# 2024 IBS-DIMAG Workshop on Combinatorics and Geometric Measure Theory (CGMT)

## Plenary Speakers

## János Pach (Rényi Institute of Mathematics): With strings attached Slides

The intersection graph of a collection \(C\) of sets is the graph whose vertex set is \(C\) and in which two sets in \(C\) are connected by an edge if and only if they have nonempty intersection. String graphs, intersection graphs of continuous curves (``strings") in the plane have been studied intensively since the 1960s, for their exciting algorithmic and combinatorial properties and their applications in chip design, network theory, graph drawing and elsewhere. After giving a whirlwind tour of string graph theory, I will present some recent results and annoying open problems. In particular, I will sketch the proof of the following theorem, joint with Jacob Fox and Andrew Suk. Given a set \(R\) of \(n\) red curves, and and a set \(B\) of \(n\) blue curves in the plane such that any two of them meet at most once, there are subsets \(R' \subset R\) and \(B' \subset B\) with \(|R'|, |B'| \geq \Omega(n)\) with the property that either every curve in \(R'\) crosses every curve in \(B'\), or every curve in \(R'\) is disjoint from every curve in \(B'\).## Pertti Mattila (University of Helsinki): Hausdorff dimension of plane sections and general intersections

I shall discuss conditions on a general family \(P_{\lambda}:\mathbb{R}^n\to\mathbb{R}^m, \lambda \in \Lambda,\) of orthogonal projections and a measure \(\omega\) on \(\Lambda\) which guarantee that the Hausdorff dimension formula \(\dim A\cap P_{\lambda}^{-1}\{u\}=s-m\) holds for \(\omega\) almost all \(\lambda\) for measurable sets \(A\subset\mathbb{R}^n\) with positive and finite \(s\)-dimensional Hausdorff measure, \(s>m\). I shall present some families of projections where this applies. This leads to some new results on the Hausdorff dimension of intersections \(\dim A\cap (g(B)+z)\) for almost all rotations \(g\) and for positively many \(z\in\mathbb{R}^n\).## Invited Speakers

## Hong Wang (NYU): Some structure of Kakeya sets in \(R^3\) Slides

A Kakeya set in \(R^n\) is a set of points that contains a unit line segment in every direction. We study the structure of Kakeya sets in \(R^3\) and show that for any Kakeya set \(K\), there exists well-separated scales \(0 < \delta < \rho\leq 1\) so that the \delta-neighborhood of \(K\) is almost as large as the \rho-neighborhood of \(K\). As a consequence, every Kakeya set in \(R^3\) has Assouad dimension \(3\). This is joint work with Josh Zahl.## Cosmin Pohoata (Emory Univiersity): Heilbronn triangle problem Slides

The Heilbronn triangle problem is a classical problem in discrete geometry with several new connections to various topics in extremal and additive combinatorics, incidence geometry, harmonic analysis, and projection theory. In this talk, we will give an overview of some of these connections, and discuss some recent developments. Based on joint work with Alex Cohen and Dmitrii Zakharov.## Izabella Łaba (UBC): A short survey of integer tilings Slides

A set \(A\subset\mathbb{Z}\) tiles the integers by translations if there is a set \(T\subset\mathbb{Z}\) such that every integer \(n\in\mathbb{Z}\) has a unique representation \(n=a+t\) with \(a\in A\) and \(t\in T\). The main open question regarding integer tilings is the Coven-Meyerowitz conjecture, providing a tentative characterization of finite tiles. We will survey some of the recent developments and open questions in this area, including a very recent joint result with Itay Londner where we prove the Coven-Meyerowitz tiling conditions for a new class of tilings.## Venue

- Venue: Room B109, Institute for Basic Science - photos available here.

Travel Instructions: https://travel.dimag.kr/

## Organizers

- Doowon Koh (Chungbuk National University)
- Ben Lund (IBS Discrete Mathematics Group)
- Sang-il Oum (IBS Discrete Mathematics Group / KAIST Department of Mathematical Sciences)

## Schedule and Abstracts (PDF file)

## Schedule

Sunday, July 14

Arrival / Meeting / Discussion

Monday, July 15

9:00-9:30 Registration, coffee

9:30-9:40 Opening remarks

10:40-11:00 Coffee break

11:00-11:30 Bochen Liu

11:30-12:00 Giorgis Petridis

12:00-14:00 Lunch

14:00-14:30 Manik Dhar

14:30-15:00 Jonathan Pakianathan

15:00-15:30 Coffee break

15:30-17:00 Open Problem Session

Tuesday, July 16

10:30-11:00 Coffee break

11:00-11:30 Mihalis Kolountzakis Slides

11:30-12:00 Alexander Clifton Slides

12:00-14:00 Lunch

14:00-14:30 Eyvindur Palsson Slides

14:30-15:00 Steven Senger Slides

15:00-15:30 Coffee break

15:30-17:00 Small group collaboration

Wednesday, July 17

9:30-10:30 Pertti Mattila

Plenary lecture

10:30-11:00 Coffee break

11:00-11:30 Wei-Hsuan Yu Slides

11:30-12:00 Hai Long Dao

12:00-14:00 Lunch

14:00-18:30 Excursion (Gongju National Museum, Gapsa Temple)

18:30 Banquet (near Gapsa Temple)

Thursday, July 18

10:30-11:00 Coffee break

11:00-11:30 Alan Chang

11:30-12:00 Terry Harris Slides

12:00-14:00 Lunch

14:00-14:30 Charlotte Aten Slides

14:30-15:00 Semin Yoo

15:00-15:30 Coffee break

15:30-17:00 Small group collaboration

Friday, July 19

10:30-11:00 Coffee break

11:00-11:30 Andreas Holmsen Slides

12:00-14:00 Lunch

14:00-14:30 Olivine Silier

14:30-15:00 Matthew Kroeker

15:00-15:30 Coffee break

15:30-17:00 Small group collaboration

## Restaurant guide

We do NOT plan to provide lunches for participants. Banquet will be provided on Wednesday evening after the excursion around Gapsa temple.

- Restaurant Guide around IBS (PDF file)
- IBS cafeteria is open for lunch (11:30-13:30) and dinner (17:30-19:30) on weekdays. Lunch costs 5,000 KRW and dinner costs 4,500 KRW. One can pay by cash or credit card. The cafeteria is located on the ground floor of the main building, very close to the workshop venue.
- Shinsaegae Department Store (10 minutes walk from IBS) has a food court on the B1 floor; more than 10 Korean restaurants, 7 Japanese restaurants, 4 Western restaurants, 4 Chinese restaurants, 1 Vietnamese restaurant. Open daily from 10:30 to 20:00. The 5th floor has several restaurants too with longer opening hours (open until 21:30).
- Starbucks on the 38th floor of the EXPO tower (having the Onoma Hotel adjacent to IBS). Open at 8 am daily. Good for breakfast with great view.
- More local information.