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PRODID:-//Discrete Mathematics Group - ECPv6.15.20//NONSGML v1.0//EN
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X-ORIGINAL-URL:https://dimag.ibs.re.kr
X-WR-CALDESC:Events for Discrete Mathematics Group
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TZID:Asia/Seoul
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TZOFFSETFROM:+0900
TZOFFSETTO:+0900
TZNAME:KST
DTSTART:20230101T000000
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BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20240102T163000
DTEND;TZID=Asia/Seoul:20240102T173000
DTSTAMP:20260418T140808
CREATED:20230828T061831Z
LAST-MODIFIED:20240705T161245Z
UID:7552-1704213000-1704216600@dimag.ibs.re.kr
SUMMARY:Daniel McGinnis\, Applications of the KKM theorem to problems in discrete geometry
DESCRIPTION:We present the KKM theorem and a recent proof method utilizing it that has proven to be very useful for problems in discrete geometry. For example\, the method was used to show that for a planar family of convex sets with the property that every three sets are pierced by a line\, there are three lines whose union intersects each set in the family. This was previously a long-unsolved problem posed by Eckhoff. We go over a couple of examples demonstrating the method and propose a potential future research direction to push the method even further.
URL:https://dimag.ibs.re.kr/event/2024-01-02/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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DTSTART;TZID=Asia/Seoul:20240111T163000
DTEND;TZID=Asia/Seoul:20240111T173000
DTSTAMP:20260418T140808
CREATED:20231116T155919Z
LAST-MODIFIED:20240705T155117Z
UID:7919-1704990600-1704994200@dimag.ibs.re.kr
SUMMARY:Jinyoung Park (박진영)\, Dedekind's Problem and beyond
DESCRIPTION:The Dedekind’s Problem asks the number of monotone Boolean functions\, a(n)\, on n variables. Equivalently\, a(n) is the number of antichains in the n-dimensional Boolean lattice $[2]^n$. While the exact formula for the Dedekind number a(n) is still unknown\, its asymptotic formula has been well-studied. Since any subsets of a middle layer of the Boolean lattice is an antichain\, the logarithm of a(n) is trivially bounded below by the size of the middle layer. In the 1960’s\, Kleitman proved that this trivial lower bound is optimal in the logarithmic scale\, and the actual asymptotics was also proved by Korshunov in 1980’s. In this talk\, we will discuss recent developments on some variants of Dedekind’s Problem. Based on joint works with Matthew Jenssen\, Alex Malekshahian\, Michail Sarantis\, and Prasad Tetali.
URL:https://dimag.ibs.re.kr/event/2024-01-11/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20240116T163000
DTEND;TZID=Asia/Seoul:20240116T173000
DTSTAMP:20260418T140808
CREATED:20231211T010749Z
LAST-MODIFIED:20240705T155059Z
UID:8007-1705422600-1705426200@dimag.ibs.re.kr
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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BEGIN:VEVENT
DTSTART;TZID=Asia/Seoul:20240123T163000
DTEND;TZID=Asia/Seoul:20240123T173000
DTSTAMP:20260418T140808
CREATED:20231120T123044Z
LAST-MODIFIED:20240707T072547Z
UID:7930-1706027400-1706031000@dimag.ibs.re.kr
SUMMARY:Zichao Dong\, Convex polytopes in non-elongated point sets in $\mathbb{R}^d$
DESCRIPTION:For any finite point set $P \subset \mathbb{R}^d$\, we denote by $\text{diam}(P)$ the ratio of the largest to the smallest distances between pairs of points in $P$. Let $c_{d\, \alpha}(n)$ be the largest integer $c$ such that any $n$-point set $P \subset \mathbb{R}^d$ in general position\, satisfying $\text{diam}(P) < \alpha\sqrt[d]{n}$ (informally speaking\, `non-elongated’)\, contains a convex $c$-polytope. Valtr proved that $c_{2\, \alpha}(n) \approx \sqrt[3]{n}$\, which is asymptotically tight in the plane. We generalize the results by establishing $c_{d\, \alpha}(n) \approx n^{\frac{d-1}{d+1}}$. Along the way we generalize the definitions and analysis of convex cups and caps to higher dimensions\, which may be of independent interest. Joint work with Boris Bukh.
URL:https://dimag.ibs.re.kr/event/2024-01-23/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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