CAREER: Set-Systems: Probabilistic, Geometric and Extremal Perspectives

In this project, the PI will study various topics in discrete mathematics and probability, two areas which have grown significantly in both depth and breadth in the 21st century, resulting in methods that apply to a number of scientific disciplines. These include applications, and many significant breakthroughs, in theoretical computer science and statistical physics, not to mention a number of other areas of mathematics. The approaches and techniques resulting from this project will have a significant impact on the development of these areas and will also be applicable in other branches of mathematics and theoretical computer science. This project also puts forward a comprehensive plan for training undergraduate and graduate students. The broad research goals of this project involve studying set-systems in the Boolean hypercube from various points of view: combinatorial, probabilistic and geometric. Up-sets and down-sets in particular appear as basic building blocks in a number of different areas of mathematics: they encode many events of interest in probability spaces arising in probabilistic combinatorics and statistical physics, they arise in extremal set theory, with intersecting families furnishing an important class of examples, they appear in convex geometry as discrete analogues of half-spaces, and viewed as simplicial complexes, they are central objects in simplicial topology. Given the prevalence of these objects across multiple areas, a better understanding of their various facets will spur developments across these fields. By targeting a careful selection of basic problems about set-systems arising from combinatorics, probability and geometry, the PI aims to develop new analytic, combinatorial, probabilistic and geometric techniques to study these objects. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.