IBS-KAIST 세미나 - 이산수학: Applications of the KKM theorem to problems in discrete geometry
by Daniel McGinnis(Iowa State University)
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.
In this talk, we derive second-order expressions for both the one- and two-particle reduced density matrices of the Gibbs state at fixed positive temperatures. We consider a translation-invariant system of N bosons in a three-dimensional torus. These bosons interact through a repulsive two-body potential with a scattering length of order 1/N in the large N limit. This analysis provides a justification of Bogoliubov's prediction regarding the fluctuations around the condensate. The talk will primarily introduce basic concepts and settings, ensuring accessibility for all attendees. This work is a joint effort with Christian Brennecke and Phan Thành Nam.
