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# 2023 Mini-Workshop on Discrete Geometry

## August 9 Wednesday @ 10:00 AM - 4:00 PM KST

2023 Mini-Workshop on Discrete Geometry will be held on August 9th at Room B332, Institute for Basic Science (IBS), Daejeon, Republic of Korea.

The workshop consists of three presentations on recent results and an open problem session.

Researchers who are highly interested in this field are welcome to attend.

## Tentative schedule

- 10:00-10:50 Michael Dobbins (SUNY Binghamton):
*Colorful intersections, Tverberg partitions, and geometric transversals* - 11:00-11:30 Andreas Holmsen (KAIST):
*The topology of the complex of ordered partitions* - 11:30-13:30 Lunch and free discussion
- 13:30-14:10 Minki Kim
^{김민기}(GIST):*Some variants of the colorful Helly theorems* - 14:20-16:00 Open problem session
*16:30-17:30 Discrete Math Seminar: R. Amzi Jeffs (Carnegie Mellon University)*

## Organizers

- Andreas Holmsen (KAIST)
- Jinha Kim
^{김진하}(IBS Discrete Mathematics Group) - Minki Kim
^{김민기}(GIST) - Sang-il Oum
^{엄상일}(IBS Discrete Mathematics Group / KAIST)

## Abstracts

#### Michael Dobbins (SUNY Binghamton): *Colorful intersections, Tverberg partitions, and geometric transversals*

Given three red convex sets and three blue convex sets in three-dimensional space, suppose every red set intersects every blue set. Montejano’s theorem says there is a line that intersects all the red sets or all the blue sets. This was generalized to k-transversals in $\mathbb R^d$ by Montejano and Karasev using sophisticated algebraic and topological tools. Here we give further generalizations based on more accessible methods such as the test-map/configuration space scheme, Sarkaria’s tensor method, and discrete Morse theory.

#### Andreas Holmsen (KAIST): *The topology of the complex of ordered partitions*

The set of partitions of {1,…,n} into k nonempty ordered parts can be equipped with a natural cell-complex structure which we denote by P(n,k). Here we use discrete Morse theory to show that P(n,k) is homotopy equivalent to a wedge of (n-k)-dimensional spheres, where the number of spheres is given in terms of Stirling numbers of the second kind. Our result has applications related to geometric transversals and Tverberg partitions.

#### Minki Kim (GIST): *Some variants of the colorful Helly theorems*

Given a finite family $F$ of nonempty sets, the nerve of $F$ is the simplicial complex on $F$ whose simplices are precisely the intersecting subfamilies of $F$. The colorful Helly theorem, which generalizes Helly’s theorem, asserts the following: if $X$ is the nerve of the disjoint union of $d+1$ many finite families $F_1,\ldots,F_{d+1}$ of convex sets in $\mathbb{R}^d$ where each $F_i$ is not a simplex in $X$, then there is a colorful $(d+1)$-tuple that is not a simplex of $X$. It was shown by Kalai and Meshulam that the same statement holds for $d$-collapsible/Leray complexes. In this talk, I will explain the notion of $d$-collapsibility and $d$-Lerayness of simplicial complexes, and present recent results on variants of the colorful Helly theorem and applications. This is based on joint work with Alan Lew.