The study of the interplay between combinatorics, geometry and algebra goes back to antiquity, with
regular convex polyhedra already being carved out of stone in the Neolithic period and often conflated
with religious mysticism. Contemporary study of combinatorial structures is often motivated by the
importance of optimization in modern economy, as well as the ubiquity of combinatorial problems in
pure mathematics. It has also, however, become a vital subject in itself.
A popular and central subject of modern combinatorics, polytope theory, experienced its first
serious developments by Euler 1758, and later Legendre and Cauchy in the late 18th century and
early 19th century. With the dawn of the 19th century, meticulous study of polytopes gained traction.
Combinatorialists such as Sommerville were interested in enumerative problems, Schl¨ afli studied the
geometric properties of polytopes (including highly symmetric polytopes) and Poincar´ e introduced
topologists to homology and simplicial homology, generating interest in triangulated manifolds and
other combinatorial decompositions of topological spaces.
Modern combinatorialists around McMullen and Stanley discovered the relation between enumerative
questions on polytopes and toric algebraic geometry, and commutative algebra. Meanwhile, the
polyhedral study of 3-manifolds is one of the pillars of the geometrization program of Thurston.
This research, together with enumerative problems in algebraic geometry, also created the field of
combinatorial topology , where special decompositions of topological and algebraic spaces are studied
and analyzed with combinatorial methods.
A key phenomenon I wish to investigate is the interplay between analytic and discrete structures.
Indeed, algebraic and analytic tools directed towards combinatorial questions have been well-studied,
as described above. However, combinatorics should also be understood to hold the capacity to reverse
this approach, and to improve our understanding of geometry.
The approach promises both challenges and opportunities. Combinatorial structures are often
harder to study than smooth structures, for example, as they often reduce problems to the minimal
assumptions reasonable and possible. As such, many problems have applications to foundational
mathematics.
I will roughly divide the contributions and program into three areas: Metric theory of discrete
structures, topological aspects of combinatorial geometry, and algebraic tools and results, though these
distinctions are not always sharp. We will not
cover every aspect of the research and sacrifice verbosity to brevity and focus.
