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Combinatorics with an analytic structure

Periodic Reporting for period 4 - CASe (Combinatorics with an analytic structure)

Reporting period: 2021-06-01 to 2022-05-31

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.
Key advances were made far beyond our initial expectations. I will give a brief overview of research carried out, as it was quite diverse and extensive:

In combinatorial topology, Andrea Bianchi has provided significant progress on combinatorial structure of Hurwitz spaces (2112.10864) and Peter Patzt worked on combinatorics and geometry of groups; he resolved a problem of Margalit concerning congruence conditions of Burau representations (arXiv:2209.09889) and studied codimension two homology of SLn(Z) (arXiv:2204.11967) (resolving a conjecture of Church-Farb-Putman).

Hailun Zheng worked on combinatorial models for Chow rings, known as stresses in the combinatorial community, and proved several impressive results, among them a conjecture of Stanley (published in Algebraic Combinatorics, 2021) and improved the bound on the number of triangulated neighborly spheres (published in Mathematische Annalen).

Patrick Schnider works on problems in combinatorial topology, and proved several interesting results concerning the art gallery problem (arXiv:2108.04007) and epsilon nets (2002.08693).

Rémi Cocou Avohou has worked on face vectors of polytopes and on Delta matroids, in particular making progress on characterizing face vectors for arbitrary polytopes.

Quang-Nhat Le made progress on the real theory of lattice polytopes, in particular exploring Fourier duality and Ehrhart period collapse in detail, which is of relevance in symplectic geometry (1808.00146 "Fourier transforms of polytopes, solid angle sums, and discrete volume" published in Mathematische Annalen).

Lukas Kuehne and Geva Yashfe resolved long standing and deep conjectures in concerning the approximability of matroids using subspace arrangements; this poses a big leap forward and introduces previously unexplored techniques ("Von Staudt Constructions for Skew-Linear and Multilinear Matroids" published in Israel Journal of Mathematics) and " On entropic and almost multilinear representability of matroids ", (arXiv:2206.03465).

Sergey Avvakumov proved several central conjectures in combinatorial topology, and contributed to significant progress in others. Let me highlight "A subexponential size triangulation of RPn", published in Combinatorica, which provided the first and to date only construction for subexponentially sized real projective space.

Arindam Biswas focused on spectral theory of graphs and groups, providing significant progress on expansion in Cayley graphs (2103.05935)

Which brings me to the PI: Several important advances were made during the project duration: We resolved the g-conjecture and Gruenbaum conjecture (arXiv:1812.10454) the semistable reduction conjecture (1810.03131) and proved a Lefschetz property for cycles (2101.07245). This provided great progress beyond the state of the art in combinatorics and algebraic geometry, and continues to provide new techniques towards new results.
The progress beyond the state of the art can, focused on the most important results, be summarized thusly.

The work of Kuehne and Yashfe provided an unprecedented breakthrough on long standing conjectures; while linear realizability of matroids was known to be hard, but possible, to check, realization as multilinear arrangements turned out to be significantly more mysterious. They finally resolved this problem and proved undecidability, therefore providing significant progress on a problem with applied relevance in computer science, in particular to so called secret sharing schemes.

The work of the PI on combinatorial Lefschetz theory, culminating in recent work with Papadakis, Petrotou and Steinmeyer, has for the first time extended central results of algebraic geometry to a combinatorial setting, involving in particular Lefschetz theory in positive characteristic. The siginificant progress here is the notion of non-degeneracy of Poincare pairings in ideals and its relation to the Lefschetz property. An important corollary pertains to combinatorial topology: in a simplicial complex embedding into 2n dimensional euclidean space, the ratio of n-simplices to n-1-simplices is bounded from above by a constant. This had been conjectured for several decades, but no attempt came close to a proof.

The work of PI on polysimplicial subdivisions resulted in the resolution of the long-standing semistable reduction conjecture put forward by Deligne and Mumford in The irreducibility of the space of curves of given genus in 1969. Further work on triangulations of manifolds resulted in the new construction of small triangulations of real projective space (with Avvakumov (funded by this project) and Roman Karasev).