[{"abstract":[{"lang":"eng","text":"We present solutions to several problems originating from geometry and discrete mathematics: existence of equipartitions, maps without Tverberg multiple points, and inscribing quadrilaterals. Equivariant obstruction theory is the natural topological approach to these type of questions. However, for the specific problems we consider it had yielded only partial or no results. We get our results by complementing equivariant obstruction theory with other techniques from topology and geometry."}],"alternative_title":["ISTA Thesis"],"type":"dissertation","oa_version":"Published Version","file":[{"creator":"savvakum","content_type":"application/zip","file_size":1061740,"file_name":"source.zip","access_level":"closed","date_updated":"2020-07-27T12:44:51Z","date_created":"2020-07-27T12:44:51Z","file_id":"8178","relation":"source_file"},{"success":1,"date_created":"2020-07-27T12:46:53Z","date_updated":"2020-07-27T12:46:53Z","relation":"main_file","file_id":"8179","file_size":1336501,"content_type":"application/pdf","creator":"savvakum","access_level":"open_access","file_name":"thesis_pdfa.pdf"}],"title":"Topological methods in geometry and discrete mathematics","status":"public","ddc":["514"],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","_id":"8156","has_accepted_license":"1","article_processing_charge":"No","day":"24","date_published":"2020-07-24T00:00:00Z","page":"119","citation":{"short":"S. Avvakumov, Topological Methods in Geometry and Discrete Mathematics, Institute of Science and Technology Austria, 2020.","mla":"Avvakumov, Sergey. Topological Methods in Geometry and Discrete Mathematics. Institute of Science and Technology Austria, 2020, doi:10.15479/AT:ISTA:8156.","chicago":"Avvakumov, Sergey. “Topological Methods in Geometry and Discrete Mathematics.” Institute of Science and Technology Austria, 2020. https://doi.org/10.15479/AT:ISTA:8156.","ama":"Avvakumov S. Topological methods in geometry and discrete mathematics. 2020. doi:10.15479/AT:ISTA:8156","apa":"Avvakumov, S. (2020). Topological methods in geometry and discrete mathematics. Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:8156","ieee":"S. Avvakumov, “Topological methods in geometry and discrete mathematics,” Institute of Science and Technology Austria, 2020.","ista":"Avvakumov S. 2020. Topological methods in geometry and discrete mathematics. 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We provide a quantitative estimate on the difference between the grand-canonical Levy–Lieb energy of a given density (the lowest possible energy of all quantum states having this density) and the integral over the uniform electron gas energy of this density. The error involves gradient terms and justifies the use of the local density approximation in the situation where the density is very flat on sufficiently large regions in space."}],"issue":"1","status":"public","title":" The local density approximation in density functional theory","intvolume":" 2","_id":"14891","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Preprint","scopus_import":"1","day":"01","article_processing_charge":"No","article_type":"original","page":"35-73","publication":"Pure and Applied Analysis","citation":{"chicago":"Lewin, Mathieu, Elliott H. Lieb, and Robert Seiringer. “ The Local Density Approximation in Density Functional Theory.” Pure and Applied Analysis. Mathematical Sciences Publishers, 2020. https://doi.org/10.2140/paa.2020.2.35.","mla":"Lewin, Mathieu, et al. “ The Local Density Approximation in Density Functional Theory.” Pure and Applied Analysis, vol. 2, no. 1, Mathematical Sciences Publishers, 2020, pp. 35–73, doi:10.2140/paa.2020.2.35.","short":"M. Lewin, E.H. Lieb, R. Seiringer, Pure and Applied Analysis 2 (2020) 35–73.","ista":"Lewin M, Lieb EH, Seiringer R. 2020. The local density approximation in density functional theory. Pure and Applied Analysis. 2(1), 35–73.","apa":"Lewin, M., Lieb, E. H., & Seiringer, R. (2020). The local density approximation in density functional theory. Pure and Applied Analysis. Mathematical Sciences Publishers. https://doi.org/10.2140/paa.2020.2.35","ieee":"M. Lewin, E. H. Lieb, and R. Seiringer, “ The local density approximation in density functional theory,” Pure and Applied Analysis, vol. 2, no. 1. Mathematical Sciences Publishers, pp. 35–73, 2020.","ama":"Lewin M, Lieb EH, Seiringer R. The local density approximation in density functional theory. 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Imaging and related analyses were facilitated by The Waitt Advanced Biophotonics Center Core at the Salk Institute, supported by grants from NIH-NCI CCSG (P30 014195) and NINDS Neuroscience Center (NS072031). The authors would like to additionally thank Drs. Jane Dodd, Robert Brownstone, and Laskaro Zagoraiou for helpful comments on the manuscript. This manuscript is dedicated to Tom Jessell, an inspirational scientist, friend and mentor.","year":"2020","has_accepted_license":"1","article_processing_charge":"Yes (via OA deal)","day":"01","scopus_import":"1","date_published":"2020-12-01T00:00:00Z","page":"81-95","article_type":"original","citation":{"chicago":"Salamatina, Alina, Jerry H Yang, Susan Brenner-Morton, Jay B Bikoff, Linjing Fang, Christopher R Kintner, Thomas M Jessell, and Lora B. Sweeney. “Differential Loss of Spinal Interneurons in a Mouse Model of ALS.” Neuroscience. Elsevier, 2020. https://doi.org/10.1016/j.neuroscience.2020.08.011.","mla":"Salamatina, Alina, et al. “Differential Loss of Spinal Interneurons in a Mouse Model of ALS.” Neuroscience, vol. 450, Elsevier, 2020, pp. 81–95, doi:10.1016/j.neuroscience.2020.08.011.","short":"A. Salamatina, J.H. Yang, S. Brenner-Morton, J.B. Bikoff, L. Fang, C.R. Kintner, T.M. Jessell, L.B. Sweeney, Neuroscience 450 (2020) 81–95.","ista":"Salamatina A, Yang JH, Brenner-Morton S, Bikoff JB, Fang L, Kintner CR, Jessell TM, Sweeney LB. 2020. Differential loss of spinal interneurons in a mouse model of ALS. Neuroscience. 450, 81–95.","apa":"Salamatina, A., Yang, J. H., Brenner-Morton, S., Bikoff, J. B., Fang, L., Kintner, C. R., … Sweeney, L. B. (2020). Differential loss of spinal interneurons in a mouse model of ALS. Neuroscience. Elsevier. https://doi.org/10.1016/j.neuroscience.2020.08.011","ieee":"A. Salamatina et al., “Differential loss of spinal interneurons in a mouse model of ALS,” Neuroscience, vol. 450. Elsevier, pp. 81–95, 2020.","ama":"Salamatina A, Yang JH, Brenner-Morton S, et al. Differential loss of spinal interneurons in a mouse model of ALS. Neuroscience. 2020;450:81-95. doi:10.1016/j.neuroscience.2020.08.011"},"publication":"Neuroscience","abstract":[{"lang":"eng","text":"Amyotrophic lateral sclerosis (ALS) leads to a loss of specific motor neuron populations in the spinal cord and cortex. Emerging evidence suggests that interneurons may also be affected, but a detailed characterization of interneuron loss and its potential impacts on motor neuron loss and disease progression is lacking. To examine this issue, the fate of V1 inhibitory neurons during ALS was assessed in the ventral spinal cord using the SODG93A mouse model. The V1 population makes up ∼30% of all ventral inhibitory neurons, ∼50% of direct inhibitory synaptic contacts onto motor neuron cell bodies, and is thought to play a key role in modulating motor output, in part through recurrent and reciprocal inhibitory circuits. We find that approximately half of V1 inhibitory neurons are lost in SODG93A mice at late disease stages, but that this loss is delayed relative to the loss of motor neurons and V2a excitatory neurons. We further identify V1 subpopulations based on transcription factor expression that are differentially susceptible to degeneration in SODG93A mice. At an early disease stage, we show that V1 synaptic contacts with motor neuron cell bodies increase, suggesting an upregulation of inhibition before V1 neurons are lost in substantial numbers. 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