[{"type":"journal_article","status":"public","_id":"243","author":[{"full_name":"Timothy Browning","orcid":"0000-0002-8314-0177","last_name":"Browning","id":"35827D50-F248-11E8-B48F-1D18A9856A87","first_name":"Timothy D"},{"first_name":"Roger","last_name":"Heath Brown","full_name":"Heath-Brown, Roger"}],"publist_id":"7661","title":"Quadratic polynomials represented by norm forms","date_updated":"2021-01-12T06:57:26Z","citation":{"ista":"Browning TD, Heath Brown R. 2012. Quadratic polynomials represented by norm forms. Geometric and Functional Analysis. 22(5), 1124–1190.","chicago":"Browning, Timothy D, and Roger Heath Brown. “Quadratic Polynomials Represented by Norm Forms.” Geometric and Functional Analysis. Springer Basel, 2012. https://doi.org/10.1007/s00039-012-0168-5.","ama":"Browning TD, Heath Brown R. Quadratic polynomials represented by norm forms. Geometric and Functional Analysis. 2012;22(5):1124-1190. doi:10.1007/s00039-012-0168-5","apa":"Browning, T. D., & Heath Brown, R. (2012). Quadratic polynomials represented by norm forms. Geometric and Functional Analysis. Springer Basel. https://doi.org/10.1007/s00039-012-0168-5","short":"T.D. Browning, R. Heath Brown, Geometric and Functional Analysis 22 (2012) 1124–1190.","ieee":"T. D. Browning and R. Heath Brown, “Quadratic polynomials represented by norm forms,” Geometric and Functional Analysis, vol. 22, no. 5. Springer Basel, pp. 1124–1190, 2012.","mla":"Browning, Timothy D., and Roger Heath Brown. “Quadratic Polynomials Represented by Norm Forms.” Geometric and Functional Analysis, vol. 22, no. 5, Springer Basel, 2012, pp. 1124–90, doi:10.1007/s00039-012-0168-5."},"extern":1,"publisher":"Springer Basel","quality_controlled":0,"intvolume":" 22","month":"08","abstract":[{"lang":"eng","text":"Let P(t) ∈ ℚ[t] be an irreducible quadratic polynomial and suppose that K is a quartic extension of ℚ containing the roots of P(t). Let N K/ℚ(X) be a full norm form for the extension K/ℚ. We show that the variety P(t) =N K/ℚ(X)≠ 0 satisfies the Hasse principle and weak approximation. The proof uses analytic methods."}],"page":"1124 - 1190","date_created":"2018-12-11T11:45:24Z","issue":"5","date_published":"2012-08-25T00:00:00Z","volume":22,"doi":"10.1007/s00039-012-0168-5","publication_status":"published","year":"2012","publication":"Geometric and Functional Analysis","day":"25"},{"abstract":[{"lang":"eng","text":"We present an algorithm for computing [X, Y], i.e., all homotopy classes of continuous maps X → Y, where X, Y are topological spaces given as finite simplicial complexes, Y is (d - 1)-connected for some d ≥ 2 (for example, Y can be the d-dimensional sphere S d), and dim X ≤ 2d - 2. These conditions on X, Y guarantee that [X, Y] has a natural structure of a finitely generated Abelian group, and the algorithm finds generators and relations for it. We combine several tools and ideas from homotopy theory (such as Postnikov systems, simplicial sets, and obstruction theory) with algorithmic tools from effective algebraic topology (objects with effective homology). We hope that a further extension of the methods developed here will yield an algorithm for computing, in some cases of interest, the ℤ 2-index, which is a quantity playing a prominent role in Borsuk-Ulam style applications of topology in combinatorics and geometry, e.g., in topological lower bounds for the chromatic number of a graph. In a certain range of dimensions, deciding the embeddability of a simplicial complex into ℝ d also amounts to a ℤ 2-index computation. This is the main motivation of our work. We believe that investigating the computational complexity of questions in homotopy theory and similar areas presents a fascinating research area, and we hope that our work may help bridge the cultural gap between algebraic topology and theoretical computer science."}],"main_file_link":[{"url":"http://arxiv.org/abs/1105.6257","open_access":"0"}],"publisher":"SIAM","quality_controlled":0,"month":"01","publication_status":"published","year":"2012","day":"01","page":"1 - 10","date_created":"2018-12-11T11:57:40Z","date_published":"2012-01-01T00:00:00Z","_id":"2440","conference":{"name":"SODA: Symposium on Discrete Algorithms"},"type":"conference","status":"public","citation":{"mla":"Čadek, Martin, et al. Computing All Maps into a Sphere. SIAM, 2012, pp. 1–10.","short":"M. Čadek, M. Krcál, J. Matoušek, F. Sergeraert, L. Vokřínek, U. Wagner, in:, SIAM, 2012, pp. 1–10.","ieee":"M. Čadek, M. Krcál, J. Matoušek, F. Sergeraert, L. Vokřínek, and U. Wagner, “Computing all maps into a sphere,” presented at the SODA: Symposium on Discrete Algorithms, 2012, pp. 1–10.","ama":"Čadek M, Krcál M, Matoušek J, Sergeraert F, Vokřínek L, Wagner U. Computing all maps into a sphere. In: SIAM; 2012:1-10.","apa":"Čadek, M., Krcál, M., Matoušek, J., Sergeraert, F., Vokřínek, L., & Wagner, U. (2012). Computing all maps into a sphere (pp. 1–10). Presented at the SODA: Symposium on Discrete Algorithms, SIAM.","chicago":"Čadek, Martin, Marek Krcál, Jiří Matoušek, Francis Sergeraert, Lukáš Vokřínek, and Uli Wagner. “Computing All Maps into a Sphere,” 1–10. SIAM, 2012.","ista":"Čadek M, Krcál M, Matoušek J, Sergeraert F, Vokřínek L, Wagner U. 2012. Computing all maps into a sphere. SODA: Symposium on Discrete Algorithms, 1–10."},"date_updated":"2021-01-12T06:57:30Z","extern":1,"author":[{"first_name":"Martin","last_name":"Čadek","full_name":"Čadek, Martin"},{"id":"33E21118-F248-11E8-B48F-1D18A9856A87","first_name":"Marek","last_name":"Krcál","full_name":"Marek Krcál"},{"last_name":"Matoušek","full_name":"Matoušek, Jiří","first_name":"Jiří"},{"first_name":"Francis","full_name":"Sergeraert, Francis","last_name":"Sergeraert"},{"full_name":"Vokřínek, Lukáš","last_name":"Vokřínek","first_name":"Lukáš"},{"id":"36690CA2-F248-11E8-B48F-1D18A9856A87","first_name":"Uli","full_name":"Uli Wagner","orcid":"0000-0002-1494-0568","last_name":"Wagner"}],"publist_id":"4466","title":"Computing all maps into a sphere"},{"publication_status":"published","year":"2012","publication":"Discrete & Computational Geometry","day":"01","page":"245 - 265","date_created":"2018-12-11T11:57:39Z","volume":47,"doi":"10.1007/s00454-011-9368-2","issue":"2","date_published":"2012-03-01T00:00:00Z","abstract":[{"text":"The colored Tverberg theorem asserts that for eve;ry d and r there exists t=t(d,r) such that for every set C ⊂ ℝ d of cardinality (d + 1)t, partitioned into t-point subsets C 1, C 2,...,C d+1 (which we think of as color classes; e. g., the points of C 1 are red, the points of C 2 blue, etc.), there exist r disjoint sets R 1, R 2,...,R r⊆C that are rainbow, meaning that {pipe}R i∩C j{pipe}≤1 for every i,j, and whose convex hulls all have a common point. All known proofs of this theorem are topological. We present a geometric version of a recent beautiful proof by Blagojević, Matschke, and Ziegler, avoiding a direct use of topological methods. The purpose of this de-topologization is to make the proof more concrete and intuitive, and accessible to a wider audience.","lang":"eng"}],"acknowledgement":"We would like to thank Marek Krcál for useful discussions at initial stages of this research. We also thank Günter M. Ziegler for valuable comments, and Peter Landweber and two anonymous referees for detailed comments and corrections that greatly helped to improve the presentation. In particular, we are indebted to one of the referees for pointing out to us reference [19]. M. Tancer is supported by the grants SVV-2010-261313 (Discrete Methods and Algorithms) and GAUK 49209. U. Wagner’s research is supported by the Swiss National Science Foundation (SNF Projects 200021- 125309 and 200020-125027). ","quality_controlled":0,"publisher":"Springer","intvolume":" 47","month":"03","citation":{"mla":"Matoušek, Jiří, et al. “A Geometric Proof of the Colored Tverberg Theorem.” Discrete & Computational Geometry, vol. 47, no. 2, Springer, 2012, pp. 245–65, doi:10.1007/s00454-011-9368-2.","short":"J. Matoušek, M. Tancer, U. Wagner, Discrete & Computational Geometry 47 (2012) 245–265.","ieee":"J. Matoušek, M. Tancer, and U. Wagner, “A geometric proof of the colored Tverberg theorem,” Discrete & Computational Geometry, vol. 47, no. 2. Springer, pp. 245–265, 2012.","apa":"Matoušek, J., Tancer, M., & Wagner, U. (2012). A geometric proof of the colored Tverberg theorem. Discrete & Computational Geometry. Springer. https://doi.org/10.1007/s00454-011-9368-2","ama":"Matoušek J, Tancer M, Wagner U. A geometric proof of the colored Tverberg theorem. Discrete & Computational Geometry. 2012;47(2):245-265. doi:10.1007/s00454-011-9368-2","chicago":"Matoušek, Jiří, Martin Tancer, and Uli Wagner. “A Geometric Proof of the Colored Tverberg Theorem.” Discrete & Computational Geometry. Springer, 2012. https://doi.org/10.1007/s00454-011-9368-2.","ista":"Matoušek J, Tancer M, Wagner U. 2012. A geometric proof of the colored Tverberg theorem. Discrete & Computational Geometry. 47(2), 245–265."},"date_updated":"2021-01-12T06:57:29Z","extern":1,"publist_id":"4468","author":[{"last_name":"Matoušek","full_name":"Matoušek, Jiří","first_name":"Jiří"},{"first_name":"Martin","id":"38AC689C-F248-11E8-B48F-1D18A9856A87","full_name":"Martin Tancer","orcid":"0000-0002-1191-6714","last_name":"Tancer"},{"id":"36690CA2-F248-11E8-B48F-1D18A9856A87","first_name":"Uli","last_name":"Wagner","orcid":"0000-0002-1494-0568","full_name":"Uli Wagner"}],"title":"A geometric proof of the colored Tverberg theorem","_id":"2438","type":"journal_article","status":"public"},{"_id":"244","status":"public","type":"journal_article","extern":1,"citation":{"chicago":"Browning, Timothy D, and Alan Haynes. “Incomplete Kloosterman Sums and Multiplicative Inverses in Short Intervals.” International Journal of Number Theory. World Scientific Publishing, 2012. https://doi.org/ https://doi.org/10.1142/S1793042112501448.","ista":"Browning TD, Haynes A. 2012. Incomplete kloosterman sums and multiplicative inverses in short intervals. International Journal of Number Theory. 9(2), 481–486.","mla":"Browning, Timothy D., and Alan Haynes. “Incomplete Kloosterman Sums and Multiplicative Inverses in Short Intervals.” International Journal of Number Theory, vol. 9, no. 2, World Scientific Publishing, 2012, pp. 481–86, doi: https://doi.org/10.1142/S1793042112501448.","ama":"Browning TD, Haynes A. Incomplete kloosterman sums and multiplicative inverses in short intervals. International Journal of Number Theory. 2012;9(2):481-486. doi: https://doi.org/10.1142/S1793042112501448","apa":"Browning, T. D., & Haynes, A. (2012). Incomplete kloosterman sums and multiplicative inverses in short intervals. International Journal of Number Theory. World Scientific Publishing. https://doi.org/ https://doi.org/10.1142/S1793042112501448","short":"T.D. Browning, A. Haynes, International Journal of Number Theory 9 (2012) 481–486.","ieee":"T. D. Browning and A. Haynes, “Incomplete kloosterman sums and multiplicative inverses in short intervals,” International Journal of Number Theory, vol. 9, no. 2. World Scientific Publishing, pp. 481–486, 2012."},"date_updated":"2021-01-12T06:57:30Z","title":"Incomplete kloosterman sums and multiplicative inverses in short intervals","author":[{"id":"35827D50-F248-11E8-B48F-1D18A9856A87","first_name":"Timothy D","last_name":"Browning","orcid":"0000-0002-8314-0177","full_name":"Timothy Browning"},{"full_name":"Haynes, Alan K","last_name":"Haynes","first_name":"Alan"}],"publist_id":"7660","acknowledgement":"EP/E053262/1\tEngineering and Physical Sciences Research Council\tEPSRC,\nEP/J00149X/1\tEngineering and Physical Sciences Research Council\tEPSRC\t","abstract":[{"lang":"eng","text":"We investigate the solubility of the congruence xy ≡ 1 (mod p), where p is a prime and x, y are restricted to lie in suitable short intervals. Our work relies on a mean value theorem for incomplete Kloosterman sums."}],"intvolume":" 9","month":"11","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1204.6374"}],"oa":1,"publisher":"World Scientific Publishing","quality_controlled":0,"publication":"International Journal of Number Theory","day":"30","year":"2012","publication_status":"published","date_created":"2018-12-11T11:45:24Z","volume":9,"issue":"2","date_published":"2012-11-30T00:00:00Z","doi":" https://doi.org/10.1142/S1793042112501448","page":"481 - 486"},{"title":"Absolute approximation of Tukey depth: Theory and experiments","author":[{"full_name":"Chen, Dan","last_name":"Chen","first_name":"Dan"},{"full_name":"Morin, Pat","last_name":"Morin","first_name":"Pat"},{"id":"36690CA2-F248-11E8-B48F-1D18A9856A87","first_name":"Uli","last_name":"Wagner","orcid":"0000-0002-1494-0568","full_name":"Uli Wagner"}],"publist_id":"4467","extern":1,"date_updated":"2021-01-12T06:57:29Z","citation":{"chicago":"Chen, Dan, Pat Morin, and Uli Wagner. “Absolute Approximation of Tukey Depth: Theory and Experiments.” Computational Geometry: Theory and Applications. Elsevier, 2012. https://doi.org/10.1016/j.comgeo.2012.03.001.","ista":"Chen D, Morin P, Wagner U. 2012. Absolute approximation of Tukey depth: Theory and experiments. Computational Geometry: Theory and Applications. 46(5), 566–573.","mla":"Chen, Dan, et al. “Absolute Approximation of Tukey Depth: Theory and Experiments.” Computational Geometry: Theory and Applications, vol. 46, no. 5, Elsevier, 2012, pp. 566–73, doi:10.1016/j.comgeo.2012.03.001.","ama":"Chen D, Morin P, Wagner U. Absolute approximation of Tukey depth: Theory and experiments. Computational Geometry: Theory and Applications. 2012;46(5):566-573. doi:10.1016/j.comgeo.2012.03.001","apa":"Chen, D., Morin, P., & Wagner, U. (2012). Absolute approximation of Tukey depth: Theory and experiments. Computational Geometry: Theory and Applications. Elsevier. https://doi.org/10.1016/j.comgeo.2012.03.001","short":"D. Chen, P. Morin, U. Wagner, Computational Geometry: Theory and Applications 46 (2012) 566–573.","ieee":"D. Chen, P. Morin, and U. Wagner, “Absolute approximation of Tukey depth: Theory and experiments,” Computational Geometry: Theory and Applications, vol. 46, no. 5. Elsevier, pp. 566–573, 2012."},"status":"public","type":"journal_article","_id":"2439","volume":46,"issue":"5","doi":"10.1016/j.comgeo.2012.03.001","date_published":"2012-07-01T00:00:00Z","date_created":"2018-12-11T11:57:40Z","page":"566 - 573","day":"01","publication":"Computational Geometry: Theory and Applications","publication_status":"published","year":"2012","month":"07","intvolume":" 46","publisher":"Elsevier","quality_controlled":0,"abstract":[{"lang":"eng","text":"A Monte Carlo approximation algorithm for the Tukey depth problem in high dimensions is introduced. The algorithm is a generalization of an algorithm presented by Rousseeuw and Struyf (1998) . The performance of this algorithm is studied both analytically and experimentally."}]}]