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DTSTART:20230101T000000
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DTSTART;TZID=Asia/Seoul:20240116T163000
DTEND;TZID=Asia/Seoul:20240116T173000
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SUMMARY:Matthew Kroeker\, Average flat-size in complex-representable matroids
DESCRIPTION:Melchior’s Inequality (1941) implies that\, in a rank-3 real-representable matroid\, the average number of points in a line is less than three. This was extended to the complex-representable matroids by Hirzebruch in 1983 with the slightly weaker bound of four. In this talk\, we discuss and sketch the proof of the recent result that\, in a rank-4 complex-representable matroid which is not the direct-sum of two lines\, the average number of points in a plane is bounded above by an absolute constant. Consequently\, the average number of points in a flat in a rank-4 complex-representable matroid is bounded above by an absolute constant. Extensions of these results to higher dimensions will also be discussed. In particular\, the following quantities are bounded in terms of k and r respectively: the average number of points in a rank-k flat in a complex-representable matroid of rank at least 2k-1\, and the average number of points in a flat in a rank-r complex-representable matroid. Our techniques rely on a theorem of Ben Lund which approximates the number of flats of a given rank. \nThis talk is based on joint work with Rutger Campbell and Jim Geelen.
URL:https://dimag.ibs.re.kr/event/2024-01-16/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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