Welcome Prof. Andreas Holmsen from KAIST, a new Visiting Research Fellow in the IBS Discrete Mathematics Group

The IBS Discrete Mathematics Group welcomes Prof. Andreas Holmsen from the Department of Mathematical Sciences, KAIST, Daejeon, Korea. He will visit the IBS Discrete Mathematics Group for 1 year from February 1, 2024 during his sabbatical leave from KAIST. He received his Ph.D. from the University of Bergen in 2004 under the supervision of Prof. Helge Tverberg and was at the University of Bergen, New York University, City College of New York before coming to KAIST.

The 2023 Mini-Workshop on Discrete Geometry was held on August 9 with three talks by Michael Dobbins, Andreas Holmsen, and Minki Kim

On August 9, 2023, the 2023 Mini-Workshop on Discrete Geometry was held at the IBS. This workshop consisted of three talks by Michael Dobbins from SUNY Binghamton, Andreas Holmsen from KAIST, and Minki Kim from GIST. Here is the list of talks.

  • Michael Dobbins (SUNY Binghamton): Colorful intersections, Tverberg partitions, and geometric transversals
  • Andreas Holmsen (KAIST): The topology of the complex of ordered partitions
  • Minki Kim김민기 (GIST): Some variants of the colorful Helly theorems

Andreas Holmsen, A colorful version of the Goodman-Pollack-Wenger transversal theorem

Hadwiger’s transversal theorem gives necessary and sufficient conditions for the existence of a line transversal to a family of pairwise disjoint convex sets in the plane. These conditions were subsequently generalized to hyperplane transversals in $\mathbb{R}^d$ by Goodman, Pollack, and Wenger. Here we establish a colorful extension of their theorem, which proves a conjecture of Arocha, Bracho, and Montejano. The proof uses topological methods, in particular the Borsuk-Ulam theorem. The same methods also allow us to generalize some colorful transversal theorems of Montejano and Karasev.

Andreas Holmsen, Some recent results on geometric transversals

A geometric transversal to a family of convex sets in $\mathbb R^d$ is an affine flat that intersects the members of the family. While there exists a far-reaching theory concerning 0-dimensional transversals (intersection patterns of convex sets), much less is known when it comes to higher-dimensional transversals. In this talk, I will present some new and old results and problems regarding geometric transversals, based on joint work with Otfried Cheong and Xavier Goaoc.

Andreas Holmsen, Discrete geometry in convexity spaces

The notion of convexity spaces provides a purely combinatorial framework for certain problems in discrete geometry. In the last ten years, we have seen some progress on several open problems in the area, and in this talk, I will focus on the recent results relating to Tverberg’s theorem and the Alon-Kleitman (p,q) theorem.

Andreas Holmsen, Fractional Helly and topological complexity

The fractional Helly theorem is a simple yet remarkable generalization of Helly’s classical theorem on the intersection of convex sets, and it is of considerable interest to extend the fractional Helly theorem beyond the setting of convexity. In this talk I will discuss a recent result which shows that the fractional Helly theorem holds for families of subsets of $\mathbb R^d$ which satisfy only very weak topological assumptions. The proofs combine a number of tools such as homological minors, stair-convexity, supersaturation in hypergraphs, Radon dimension, and Ramsey-type arguments. This is joint work with Xavier Goaoc and Zuzana Patáková.

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