New Probes into Discrete and Convex Geometry

János Pach is a research professor at the Alfréd Rényi Institute, Budapest. His main fields of interest are discrete and computational geometry, convexity, and combinatorics. He has written more than 350 research papers. His books, "Research Problems in Discrete Geometry'' (with Brass and Moser) and "Combinatorial Geometry" (with Agarwal) were translated into Japanese, Russian, and Chinese. He is the co-editor-in-chief of Discrete & Computational Geometry. He received the Lester Ford Award from the Mathematical Association of America (1990), the Rényi Prize and the Academy Award from the Hungarian Academy of Sciences (1993,1998), and the Szele Prize from the Bolyai Mathematical Society (2019). He was elected the ACM Fellow (2011) and was a member of Academia Europaea (2014) and the Hungarian Academy of Sciences (2022). He was an invited speaker at the International Congress of Mathematicians (2014) and a plenary speaker at the European Congress of Mathematics (2021).Géza Tóth is a research professor and the head of the Department of Geometry at the Alfréd Rényi Institute, Budapest, working on problems in discrete and computational geometry and combinatorics. He received his PhD degree at Courant Institute, New York University in 1997, as a student of János Pach, winning the Best Dissertation Prize in mathematics and the Sokol Award, the highest university-wide prize for fresh PhDs. He has written more than 100 research articles and he is the co-editor-in-chief of Studia Scientiarum Mathematicarum Hungarica and the editor of Computational Geometry and Acta Mathematica Hungarica. Among other distinctions, he received the Erdős Prize of the Hungarian Academy of Sciences in 2008 and the Rényi Prize of the Rényi Institute in 2010.

• Chapter 1. Efficient triangulations of manifolds.- Chapter 2. Short path and short chain problems in the plane.- Chapter 3. On separability in discrete geometry.- Chapter 4. The Brascamp-Lieb inequality in Convex Geometry and in the Theory of Algorithms.- Chapter 5. Chromatic Topological Data Analysis.- Chapter 6. Free Sets in Planar Graphs: History and Applications.- Chapter 7. Lattice and Non-lattice Piercing of Axis-Parallel Rectangles.- Chapter 8. Why do adjacent crossings matter?.- Chapter 9. Helly type problems in convexity spaces.- Chapter 10. Centroids and equilibrium points of convex bodies.- Chapter 11. Using the KKM theorem.- Chapter 12. New combinatorial challenges and variations arising from Tverberg’s theorem.- Chapter 13. Helly-type problems from a topological perspective.- Chapter 14. No-dimensional Tverberg-type problems.- Chapter 15. On flotation, stability and related questions: A survey.- Chapter 16. A survey of Zarankiewicz problems in geometry.- Chapter 17. Coloring Geometric Hypergraphs: A Survey.