[{"date_updated":"2023-08-03T11:56:30Z","ddc":["510"],"file_date_updated":"2023-02-02T07:32:48Z","department":[{"_id":"HeEd"}],"_id":"11545","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"article_type":"original","type":"journal_article","keyword":["Algebra and Number Theory"],"status":"public","publication_status":"published","publication_identifier":{"issn":["0021-8693"]},"language":[{"iso":"eng"}],"file":[{"file_id":"12473","checksum":"82abaee3d7837f703e499a9ecbb25b7c","success":1,"content_type":"application/pdf","access_level":"open_access","relation":"main_file","date_created":"2023-02-02T07:32:48Z","file_name":"2022_JournalAlgebra_Brown.pdf","date_updated":"2023-02-02T07:32:48Z","file_size":582962,"creator":"dernst"}],"ec_funded":1,"issue":"11","volume":609,"abstract":[{"text":"We classify contravariant pairings between standard Whittaker modules and Verma modules over a complex semisimple Lie algebra. These contravariant pairings are useful in extending several classical techniques for category O to the Miličić–Soergel category N . We introduce a class of costandard modules which generalize dual Verma modules, and describe canonical maps from standard to costandard modules in terms of contravariant pairings.\r\nWe show that costandard modules have unique irreducible submodules and share the same composition factors as the corresponding standard Whittaker modules. We show that costandard modules give an algebraic characterization of the global sections of costandard twisted Harish-Chandra sheaves on the associated flag variety, which are defined using holonomic duality of D-modules. We prove that with these costandard modules, blocks of category\r\nN have the structure of highest weight categories and we establish a BGG reciprocity theorem for N .","lang":"eng"}],"oa_version":"Published Version","scopus_import":"1","intvolume":" 609","month":"11","citation":{"ista":"Brown A, Romanov A. 2022. Contravariant pairings between standard Whittaker modules and Verma modules. Journal of Algebra. 609(11), 145–179.","chicago":"Brown, Adam, and Anna Romanov. “Contravariant Pairings between Standard Whittaker Modules and Verma Modules.” Journal of Algebra. Elsevier, 2022. https://doi.org/10.1016/j.jalgebra.2022.06.017.","ama":"Brown A, Romanov A. Contravariant pairings between standard Whittaker modules and Verma modules. Journal of Algebra. 2022;609(11):145-179. doi:10.1016/j.jalgebra.2022.06.017","apa":"Brown, A., & Romanov, A. (2022). Contravariant pairings between standard Whittaker modules and Verma modules. Journal of Algebra. Elsevier. https://doi.org/10.1016/j.jalgebra.2022.06.017","short":"A. Brown, A. Romanov, Journal of Algebra 609 (2022) 145–179.","ieee":"A. Brown and A. Romanov, “Contravariant pairings between standard Whittaker modules and Verma modules,” Journal of Algebra, vol. 609, no. 11. Elsevier, pp. 145–179, 2022.","mla":"Brown, Adam, and Anna Romanov. “Contravariant Pairings between Standard Whittaker Modules and Verma Modules.” Journal of Algebra, vol. 609, no. 11, Elsevier, 2022, pp. 145–79, doi:10.1016/j.jalgebra.2022.06.017."},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","article_processing_charge":"Yes (via OA deal)","external_id":{"isi":["000861841100004"]},"author":[{"id":"70B7FDF6-608D-11E9-9333-8535E6697425","first_name":"Adam","full_name":"Brown, Adam","last_name":"Brown"},{"first_name":"Anna","full_name":"Romanov, Anna","last_name":"Romanov"}],"title":"Contravariant pairings between standard Whittaker modules and Verma modules","project":[{"_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411"}],"year":"2022","has_accepted_license":"1","isi":1,"publication":"Journal of Algebra","day":"01","page":"145-179","date_created":"2022-07-08T11:40:07Z","doi":"10.1016/j.jalgebra.2022.06.017","date_published":"2022-11-01T00:00:00Z","acknowledgement":"We thank Catharina Stroppel and Jens Niklas Eberhardt for interesting discussions. The first author acknowledges the support of the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411. The second author is supported by the National Science Foundation Award No. 1803059 and the Australian Research Council grant DP170101579.","oa":1,"publisher":"Elsevier","quality_controlled":"1"},{"acknowledgement":"We would like to thank Peter Trapa for useful discussions, and Dragan Milicic and Arun Ram for valuable feedback on the structure of the paper. The first author acknowledges the support of the European Unions Horizon 2020 research and innovation programme under the Marie Skodowska-Curie Grant Agreement No. 754411. The second author is\r\nsupported by the National Science Foundation Award No. 1803059.","oa":1,"quality_controlled":"1","publisher":"American Mathematical Society","year":"2021","isi":1,"publication":"Proceedings of the American Mathematical Society","day":"01","page":"37-52","date_created":"2020-11-19T10:17:40Z","date_published":"2021-01-01T00:00:00Z","doi":"10.1090/proc/15205","project":[{"grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"}],"citation":{"mla":"Brown, Adam, and Anna Romanov. “Contravariant Forms on Whittaker Modules.” Proceedings of the American Mathematical Society, vol. 149, no. 1, American Mathematical Society, 2021, pp. 37–52, doi:10.1090/proc/15205.","ama":"Brown A, Romanov A. Contravariant forms on Whittaker modules. Proceedings of the American Mathematical Society. 2021;149(1):37-52. doi:10.1090/proc/15205","apa":"Brown, A., & Romanov, A. (2021). Contravariant forms on Whittaker modules. Proceedings of the American Mathematical Society. American Mathematical Society. https://doi.org/10.1090/proc/15205","ieee":"A. Brown and A. Romanov, “Contravariant forms on Whittaker modules,” Proceedings of the American Mathematical Society, vol. 149, no. 1. American Mathematical Society, pp. 37–52, 2021.","short":"A. Brown, A. Romanov, Proceedings of the American Mathematical Society 149 (2021) 37–52.","chicago":"Brown, Adam, and Anna Romanov. “Contravariant Forms on Whittaker Modules.” Proceedings of the American Mathematical Society. American Mathematical Society, 2021. https://doi.org/10.1090/proc/15205.","ista":"Brown A, Romanov A. 2021. Contravariant forms on Whittaker modules. Proceedings of the American Mathematical Society. 149(1), 37–52."},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","article_processing_charge":"No","external_id":{"isi":["000600416300004"],"arxiv":["1910.08286"]},"author":[{"last_name":"Brown","full_name":"Brown, Adam","id":"70B7FDF6-608D-11E9-9333-8535E6697425","first_name":"Adam"},{"first_name":"Anna","last_name":"Romanov","full_name":"Romanov, Anna"}],"title":"Contravariant forms on Whittaker modules","abstract":[{"text":"Let g be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker g-modules Y(χ,η) introduced by Kostant. We prove that the set of all contravariant forms on Y(χ,η) forms a vector space whose dimension is given by the cardinality of the Weyl group of g. We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules M(χ,η) introduced by McDowell.","lang":"eng"}],"oa_version":"Preprint","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1910.08286"}],"intvolume":" 149","month":"01","publication_status":"published","publication_identifier":{"eissn":["1088-6826"],"issn":["0002-9939"]},"language":[{"iso":"eng"}],"ec_funded":1,"volume":149,"issue":"1","_id":"8773","article_type":"original","type":"journal_article","keyword":["Applied Mathematics","General Mathematics"],"status":"public","date_updated":"2023-08-04T11:11:47Z","department":[{"_id":"HeEd"}]},{"title":"Probabilistic convergence and stability of random mapper graphs","author":[{"first_name":"Adam","id":"70B7FDF6-608D-11E9-9333-8535E6697425","last_name":"Brown","full_name":"Brown, Adam"},{"last_name":"Bobrowski","full_name":"Bobrowski, Omer","first_name":"Omer"},{"full_name":"Munch, Elizabeth","last_name":"Munch","first_name":"Elizabeth"},{"first_name":"Bei","full_name":"Wang, Bei","last_name":"Wang"}],"external_id":{"arxiv":["1909.03488"]},"article_processing_charge":"Yes (via OA deal)","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","citation":{"apa":"Brown, A., Bobrowski, O., Munch, E., & Wang, B. (2021). Probabilistic convergence and stability of random mapper graphs. Journal of Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-020-00063-x","ama":"Brown A, Bobrowski O, Munch E, Wang B. Probabilistic convergence and stability of random mapper graphs. Journal of Applied and Computational Topology. 2021;5(1):99-140. doi:10.1007/s41468-020-00063-x","short":"A. Brown, O. Bobrowski, E. Munch, B. Wang, Journal of Applied and Computational Topology 5 (2021) 99–140.","ieee":"A. Brown, O. Bobrowski, E. Munch, and B. Wang, “Probabilistic convergence and stability of random mapper graphs,” Journal of Applied and Computational Topology, vol. 5, no. 1. Springer Nature, pp. 99–140, 2021.","mla":"Brown, Adam, et al. “Probabilistic Convergence and Stability of Random Mapper Graphs.” Journal of Applied and Computational Topology, vol. 5, no. 1, Springer Nature, 2021, pp. 99–140, doi:10.1007/s41468-020-00063-x.","ista":"Brown A, Bobrowski O, Munch E, Wang B. 2021. Probabilistic convergence and stability of random mapper graphs. Journal of Applied and Computational Topology. 5(1), 99–140.","chicago":"Brown, Adam, Omer Bobrowski, Elizabeth Munch, and Bei Wang. “Probabilistic Convergence and Stability of Random Mapper Graphs.” Journal of Applied and Computational Topology. Springer Nature, 2021. https://doi.org/10.1007/s41468-020-00063-x."},"project":[{"name":"ISTplus - Postdoctoral Fellowships","grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"}],"doi":"10.1007/s41468-020-00063-x","date_published":"2021-03-01T00:00:00Z","date_created":"2021-02-11T14:41:02Z","page":"99-140","day":"01","publication":"Journal of Applied and Computational Topology","has_accepted_license":"1","year":"2021","quality_controlled":"1","publisher":"Springer Nature","oa":1,"acknowledgement":"AB was supported in part by the European Union’s Horizon 2020 research and innovation\r\nprogramme under the Marie Sklodowska-Curie GrantAgreement No. 754411 and NSF IIS-1513616. OB was supported in part by the Israel Science Foundation, Grant 1965/19. BW was supported in part by NSF IIS-1513616 and DBI-1661375. EM was supported in part by NSF CMMI-1800466, DMS-1800446, and CCF-1907591.We would like to thank the Institute for Mathematics and its Applications for hosting a workshop titled Bridging Statistics and Sheaves in May 2018, where this work was conceived.\r\nOpen Access funding provided by Institute of Science and Technology (IST Austria).","department":[{"_id":"HeEd"}],"file_date_updated":"2021-02-11T14:43:59Z","ddc":["510"],"date_updated":"2023-09-05T15:37:56Z","status":"public","article_type":"original","type":"journal_article","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"_id":"9111","issue":"1","volume":5,"ec_funded":1,"file":[{"date_updated":"2021-02-11T14:43:59Z","file_size":2090265,"creator":"dernst","date_created":"2021-02-11T14:43:59Z","file_name":"2020_JourApplCompTopology_Brown.pdf","content_type":"application/pdf","access_level":"open_access","relation":"main_file","checksum":"3f02e9d47c428484733da0f588a3c069","file_id":"9112","success":1}],"language":[{"iso":"eng"}],"publication_identifier":{"issn":["2367-1726"],"eissn":["2367-1734"]},"publication_status":"published","month":"03","intvolume":" 5","scopus_import":"1","oa_version":"Published Version","abstract":[{"lang":"eng","text":"We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space X equipped with a continuous function f:X→R. We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line R. We then introduce a variant of the classic mapper graph of Singh et al. (in: Eurographics symposium on point-based graphics, 2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of (X,f) when it is applied to points randomly sampled from a probability density function concentrated on (X,f). Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (In: 32nd international symposium on computational geometry, volume 51 of Leibniz international proceedings in informatics (LIPIcs), Dagstuhl, Germany, pp 53:1–53:16, 2016), we first show that the mapper graph of (X,f), a constructible R-space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of (X,f) to the mapper of a super-level set of a probability density function concentrated on (X,f). Finally, building on the approach of Bobrowski et al. (Bernoulli 23(1):288–328, 2017b), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data."}]},{"project":[{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"}],"external_id":{"isi":["000536324700001"],"arxiv":["1712.07734"]},"article_processing_charge":"Yes (via OA deal)","author":[{"first_name":"Adam","id":"70B7FDF6-608D-11E9-9333-8535E6697425","last_name":"Brown","full_name":"Brown, Adam"},{"first_name":"Bei","last_name":"Wang","full_name":"Wang, Bei"}],"title":"Sheaf-theoretic stratification learning from geometric and topological perspectives","citation":{"mla":"Brown, Adam, and Bei Wang. “Sheaf-Theoretic Stratification Learning from Geometric and Topological Perspectives.” Discrete and Computational Geometry, vol. 65, Springer Nature, 2021, pp. 1166–98, doi:10.1007/s00454-020-00206-y.","ama":"Brown A, Wang B. Sheaf-theoretic stratification learning from geometric and topological perspectives. Discrete and Computational Geometry. 2021;65:1166-1198. doi:10.1007/s00454-020-00206-y","apa":"Brown, A., & Wang, B. (2021). Sheaf-theoretic stratification learning from geometric and topological perspectives. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00206-y","ieee":"A. Brown and B. Wang, “Sheaf-theoretic stratification learning from geometric and topological perspectives,” Discrete and Computational Geometry, vol. 65. Springer Nature, pp. 1166–1198, 2021.","short":"A. Brown, B. Wang, Discrete and Computational Geometry 65 (2021) 1166–1198.","chicago":"Brown, Adam, and Bei Wang. “Sheaf-Theoretic Stratification Learning from Geometric and Topological Perspectives.” Discrete and Computational Geometry. Springer Nature, 2021. https://doi.org/10.1007/s00454-020-00206-y.","ista":"Brown A, Wang B. 2021. Sheaf-theoretic stratification learning from geometric and topological perspectives. Discrete and Computational Geometry. 65, 1166–1198."},"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","oa":1,"quality_controlled":"1","publisher":"Springer Nature","acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria). This work was partially supported by NSF IIS-1513616 and NSF ABI-1661375. The authors would like to thank the anonymous referees for their insightful comments.","page":"1166-1198","date_created":"2020-05-30T10:26:04Z","date_published":"2021-06-01T00:00:00Z","doi":"10.1007/s00454-020-00206-y","year":"2021","has_accepted_license":"1","isi":1,"publication":"Discrete and Computational Geometry","day":"01","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"article_type":"original","type":"journal_article","status":"public","_id":"7905","file_date_updated":"2020-11-25T09:06:41Z","department":[{"_id":"HeEd"}],"date_updated":"2024-03-07T15:01:58Z","ddc":["510"],"scopus_import":"1","intvolume":" 65","month":"06","abstract":[{"text":"We investigate a sheaf-theoretic interpretation of stratification learning from geometric and topological perspectives. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the given sheaf is constructible. In particular, when we choose to work with the local homology sheaf, our algorithm gives an alternative to the local homology transfer algorithm given in Bendich et al. (Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1355–1370, ACM, New York, 2012), and the cohomology stratification algorithm given in Nanda (Found. Comput. Math. 20(2), 195–222, 2020). Additionally, we give examples of stratifications based on the geometric techniques of Breiding et al. (Rev. Mat. Complut. 31(3), 545–593, 2018), illustrating how the sheaf-theoretic approach can be used to study stratifications from both topological and geometric perspectives. This approach also points toward future applications of sheaf theory in the study of topological data analysis by illustrating the utility of the language of sheaf theory in generalizing existing algorithms.","lang":"eng"}],"oa_version":"Published Version","volume":65,"publication_status":"published","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"language":[{"iso":"eng"}],"file":[{"content_type":"application/pdf","relation":"main_file","access_level":"open_access","success":1,"file_id":"8803","checksum":"487a84ea5841b75f04f66d7ebd71b67e","file_size":1013730,"date_updated":"2020-11-25T09:06:41Z","creator":"dernst","file_name":"2020_DiscreteCompGeometry_Brown.pdf","date_created":"2020-11-25T09:06:41Z"}]},{"title":"Arakawa-Suzuki functors for Whittaker modules","external_id":{"arxiv":["1805.04676"],"isi":["000487176300011"]},"article_processing_charge":"No","author":[{"last_name":"Brown","full_name":"Brown, Adam","id":"70B7FDF6-608D-11E9-9333-8535E6697425","first_name":"Adam"}],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","citation":{"mla":"Brown, Adam. “Arakawa-Suzuki Functors for Whittaker Modules.” Journal of Algebra, vol. 538, Elsevier, 2019, pp. 261–89, doi:10.1016/j.jalgebra.2019.07.027.","ieee":"A. Brown, “Arakawa-Suzuki functors for Whittaker modules,” Journal of Algebra, vol. 538. Elsevier, pp. 261–289, 2019.","short":"A. Brown, Journal of Algebra 538 (2019) 261–289.","apa":"Brown, A. (2019). Arakawa-Suzuki functors for Whittaker modules. Journal of Algebra. Elsevier. https://doi.org/10.1016/j.jalgebra.2019.07.027","ama":"Brown A. Arakawa-Suzuki functors for Whittaker modules. Journal of Algebra. 2019;538:261-289. doi:10.1016/j.jalgebra.2019.07.027","chicago":"Brown, Adam. “Arakawa-Suzuki Functors for Whittaker Modules.” Journal of Algebra. Elsevier, 2019. https://doi.org/10.1016/j.jalgebra.2019.07.027.","ista":"Brown A. 2019. Arakawa-Suzuki functors for Whittaker modules. Journal of Algebra. 538, 261–289."},"date_created":"2019-08-22T07:54:13Z","date_published":"2019-11-15T00:00:00Z","doi":"10.1016/j.jalgebra.2019.07.027","page":"261-289","publication":"Journal of Algebra","day":"15","year":"2019","isi":1,"oa":1,"quality_controlled":"1","publisher":"Elsevier","department":[{"_id":"HeEd"}],"date_updated":"2023-08-29T07:11:47Z","status":"public","article_type":"original","type":"journal_article","_id":"6828","volume":538,"language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"issn":["0021-8693"]},"intvolume":" 538","month":"11","main_file_link":[{"url":"https://arxiv.org/abs/1805.04676","open_access":"1"}],"oa_version":"Preprint","abstract":[{"lang":"eng","text":"In this paper we construct a family of exact functors from the category of Whittaker modules of the simple complex Lie algebra of type to the category of finite-dimensional modules of the graded affine Hecke algebra of type . Using results of Backelin [2] and of Arakawa-Suzuki [1], we prove that these functors map standard modules to standard modules (or zero) and simple modules to simple modules (or zero). Moreover, we show that each simple module of the graded affine Hecke algebra appears as the image of a simple Whittaker module. Since the Whittaker category contains the BGG category as a full subcategory, our results generalize results of Arakawa-Suzuki [1], which in turn generalize Schur-Weyl duality between finite-dimensional representations of and representations of the symmetric group ."}]}]