## Periodic Reporting for period 3 - CASe (Combinatorics with an analytic structure)

Reporting period: 2020-01-01 to 2021-05-31

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. I will refer to the publication list and arxiv preprints, and will not

cover every aspect of my research.

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. I will refer to the publication list and arxiv preprints, and will not

cover every aspect of my research.

We carried out research focusing on algebrao-geometric aspects of combinatorial objects, as well as in the direction of quantitative geometric properties of convex bodies. Key advances were made far beyond our initial expectations.

The first period of the StG has seen some surprising progress on the projects outlined in the DoA, and happily often with unpredicted consequences. I summarize them below.

1.1 Objectives

I subdivide into two categories (1) works that were finished during project, and appeared in refereed journals and (2) research in progress, including publically available ones.

In category (1), we have the publications

[ABG] Adiprasito, Karim A.; Björner, Anders; Goodarzi, Afshin Face numbers of sequentially Cohen-Macaulay complexes and Betti numbers of componentwise linear ideals. J. Eur. Math. Soc. (JEMS) 19 (2017), no. 12, 3851–3865.

[A] Adiprasito, Karim Toric chordality. J. Math. Pures Appl. (9) 108 (2017), no. 5, 783–807.

[AHK] Karim Adiprasito, June Huh, Eric Katz; Hodge theory for combinatorial geometries (to appear in Annals of Mathematics)

which consitute critical advances in the algebraic theory of combinatorics, each solving important problems and introducing critical new methods to the field of algebraic geometry as well as combinatorics (Section III of DoA). [ABG] provides a generalization of results of Macaulay and Stanley beyond purity, a goal that has eluded researchers for more than 30 years prior to this result. [A] discusses relations between combinatorial properties of fans and topolgical qualities of the related toric varieties, and [AHK] solves the classical Rota Conjecture.

Moreover,

Adiprasito, Karim A note on the simplex-cosimplex problem. European J. Combin. 66 (2017), 5–12.

Adiprasito, Karim A.; Benedetti, Bruno; Lutz, Frank H. Extremal examples of collapsible complexes and random discrete Morse theory. Discrete Comput. Geom. 57 (2017), no. 4, 824–853.

Karim A. Adiprasito, Philip Brinkmann, Arnau Padrol, Pavel Paták, Zuzana Patáková, Raman Sanyal; Colorful Simplicial Depth, Minkowski Sums, and Generalized Gale Transforms, to appear in International Mathematics Research Notices

provide advances in the area of topological and geometric combinatorics, introducing new geometric techniques to discrete problems, and contribute to the goals of Section I.

Concerning (2), I made several intriguing advances concerning algebraic aspects of combinatorial objects. This lead to, in the beginning of the project, the solution of the Gruenbaum conjecture and a first proof of the Hard Lefschetz theorem for sufficiently generic toric varieties beyond projectivity (where sufficiently generic means under generic among those varieties with fixed equivariant cohomology). A particularly striking corollary coming out of this theory is that for a 2-dimensional simplicial complex PL embedded in $\mathbb{R}^4$, the number of triangles is at most 4 times the number of edges, generalizing a classical result of Descartes for planar graphs (in preparation). The algebraic part of this work is a advance on \textbf{Combinatorial Lefschetz and Hodge theory} (Section III of DoA) I could only dream of when the project started.

Further advances in this area include counterexamples to the tropical Lefschetz conjecture of Nisse and Sottile (arxiv:1711.02045) and solutions to several conjectures concerning quantitative aspects of toric varieties (rationally smooth as well as singular, partly contained in arXiv:1805.03267 arXiv:1711.07218 arXiv:1806.03322 which builds on Toric chordality (J. Math. Pures Appl. (9) 108, No. 5, 783-807 (2017).) as well as articles in preparation.)

These often connect approximation theory and algebraic geometry in interesting ways, and provide significant progress towards \textbf{Main Objective B.2.} of the proposal.

We made breakthroughs concerning dimensionless Helly type theorems (with Imre Barany, Nabil Mustafa, arxiv:1806.08725) bounds on the number of triangulations of space forms (with Benedetti, arXiv:1710.00130) (this is towards \textbf{Section I} of the DoA) as well as several other articles towards that goal (arXiv:1709.07930).

Related ERC supported final aspects of a work with Eran Nevo and Martin Tancer concerning extremal Betti numbers of simplicial complexes with forbidden induced minors ("On Betti numbers of flag complexes with forbidden induced subgraphs" submitted to Math. Proc. Cam. Phil. Soc., positive report)

In addition, we proved a conjecture of Berkovich concerning polystable reduction of log-schemes (with Michael Temkin, Gaku Liu, Igor Pak, arXiv:1806.09168) using interesting combinatorial methods: We proved the existence polystable log modifications for log varieties.

The project will reach and surpass some of its main goals, in particular in pertaining to the g-conjecture, and the combinatorial standard conjectures.

1.1 Objectives

I subdivide into two categories (1) works that were finished during project, and appeared in refereed journals and (2) research in progress, including publically available ones.

In category (1), we have the publications

[ABG] Adiprasito, Karim A.; Björner, Anders; Goodarzi, Afshin Face numbers of sequentially Cohen-Macaulay complexes and Betti numbers of componentwise linear ideals. J. Eur. Math. Soc. (JEMS) 19 (2017), no. 12, 3851–3865.

[A] Adiprasito, Karim Toric chordality. J. Math. Pures Appl. (9) 108 (2017), no. 5, 783–807.

[AHK] Karim Adiprasito, June Huh, Eric Katz; Hodge theory for combinatorial geometries (to appear in Annals of Mathematics)

which consitute critical advances in the algebraic theory of combinatorics, each solving important problems and introducing critical new methods to the field of algebraic geometry as well as combinatorics (Section III of DoA). [ABG] provides a generalization of results of Macaulay and Stanley beyond purity, a goal that has eluded researchers for more than 30 years prior to this result. [A] discusses relations between combinatorial properties of fans and topolgical qualities of the related toric varieties, and [AHK] solves the classical Rota Conjecture.

Moreover,

Adiprasito, Karim A note on the simplex-cosimplex problem. European J. Combin. 66 (2017), 5–12.

Adiprasito, Karim A.; Benedetti, Bruno; Lutz, Frank H. Extremal examples of collapsible complexes and random discrete Morse theory. Discrete Comput. Geom. 57 (2017), no. 4, 824–853.

Karim A. Adiprasito, Philip Brinkmann, Arnau Padrol, Pavel Paták, Zuzana Patáková, Raman Sanyal; Colorful Simplicial Depth, Minkowski Sums, and Generalized Gale Transforms, to appear in International Mathematics Research Notices

provide advances in the area of topological and geometric combinatorics, introducing new geometric techniques to discrete problems, and contribute to the goals of Section I.

Concerning (2), I made several intriguing advances concerning algebraic aspects of combinatorial objects. This lead to, in the beginning of the project, the solution of the Gruenbaum conjecture and a first proof of the Hard Lefschetz theorem for sufficiently generic toric varieties beyond projectivity (where sufficiently generic means under generic among those varieties with fixed equivariant cohomology). A particularly striking corollary coming out of this theory is that for a 2-dimensional simplicial complex PL embedded in $\mathbb{R}^4$, the number of triangles is at most 4 times the number of edges, generalizing a classical result of Descartes for planar graphs (in preparation). The algebraic part of this work is a advance on \textbf{Combinatorial Lefschetz and Hodge theory} (Section III of DoA) I could only dream of when the project started.

Further advances in this area include counterexamples to the tropical Lefschetz conjecture of Nisse and Sottile (arxiv:1711.02045) and solutions to several conjectures concerning quantitative aspects of toric varieties (rationally smooth as well as singular, partly contained in arXiv:1805.03267 arXiv:1711.07218 arXiv:1806.03322 which builds on Toric chordality (J. Math. Pures Appl. (9) 108, No. 5, 783-807 (2017).) as well as articles in preparation.)

These often connect approximation theory and algebraic geometry in interesting ways, and provide significant progress towards \textbf{Main Objective B.2.} of the proposal.

We made breakthroughs concerning dimensionless Helly type theorems (with Imre Barany, Nabil Mustafa, arxiv:1806.08725) bounds on the number of triangulations of space forms (with Benedetti, arXiv:1710.00130) (this is towards \textbf{Section I} of the DoA) as well as several other articles towards that goal (arXiv:1709.07930).

Related ERC supported final aspects of a work with Eran Nevo and Martin Tancer concerning extremal Betti numbers of simplicial complexes with forbidden induced minors ("On Betti numbers of flag complexes with forbidden induced subgraphs" submitted to Math. Proc. Cam. Phil. Soc., positive report)

In addition, we proved a conjecture of Berkovich concerning polystable reduction of log-schemes (with Michael Temkin, Gaku Liu, Igor Pak, arXiv:1806.09168) using interesting combinatorial methods: We proved the existence polystable log modifications for log varieties.

The project will reach and surpass some of its main goals, in particular in pertaining to the g-conjecture, and the combinatorial standard conjectures.