9</a>","mla":"Wagner, Uli, and Pascal Wild. “Coboundary Expansion, Equivariant Overlap, and Crossing Numbers of Simplicial Complexes.” <i>Israel Journal of Mathematics</i>, vol. 256, no. 2, Springer Nature, 2023, pp. 675–717, doi:<a href=\"https://doi.org/10.1007/s11856-023-2521-9\">10.1007/s11856-023-2521-9</a>.","ama":"Wagner U, Wild P. Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes. <i>Israel Journal of Mathematics</i>. 2023;256(2):675-717. doi:<a href=\"https://doi.org/10.1007/s11856-023-2521-9\">10.1007/s11856-023-2521-9</a>","ieee":"U. Wagner and P. Wild, “Coboundary expansion, equivariant overlap, and crossing numbers of simplicial complexes,” <i>Israel Journal of Mathematics</i>, vol. 256, no. 2. Springer Nature, pp. 675–717, 2023.","short":"U. Wagner, P. Wild, Israel Journal of Mathematics 256 (2023) 675–717."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"language":[{"iso":"eng"}],"department":[{"_id":"UlWa"}],"file":[{"date_updated":"2023-10-31T11:20:31Z","creator":"dernst","file_size":623787,"date_created":"2023-10-31T11:20:31Z","file_id":"14475","content_type":"application/pdf","access_level":"open_access","file_name":"2023_IsraelJourMath_Wagner.pdf","success":1,"checksum":"fbb05619fe4b650f341cc730425dd9c3","relation":"main_file"}],"month":"09","quality_controlled":"1","page":"675-717","ddc":["510"],"date_updated":"2023-12-13T13:09:07Z","_id":"14445","type":"journal_article","doi":"10.1007/s11856-023-2521-9","article_processing_charge":"Yes (via OA deal)","publisher":"Springer Nature","date_published":"2023-09-01T00:00:00Z","status":"public","publication":"Israel Journal of Mathematics","year":"2023","isi":1,"external_id":{"isi":["001081646400010"]}},{"date_updated":"2023-06-22T09:56:36Z","_id":"11777","type":"dissertation","doi":"10.15479/at:ista:11777","article_processing_charge":"No","alternative_title":["ISTA Thesis"],"publisher":"Institute of Science and Technology","page":"170","ddc":["500","516","514"],"year":"2022","ec_funded":1,"date_published":"2022-08-11T00:00:00Z","degree_awarded":"PhD","project":[{"_id":"2564DBCA-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"665385","name":"International IST Doctoral Program"}],"status":"public","date_created":"2022-08-10T15:51:19Z","author":[{"first_name":"Pascal","full_name":"Wild, Pascal","id":"4C20D868-F248-11E8-B48F-1D18A9856A87","last_name":"Wild"}],"day":"11","title":"High-dimensional expansion and crossing numbers of simplicial complexes","oa_version":"Published Version","publication_status":"published","publication_identifier":{"issn":["2663-337X"],"isbn":["978-3-99078-021-3"]},"file_date_updated":"2022-08-11T16:09:19Z","has_accepted_license":"1","abstract":[{"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.","lang":"eng"}],"department":[{"_id":"GradSch"},{"_id":"UlWa"}],"file":[{"file_id":"11780","file_size":16828,"date_created":"2022-08-10T15:34:04Z","date_updated":"2022-08-10T15:34:04Z","creator":"pwild","relation":"supplementary_material","checksum":"f5f3af1fb7c8a24b71ddc88ad7f7c5b4","description":"Code for computer-assisted proofs in Section 8.4.7 in Thesis","file_name":"flags.py","access_level":"open_access","content_type":"text/x-python"},{"relation":"supplementary_material","description":"Code for proof of Lemma 8.20 in Thesis","checksum":"1f7c12dfe3bdaa9b147e4fbc3d34e3d5","file_name":"lowerbound.cpp","content_type":"text/x-c++src","access_level":"open_access","file_id":"11781","date_created":"2022-08-10T15:34:10Z","file_size":12226,"creator":"pwild","date_updated":"2022-08-10T15:34:10Z"},{"file_id":"11782","date_updated":"2022-08-10T15:34:17Z","creator":"pwild","file_size":3240,"date_created":"2022-08-10T15:34:17Z","description":"Code for proof of Proposition 7.9 in Thesis","checksum":"4cf81455c49e5dec3b9b2e3980137eeb","relation":"supplementary_material","content_type":"text/x-python","access_level":"open_access","file_name":"upperbound.py"},{"checksum":"4e96575b10cbe4e0d0db2045b2847774","relation":"main_file","content_type":"application/pdf","access_level":"open_access","file_name":"finalthesisPascalWildPDFA.pdf","title":"High-Dimensional Expansion and Crossing Numbers of Simplicial Complexes","file_id":"11809","creator":"pwild","date_updated":"2022-08-11T16:08:33Z","date_created":"2022-08-11T16:08:33Z","file_size":5086282},{"date_updated":"2022-08-11T16:09:19Z","creator":"pwild","file_size":18150068,"date_created":"2022-08-11T16:09:19Z","file_id":"11810","content_type":"application/zip","access_level":"closed","file_name":"ThesisSubmission.zip","checksum":"92d94842a1fb6dca5808448137573b2e","relation":"source_file"}],"supervisor":[{"first_name":"Uli","orcid":"0000-0002-1494-0568","last_name":"Wagner","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Uli"}],"month":"08","citation":{"short":"P. Wild, High-Dimensional Expansion and Crossing Numbers of Simplicial Complexes, Institute of Science and Technology, 2022.","ieee":"P. Wild, “High-dimensional expansion and crossing numbers of simplicial complexes,” Institute of Science and Technology, 2022.","ama":"Wild P. High-dimensional expansion and crossing numbers of simplicial complexes. 2022. doi:<a href=\"https://doi.org/10.15479/at:ista:11777\">10.15479/at:ista:11777</a>","mla":"Wild, Pascal. <i>High-Dimensional Expansion and Crossing Numbers of Simplicial Complexes</i>. Institute of Science and Technology, 2022, doi:<a href=\"https://doi.org/10.15479/at:ista:11777\">10.15479/at:ista:11777</a>.","apa":"Wild, P. (2022). <i>High-dimensional expansion and crossing numbers of simplicial complexes</i>. Institute of Science and Technology. <a href=\"https://doi.org/10.15479/at:ista:11777\">https://doi.org/10.15479/at:ista:11777</a>","ista":"Wild P. 2022. High-dimensional expansion and crossing numbers of simplicial complexes. Institute of Science and Technology.","chicago":"Wild, Pascal. “High-Dimensional Expansion and Crossing Numbers of Simplicial Complexes.” Institute of Science and Technology, 2022. <a href=\"https://doi.org/10.15479/at:ista:11777\">https://doi.org/10.15479/at:ista:11777</a>."},"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","oa":1,"language":[{"iso":"eng"}]}]