[{"issue":"6","department":[{"_id":"JaMa"}],"status":"public","citation":{"short":"L. Dello Schiavo, J. Maas, F. Pedrotti, Transactions of the American Mathematical Society 377 (2024) 3779–3804.","mla":"Dello Schiavo, Lorenzo, et al. “Local Conditions for Global Convergence of Gradient Flows and Proximal Point Sequences in Metric Spaces.” <i>Transactions of the American Mathematical Society</i>, vol. 377, no. 6, American Mathematical Society, 2024, pp. 3779–804, doi:<a href=\"https://doi.org/10.1090/tran/9156\">10.1090/tran/9156</a>.","ista":"Dello Schiavo L, Maas J, Pedrotti F. 2024. Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces. Transactions of the American Mathematical Society. 377(6), 3779–3804.","chicago":"Dello Schiavo, Lorenzo, Jan Maas, and Francesco Pedrotti. “Local Conditions for Global Convergence of Gradient Flows and Proximal Point Sequences in Metric Spaces.” <i>Transactions of the American Mathematical Society</i>. American Mathematical Society, 2024. <a href=\"https://doi.org/10.1090/tran/9156\">https://doi.org/10.1090/tran/9156</a>.","ieee":"L. Dello Schiavo, J. Maas, and F. Pedrotti, “Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces,” <i>Transactions of the American Mathematical Society</i>, vol. 377, no. 6. American Mathematical Society, pp. 3779–3804, 2024.","apa":"Dello Schiavo, L., Maas, J., &#38; Pedrotti, F. (2024). Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces. <i>Transactions of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/tran/9156\">https://doi.org/10.1090/tran/9156</a>","ama":"Dello Schiavo L, Maas J, Pedrotti F. Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces. <i>Transactions of the American Mathematical Society</i>. 2024;377(6):3779-3804. doi:<a href=\"https://doi.org/10.1090/tran/9156\">10.1090/tran/9156</a>"},"page":"3779-3804","publication":"Transactions of the American Mathematical Society","isi":1,"scopus_import":"1","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","language":[{"iso":"eng"}],"quality_controlled":"1","title":"Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces","date_updated":"2026-04-07T13:00:02Z","acknowledgement":"The authors gratefully acknowledges support by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 716117). This research was funded in part by the Austrian Science Fund (FWF) project 10.55776/ESP208. This research was funded in part by the Austrian Science Fund (FWF) project 10.55776/F65","publication_identifier":{"issn":["0002-9947"],"eissn":["1088-6850"]},"article_processing_charge":"No","oa_version":"Preprint","external_id":{"arxiv":["2304.05239"],"isi":["001203273300001"]},"ec_funded":1,"day":"01","month":"06","arxiv":1,"date_published":"2024-06-01T00:00:00Z","volume":377,"year":"2024","oa":1,"related_material":{"record":[{"status":"public","id":"17336","relation":"dissertation_contains"}]},"abstract":[{"lang":"eng","text":"This paper deals with local criteria for the convergence to a global minimiser for gradient flow trajectories and their discretisations. To obtain quantitative estimates on the speed of convergence, we consider variations on the classical Kurdyka–Łojasiewicz inequality for a large class of parameter functions. Our assumptions are given in terms of the initial data, without any reference to an equilibrium point. The main results are convergence statements for gradient flow curves and proximal point sequences to a global minimiser, together with sharp quantitative estimates on the speed of convergence. These convergence results apply in the general setting of lower semicontinuous functionals on complete metric spaces, generalising recent results for smooth functionals on Rn. While the non-smooth setting covers very general spaces, it is also useful for (non)-smooth functionals on Rn.\r\n."}],"publication_status":"published","article_type":"original","_id":"17143","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2304.05239"}],"publisher":"American Mathematical Society","date_created":"2024-06-16T22:01:06Z","author":[{"orcid":"0000-0002-9881-6870","id":"ECEBF480-9E4F-11EA-B557-B0823DDC885E","full_name":"Dello Schiavo, Lorenzo","last_name":"Dello Schiavo","first_name":"Lorenzo"},{"first_name":"Jan","last_name":"Maas","full_name":"Maas, Jan","id":"4C5696CE-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0845-1338"},{"id":"d3ac8ac6-dc8d-11ea-abe3-e2a9628c4c3c","full_name":"Pedrotti, Francesco","first_name":"Francesco","last_name":"Pedrotti"}],"project":[{"grant_number":"716117","call_identifier":"H2020","name":"Optimal Transport and Stochastic Dynamics","_id":"256E75B8-B435-11E9-9278-68D0E5697425"},{"name":"Taming Complexity in Partial Differential Systems","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","grant_number":"F6504"},{"name":"Configuration Spaces over Non-Smooth Spaces","_id":"34dbf174-11ca-11ed-8bc3-afe9d43d4b9c","grant_number":"E208"}],"doi":"10.1090/tran/9156","type":"journal_article","intvolume":"       377"},{"corr_author":"1","issue":"6","department":[{"_id":"MaKw"}],"status":"public","citation":{"ieee":"M. A. Kwan, L. Sauermann, and Y. Zhao, “Extension complexity of low-dimensional polytopes,” <i>Transactions of the American Mathematical Society</i>, vol. 375, no. 6. American Mathematical Society, pp. 4209–4250, 2022.","apa":"Kwan, M. A., Sauermann, L., &#38; Zhao, Y. (2022). Extension complexity of low-dimensional polytopes. <i>Transactions of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/tran/8614\">https://doi.org/10.1090/tran/8614</a>","ama":"Kwan MA, Sauermann L, Zhao Y. Extension complexity of low-dimensional polytopes. <i>Transactions of the American Mathematical Society</i>. 2022;375(6):4209-4250. doi:<a href=\"https://doi.org/10.1090/tran/8614\">10.1090/tran/8614</a>","mla":"Kwan, Matthew Alan, et al. “Extension Complexity of Low-Dimensional Polytopes.” <i>Transactions of the American Mathematical Society</i>, vol. 375, no. 6, American Mathematical Society, 2022, pp. 4209–50, doi:<a href=\"https://doi.org/10.1090/tran/8614\">10.1090/tran/8614</a>.","ista":"Kwan MA, Sauermann L, Zhao Y. 2022. Extension complexity of low-dimensional polytopes. Transactions of the American Mathematical Society. 375(6), 4209–4250.","chicago":"Kwan, Matthew Alan, Lisa Sauermann, and Yufei Zhao. “Extension Complexity of Low-Dimensional Polytopes.” <i>Transactions of the American Mathematical Society</i>. American Mathematical Society, 2022. <a href=\"https://doi.org/10.1090/tran/8614\">https://doi.org/10.1090/tran/8614</a>.","short":"M.A. Kwan, L. Sauermann, Y. Zhao, Transactions of the American Mathematical Society 375 (2022) 4209–4250."},"page":"4209-4250","publication":"Transactions of the American Mathematical Society","isi":1,"scopus_import":"1","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","language":[{"iso":"eng"}],"quality_controlled":"1","title":"Extension complexity of low-dimensional polytopes","acknowledgement":"The research of the first author was supported by SNSF Project 178493 and NSF Award DMS-1953990. The research of the second author supported by NSF Award DMS-1953772.\r\nThe research of the third author was supported by NSF Award DMS-1764176, NSF CAREER Award DMS-2044606, a Sloan Research Fellowship, and the MIT Solomon Buchsbaum Fund. ","date_updated":"2024-10-09T21:02:31Z","publication_identifier":{"eissn":["1088-6850"],"issn":["0002-9947"]},"article_processing_charge":"No","oa_version":"Preprint","external_id":{"isi":["000798461500001"],"arxiv":["2006.08836"]},"day":"01","month":"06","arxiv":1,"date_published":"2022-06-01T00:00:00Z","volume":375,"oa":1,"year":"2022","abstract":[{"lang":"eng","text":"Sometimes, it is possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope P is defined to be the minimum number of facets of a (possibly higher-dimensional) polytope from which P can be obtained as a (linear) projection. This notion is motivated by its relevance to combinatorial optimisation, and has been studied intensively for various specific polytopes associated with important optimisation problems. In this paper we study extension complexity as a parameter of general polytopes, more specifically considering various families of low-dimensional polytopes. First, we prove that for a fixed dimension d, the extension complexity of a random d-dimensional polytope (obtained as the convex hull of random points in a ball or on a sphere) is typically on the order of the square root of its number of vertices. Second, we prove that any cyclic n-vertex polygon (whose vertices lie on a circle) has extension complexity at most 24√n. This bound is tight up to the constant factor 24. Finally, we show that there exists an no(1)-dimensional polytope with at most n vertices and extension complexity n1−o(1). Our theorems are proved with a range of different techniques, which we hope will be of further interest."}],"publication_status":"published","article_type":"original","_id":"11443","main_file_link":[{"open_access":"1","url":" https://doi.org/10.48550/arXiv.2006.08836"}],"publisher":"American Mathematical Society","author":[{"id":"5fca0887-a1db-11eb-95d1-ca9d5e0453b3","full_name":"Kwan, Matthew Alan","orcid":"0000-0002-4003-7567","first_name":"Matthew Alan","last_name":"Kwan"},{"first_name":"Lisa","last_name":"Sauermann","full_name":"Sauermann, Lisa"},{"full_name":"Zhao, Yufei","first_name":"Yufei","last_name":"Zhao"}],"date_created":"2022-06-12T22:01:45Z","doi":"10.1090/tran/8614","type":"journal_article","intvolume":"       375"},{"author":[{"first_name":"Yik Tung","last_name":"Chan","full_name":"Chan, Yik Tung","id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","orcid":"0000-0001-8467-4106"}],"date_created":"2025-04-05T10:50:56Z","doi":"10.1090/tran/8732","type":"journal_article","intvolume":"       375","_id":"19490","publisher":"American Mathematical Society","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2004.03331","open_access":"1"}],"arxiv":1,"date_published":"2022-09-01T00:00:00Z","volume":375,"oa":1,"year":"2022","abstract":[{"text":"Abstract. We study integral points on the quadratic twists ED : y2 = x3 −\r\nD2x of the congruent number curve. We give upper bounds on the number of\r\nintegral points in each coset of 2ED(Q) in ED(Q) and show that their total is\r\n (3.8)rank ED(Q). We further show that the average number of non-torsion\r\nintegral points in this family is bounded above by 2. As an application we also\r\ndeduce from our upper bounds that the system of simultaneous Pell equations\r\naX2 − bY 2 = d, bY 2 − cZ2 = d for pairwise coprime positive integers a, b, c, d,\r\nhas at most  (3.6)ω(abcd) integer solutions.","lang":"eng"}],"article_type":"original","publication_status":"published","oa_version":"Preprint","external_id":{"arxiv":["2004.03331"]},"day":"01","OA_place":"repository","month":"09","extern":"1","title":"Integral points on the congruent number curve","date_updated":"2025-07-10T11:51:47Z","publication_identifier":{"eissn":["1088-6850"],"issn":["0002-9947"]},"article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"quality_controlled":"1","OA_type":"green","status":"public","citation":{"ama":"Chan S. Integral points on the congruent number curve. <i>Transactions of the American Mathematical Society</i>. 2022;375(9):6675-6700. doi:<a href=\"https://doi.org/10.1090/tran/8732\">10.1090/tran/8732</a>","apa":"Chan, S. (2022). Integral points on the congruent number curve. <i>Transactions of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/tran/8732\">https://doi.org/10.1090/tran/8732</a>","ieee":"S. Chan, “Integral points on the congruent number curve,” <i>Transactions of the American Mathematical Society</i>, vol. 375, no. 9. American Mathematical Society, pp. 6675–6700, 2022.","short":"S. Chan, Transactions of the American Mathematical Society 375 (2022) 6675–6700.","chicago":"Chan, Stephanie. “Integral Points on the Congruent Number Curve.” <i>Transactions of the American Mathematical Society</i>. American Mathematical Society, 2022. <a href=\"https://doi.org/10.1090/tran/8732\">https://doi.org/10.1090/tran/8732</a>.","mla":"Chan, Stephanie. “Integral Points on the Congruent Number Curve.” <i>Transactions of the American Mathematical Society</i>, vol. 375, no. 9, American Mathematical Society, 2022, pp. 6675–700, doi:<a href=\"https://doi.org/10.1090/tran/8732\">10.1090/tran/8732</a>.","ista":"Chan S. 2022. Integral points on the congruent number curve. Transactions of the American Mathematical Society. 375(9), 6675–6700."},"page":"6675-6700","publication":"Transactions of the American Mathematical Society","scopus_import":"1","issue":"9"},{"author":[{"first_name":"Gyorgy Pal","last_name":"Geher","full_name":"Geher, Gyorgy Pal"},{"full_name":"Titkos, Tamas","first_name":"Tamas","last_name":"Titkos"},{"first_name":"Daniel","last_name":"Virosztek","full_name":"Virosztek, Daniel","id":"48DB45DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-1109-5511"}],"date_created":"2020-01-29T10:20:46Z","doi":"10.1090/tran/8113","project":[{"grant_number":"846294","_id":"26A455A6-B435-11E9-9278-68D0E5697425","name":"Geometric study of Wasserstein spaces and free probability","call_identifier":"H2020"}],"type":"journal_article","intvolume":"       373","_id":"7389","publisher":"American Mathematical Society","main_file_link":[{"url":"https://arxiv.org/abs/2002.00859","open_access":"1"}],"arxiv":1,"year":"2020","oa":1,"volume":373,"date_published":"2020-08-01T00:00:00Z","publication_status":"published","article_type":"original","abstract":[{"text":"Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space W_2(R^n). It turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow. Following this line of investigation, we compute Isom(W_p(R)), the isometry group of the Wasserstein space\r\nW_p(R) for all p \\in [1,\\infty) \\setminus {2}. We show that W_2(R) is also exceptional regarding the\r\nparameter p: W_p(R) is isometrically rigid if and only if p is not equal to 2. Regarding the underlying\r\nspace, we prove that the exceptionality of p = 2 disappears if we replace R by the compact\r\ninterval [0,1]. Surprisingly, in that case, W_p([0,1]) is isometrically rigid if and only if\r\np is not equal to 1. Moreover, W_1([0,1]) admits isometries that split mass, and Isom(W_1([0,1]))\r\ncannot be embedded into Isom(W_1(R)).","lang":"eng"}],"ec_funded":1,"external_id":{"arxiv":["2002.00859"],"isi":["000551418100018"]},"oa_version":"Preprint","day":"01","month":"08","title":"Isometric study of Wasserstein spaces - the real line","publication_identifier":{"issn":["0002-9947"],"eissn":["1088-6850"]},"date_updated":"2025-07-10T11:54:32Z","article_processing_charge":"No","language":[{"iso":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","keyword":["Wasserstein space","isometric embeddings","isometric rigidity","exotic isometry flow"],"status":"public","publication":"Transactions of the American Mathematical Society","page":"5855-5883","citation":{"apa":"Geher, G. P., Titkos, T., &#38; Virosztek, D. (2020). Isometric study of Wasserstein spaces - the real line. <i>Transactions of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/tran/8113\">https://doi.org/10.1090/tran/8113</a>","ama":"Geher GP, Titkos T, Virosztek D. Isometric study of Wasserstein spaces - the real line. <i>Transactions of the American Mathematical Society</i>. 2020;373(8):5855-5883. doi:<a href=\"https://doi.org/10.1090/tran/8113\">10.1090/tran/8113</a>","ieee":"G. P. Geher, T. Titkos, and D. Virosztek, “Isometric study of Wasserstein spaces - the real line,” <i>Transactions of the American Mathematical Society</i>, vol. 373, no. 8. American Mathematical Society, pp. 5855–5883, 2020.","chicago":"Geher, Gyorgy Pal, Tamas Titkos, and Daniel Virosztek. “Isometric Study of Wasserstein Spaces - the Real Line.” <i>Transactions of the American Mathematical Society</i>. American Mathematical Society, 2020. <a href=\"https://doi.org/10.1090/tran/8113\">https://doi.org/10.1090/tran/8113</a>.","mla":"Geher, Gyorgy Pal, et al. “Isometric Study of Wasserstein Spaces - the Real Line.” <i>Transactions of the American Mathematical Society</i>, vol. 373, no. 8, American Mathematical Society, 2020, pp. 5855–83, doi:<a href=\"https://doi.org/10.1090/tran/8113\">10.1090/tran/8113</a>.","ista":"Geher GP, Titkos T, Virosztek D. 2020. Isometric study of Wasserstein spaces - the real line. Transactions of the American Mathematical Society. 373(8), 5855–5883.","short":"G.P. Geher, T. Titkos, D. Virosztek, Transactions of the American Mathematical Society 373 (2020) 5855–5883."},"scopus_import":"1","isi":1,"issue":"8","ddc":["515"],"department":[{"_id":"LaEr"}]},{"doi":"10.1090/tran/7729","date_created":"2021-06-22T09:31:45Z","author":[{"full_name":"Kwan, Matthew Alan","id":"5fca0887-a1db-11eb-95d1-ca9d5e0453b3","orcid":"0000-0002-4003-7567","first_name":"Matthew Alan","last_name":"Kwan"},{"last_name":"Sudakov","first_name":"Benny","full_name":"Sudakov, Benny"}],"type":"journal_article","intvolume":"       372","main_file_link":[{"open_access":"1","url":"https://doi.org/10.1090/tran/7729"}],"publisher":"American Mathematical Society","_id":"9585","volume":372,"date_published":"2019-10-15T00:00:00Z","oa":1,"year":"2019","arxiv":1,"abstract":[{"lang":"eng","text":"An n-vertex graph is called C-Ramsey if it has no clique or independent set of size C log n. All known constructions of Ramsey graphs involve randomness in an essential way, and there is an ongoing line of research towards showing that in fact all Ramsey graphs must obey certain “richness” properties characteristic of random graphs. More than 25 years ago, Erdős, Faudree and Sós conjectured that in any C-Ramsey graph there are Ω(n^5/2) induced subgraphs, no pair of which have the same numbers of vertices and edges. Improving on earlier results of Alon, Balogh, Kostochka and Samotij, in this paper we prove this conjecture."}],"article_type":"original","publication_status":"published","day":"15","oa_version":"Submitted Version","external_id":{"arxiv":["1712.05656"]},"month":"10","extern":"1","date_updated":"2023-02-23T14:01:50Z","publication_identifier":{"issn":["0002-9947"],"eissn":["1088-6850"]},"title":"Proof of a conjecture on induced subgraphs of Ramsey graphs","article_processing_charge":"No","quality_controlled":"1","language":[{"iso":"eng"}],"user_id":"6785fbc1-c503-11eb-8a32-93094b40e1cf","status":"public","scopus_import":"1","citation":{"ieee":"M. A. Kwan and B. Sudakov, “Proof of a conjecture on induced subgraphs of Ramsey graphs,” <i>Transactions of the American Mathematical Society</i>, vol. 372, no. 8. American Mathematical Society, pp. 5571–5594, 2019.","apa":"Kwan, M. A., &#38; Sudakov, B. (2019). Proof of a conjecture on induced subgraphs of Ramsey graphs. <i>Transactions of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/tran/7729\">https://doi.org/10.1090/tran/7729</a>","ama":"Kwan MA, Sudakov B. Proof of a conjecture on induced subgraphs of Ramsey graphs. <i>Transactions of the American Mathematical Society</i>. 2019;372(8):5571-5594. doi:<a href=\"https://doi.org/10.1090/tran/7729\">10.1090/tran/7729</a>","ista":"Kwan MA, Sudakov B. 2019. Proof of a conjecture on induced subgraphs of Ramsey graphs. Transactions of the American Mathematical Society. 372(8), 5571–5594.","mla":"Kwan, Matthew Alan, and Benny Sudakov. “Proof of a Conjecture on Induced Subgraphs of Ramsey Graphs.” <i>Transactions of the American Mathematical Society</i>, vol. 372, no. 8, American Mathematical Society, 2019, pp. 5571–94, doi:<a href=\"https://doi.org/10.1090/tran/7729\">10.1090/tran/7729</a>.","chicago":"Kwan, Matthew Alan, and Benny Sudakov. “Proof of a Conjecture on Induced Subgraphs of Ramsey Graphs.” <i>Transactions of the American Mathematical Society</i>. American Mathematical Society, 2019. <a href=\"https://doi.org/10.1090/tran/7729\">https://doi.org/10.1090/tran/7729</a>.","short":"M.A. Kwan, B. Sudakov, Transactions of the American Mathematical Society 372 (2019) 5571–5594."},"page":"5571-5594","publication":"Transactions of the American Mathematical Society","issue":"8"},{"month":"04","researchdata_availability":"no","day":"15","external_id":{"isi":["000464034200019"],"arxiv":["1705.01999"]},"oa_version":"Preprint","publication_status":"published","abstract":[{"text":"An upper bound sieve for rational points on suitable varieties isdeveloped, together with applications tocounting rational points in thin sets,to local solubility in families, and to the notion of “friable” rational pointswith respect to divisors. In the special case of quadrics, sharper estimates areobtained by developing a version of the Selberg sieve for rational points.","lang":"eng"}],"publist_id":"7746","year":"2019","oa":1,"volume":371,"date_published":"2019-04-15T00:00:00Z","arxiv":1,"main_file_link":[{"url":"https://arxiv.org/abs/1705.01999","open_access":"1"}],"publisher":"American Mathematical Society","_id":"175","type":"journal_article","intvolume":"       371","doi":"10.1090/tran/7514","date_created":"2018-12-11T11:45:01Z","author":[{"first_name":"Timothy D","last_name":"Browning","id":"35827D50-F248-11E8-B48F-1D18A9856A87","full_name":"Browning, Timothy D","orcid":"0000-0002-8314-0177"},{"full_name":"Loughran, Daniel","last_name":"Loughran","first_name":"Daniel"}],"department":[{"_id":"TiBr"}],"issue":"8","scopus_import":"1","isi":1,"publication":"Transactions of the American Mathematical Society","page":"5757-5785","citation":{"ama":"Browning TD, Loughran D. Sieving rational points on varieties. <i>Transactions of the American Mathematical Society</i>. 2019;371(8):5757-5785. doi:<a href=\"https://doi.org/10.1090/tran/7514\">10.1090/tran/7514</a>","apa":"Browning, T. D., &#38; Loughran, D. (2019). Sieving rational points on varieties. <i>Transactions of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/tran/7514\">https://doi.org/10.1090/tran/7514</a>","ieee":"T. D. Browning and D. Loughran, “Sieving rational points on varieties,” <i>Transactions of the American Mathematical Society</i>, vol. 371, no. 8. American Mathematical Society, pp. 5757–5785, 2019.","chicago":"Browning, Timothy D, and Daniel Loughran. “Sieving Rational Points on Varieties.” <i>Transactions of the American Mathematical Society</i>. American Mathematical Society, 2019. <a href=\"https://doi.org/10.1090/tran/7514\">https://doi.org/10.1090/tran/7514</a>.","mla":"Browning, Timothy D., and Daniel Loughran. “Sieving Rational Points on Varieties.” <i>Transactions of the American Mathematical Society</i>, vol. 371, no. 8, American Mathematical Society, 2019, pp. 5757–85, doi:<a href=\"https://doi.org/10.1090/tran/7514\">10.1090/tran/7514</a>.","ista":"Browning TD, Loughran D. 2019. Sieving rational points on varieties. Transactions of the American Mathematical Society. 371(8), 5757–5785.","short":"T.D. Browning, D. Loughran, Transactions of the American Mathematical Society 371 (2019) 5757–5785."},"status":"public","das_tickbox":"0","quality_controlled":"1","supplementarymaterial":"no","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","language":[{"iso":"eng"}],"article_processing_charge":"No","publication_identifier":{"issn":["0002-9947"],"eissn":["1088-6850"]},"date_updated":"2026-08-06T12:12:46Z","title":"Sieving rational points on varieties"},{"extern":"1","month":"03","OA_place":"publisher","day":"01","external_id":{"arxiv":["1008.2712"]},"oa_version":"Published Version","article_type":"original","publication_status":"published","abstract":[{"text":"We consider both the defocusing and focusing cubic nonlinear Klein–Gordon equations\r\nutt − Δu + u ± u3 =0\r\nin two space dimensions for real-valued initial data u(0) ∈ H1/x and ut(0) ∈ L2/x.\r\nWe show that in the defocusing case, solutions are global and have finite global L4/t,x spacetime bounds. In the focusing case, we characterize the dichotomy\r\nbetween this behaviour and blowup for initial data with energy less than that\r\nof the ground state.\r\nThese results rely on analogous statements for the two-dimensional cubic\r\nnonlinear Schr¨odinger equation, which are known in the defocusing case and\r\nfor spherically-symmetric initial data in the focusing case. Thus, our results\r\nare mostly unconditional.\r\nIt was previously shown by Nakanishi that spacetime bounds for Klein–\r\nGordon equations imply the same for nonlinear Schr¨odinger equations.","lang":"eng"}],"year":"2012","oa":1,"date_published":"2012-03-01T00:00:00Z","volume":364,"arxiv":1,"publisher":"American Mathematical Society","main_file_link":[{"url":"https://doi.org/10.1090/S0002-9947-2011-05536-4","open_access":"1"}],"_id":"22022","intvolume":"       364","type":"journal_article","doi":"10.1090/s0002-9947-2011-05536-4","date_created":"2026-06-19T07:31:10Z","author":[{"last_name":"Killip","first_name":"Rowan","full_name":"Killip, Rowan"},{"last_name":"Stovall","first_name":"Betsy","full_name":"Stovall, Betsy"},{"last_name":"Visan","first_name":"Monica","full_name":"Visan, Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca"}],"issue":"3","scopus_import":"1","publication":"Transactions of the American Mathematical Society","page":"1571-1631","citation":{"ista":"Killip R, Stovall B, Vişan M. 2012. Scattering for the cubic Klein-Gordon equation in two space dimensions. Transactions of the American Mathematical Society. 364(3), 1571–1631.","mla":"Killip, Rowan, et al. “Scattering for the Cubic Klein-Gordon Equation in Two Space Dimensions.” <i>Transactions of the American Mathematical Society</i>, vol. 364, no. 3, American Mathematical Society, 2012, pp. 1571–631, doi:<a href=\"https://doi.org/10.1090/s0002-9947-2011-05536-4\">10.1090/s0002-9947-2011-05536-4</a>.","chicago":"Killip, Rowan, Betsy Stovall, and Monica Vişan. “Scattering for the Cubic Klein-Gordon Equation in Two Space Dimensions.” <i>Transactions of the American Mathematical Society</i>. American Mathematical Society, 2012. <a href=\"https://doi.org/10.1090/s0002-9947-2011-05536-4\">https://doi.org/10.1090/s0002-9947-2011-05536-4</a>.","short":"R. Killip, B. Stovall, M. Vişan, Transactions of the American Mathematical Society 364 (2012) 1571–1631.","ieee":"R. Killip, B. Stovall, and M. Vişan, “Scattering for the cubic Klein-Gordon equation in two space dimensions,” <i>Transactions of the American Mathematical Society</i>, vol. 364, no. 3. American Mathematical Society, pp. 1571–1631, 2012.","apa":"Killip, R., Stovall, B., &#38; Vişan, M. (2012). Scattering for the cubic Klein-Gordon equation in two space dimensions. <i>Transactions of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/s0002-9947-2011-05536-4\">https://doi.org/10.1090/s0002-9947-2011-05536-4</a>","ama":"Killip R, Stovall B, Vişan M. Scattering for the cubic Klein-Gordon equation in two space dimensions. <i>Transactions of the American Mathematical Society</i>. 2012;364(3):1571-1631. doi:<a href=\"https://doi.org/10.1090/s0002-9947-2011-05536-4\">10.1090/s0002-9947-2011-05536-4</a>"},"status":"public","das_tickbox":"1","OA_type":"free access","quality_controlled":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"article_processing_charge":"No","publication_identifier":{"eissn":["1088-6850"],"issn":["0002-9947"]},"date_updated":"2026-06-22T09:19:43Z","title":"Scattering for the cubic Klein-Gordon equation in two space dimensions"},{"OA_type":"green","quality_controlled":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"article_processing_charge":"No","date_updated":"2026-06-22T13:18:01Z","publication_identifier":{"issn":["0002-9947"],"eissn":["1088-6850"]},"title":"The defocusing energy-supercritical nonlinear wave equation in three space dimensions","issue":"7","scopus_import":"1","citation":{"ieee":"R. Killip and M. Vişan, “The defocusing energy-supercritical nonlinear wave equation in three space dimensions,” <i>Transactions of the American Mathematical Society</i>, vol. 363, no. 7. American Mathematical Society, pp. 3893–3893, 2011.","apa":"Killip, R., &#38; Vişan, M. (2011). The defocusing energy-supercritical nonlinear wave equation in three space dimensions. <i>Transactions of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/s0002-9947-2011-05400-0\">https://doi.org/10.1090/s0002-9947-2011-05400-0</a>","ama":"Killip R, Vişan M. The defocusing energy-supercritical nonlinear wave equation in three space dimensions. <i>Transactions of the American Mathematical Society</i>. 2011;363(7):3893-3893. doi:<a href=\"https://doi.org/10.1090/s0002-9947-2011-05400-0\">10.1090/s0002-9947-2011-05400-0</a>","mla":"Killip, Rowan, and Monica Vişan. “The Defocusing Energy-Supercritical Nonlinear Wave Equation in Three Space Dimensions.” <i>Transactions of the American Mathematical Society</i>, vol. 363, no. 7, American Mathematical Society, 2011, pp. 3893–3893, doi:<a href=\"https://doi.org/10.1090/s0002-9947-2011-05400-0\">10.1090/s0002-9947-2011-05400-0</a>.","ista":"Killip R, Vişan M. 2011. The defocusing energy-supercritical nonlinear wave equation in three space dimensions. Transactions of the American Mathematical Society. 363(7), 3893–3893.","chicago":"Killip, Rowan, and Monica Vişan. “The Defocusing Energy-Supercritical Nonlinear Wave Equation in Three Space Dimensions.” <i>Transactions of the American Mathematical Society</i>. American Mathematical Society, 2011. <a href=\"https://doi.org/10.1090/s0002-9947-2011-05400-0\">https://doi.org/10.1090/s0002-9947-2011-05400-0</a>.","short":"R. Killip, M. Vişan, Transactions of the American Mathematical Society 363 (2011) 3893–3893."},"page":"3893-3893","publication":"Transactions of the American Mathematical Society","status":"public","das_tickbox":"1","publisher":"American Mathematical Society","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1001.1761","open_access":"1"}],"_id":"22040","type":"journal_article","intvolume":"       363","doi":"10.1090/s0002-9947-2011-05400-0","author":[{"full_name":"Killip, Rowan","last_name":"Killip","first_name":"Rowan"},{"full_name":"Visan, Monica","id":"056daca0-b8d1-11f0-964f-f91054abf8ca","last_name":"Visan","first_name":"Monica"}],"date_created":"2026-06-19T07:45:31Z","month":"01","extern":"1","day":"28","OA_place":"repository","oa_version":"Preprint","external_id":{"arxiv":["1001.1761"]},"abstract":[{"text":"We consider the defocusing nonlinear wave equation (mathematical formular) in the energy-supercritical regime p > 4. For even values of the power p, we show that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that solutions with bounded critical Sobolev norm are global and scatter. The impetus to consider this problem comes from recent work of Kenig and Merle who treated the case of spherically-symmetric solutions.","lang":"eng"}],"article_type":"original","publication_status":"published","volume":363,"date_published":"2011-01-28T00:00:00Z","oa":1,"year":"2011","arxiv":1}]