Tuesday, January 9, 2024

2024-01-16 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: Average flat-size in complex-representable matroids

by Matthew Kroeker(University of Waterloo)

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. This talk is based on joint work with Rutger Campbell and Jim Geelen.

2024-01-11 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: Dedekind’s Problem and beyond

by 박진영(NYU)

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.

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