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
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.
Host
