• Combinatorial Lefschetz theorems beyond positivity
  arxiv:1812:10454, local***

- It resolves the g-conjecture, that is, we can now fully characterize the possible face numbers of simplicial rational homology spheres, resolving the g-conjecture of McMullen in full generality and generalizing Stanley's earlier proof for simplicial polytopes.
- It resolves a conjecture of Grünbaum, Kalai and Sarkaria, and generalizes a result going back to Descartes and Euler: We prove that for a simplicial complex ” that embeds into 2d-dimensional euclidean space, the number of d-dimensional simplices exceeds the number of (d1)-dimensional simplices by a factor of at most d+2. This in particular implies a generalization of the celebrated crossing number inequality of Ajtai,Chvátal, Leighton, Newborn and Szemerédi to higher dimensions.

***the local version is more verbose; since the paper develops quite a bit of techniques, and non-experts are quite interested in the results, many people have questions. Mathematically, there is no change since the first arxiv version but occasionally, if more than one person asks the same question, I add some more wordy explanations. So if you have questions, let me know.

**** I updated the local version after teaching a course about it in spring of 2020. It contains also the derivation of a conjecture of Kuehnel, and fixes some bibtex issues

See also this post on Gil Kalai's blog about it, and even a post in Korean (by Sang-il Oum). I am teaching a minicourse at Sapienza about it April 19, see the link here. See also here for an overview over the proof (intended as a first read if you find ~80 pages intimidating), as well as some alternative roads to some steps of the same. It was written as a short exposition of the main ideas after the minicourse at Sapienza. You can also have a look at the Hadamard lectures at IHES I gave about this topic.

At some point, Stavros Papadakis and Vasiliki Petrotou figured out an amazing simplification of a key step in the program. This led us to jointly figure out further simplifications and generalizations, see for instance this for the latest. Moreover, you can check here for an expository article with Geva Yashfe that explained one of the innovations of the article, the partition complex.

On the flip approach to the Lefschetz property

This short note examines the conditions imposed on the Artinian reduction for the Lefschetz property to hold. It provides a rather simple obstruction for the naive "flip" approach to work in general to prove Lefschetz property, and gives a quick check to determine fatal issues with arguments in the context of this problem (though it did not keep people, including myself, from trying this approach). See for instance arxiv:2001.06594 for a recent attempt at using this approach, though it seems to fail all the same.


these are actually two papers, one with Gaku Liu and Michael Temkin (arxiv:1810.03131) , and one in addition with Igor Pak (arxiv:1806.09168). This resolves a conjecture of Abramovic and Karu, and establishes the existence of polystable/semistable modifications for log-varieties over valuation rings/surjective morphisms of complex projective varieties, generalizing classical results of Kempf, Knudsen, Mumford, Saint-Donat and Waterman to log-varieties. In particular, the combinatorial core of the argument is a generalization of the celebrated Knudsen-Mumford-Waterman result concerning unimodilar triangulations of lattice polytopes to Cayley polytopes; the construction is elementary, but rather intricate. My coauthor, friend and former postdoc Gaku realized that it was very powerful, and resolved some further questions

• Many projectively unique polytopes
  Inventiones Math, with G.M. Ziegler (arxiv:1212.5812)

We construct an infinite family of 4-polytopes whose realization spaces have dimension smaller or equal to 96. This in particular settles a problem going back to Legendre and Steinitz: To bound the dimension of the realization space of a polytope in terms of its $f$-vector. Moreover, we derive an infinite family of combinatorially distinct 69-dimensional polytopes whose real==ization is unique up to projective transformation. This answers a problem posed by Perles and Shephard in the sixties. The construction is rather cool, as it relies on discretizing a certain partial differential equation, and solving it, and is inspired by the rigidity of higher-dimensional conformal geometry. The tools developped in this paper are rather rich, and the result has recently found application in the theory of slack ideals.

(In addition to solving an old problem, this work won me the European Prize in Combinatorics in 2015)

• The universality theorem for neighborly polytopes

  Combinatorica, with Arnau Padrol (arxiv:1402.7207)

Arnau and I had some amazing collaborations over the years on realization spaces of polytopes, so let me highlight one: The celebrated universality theorem of Mnëv proves, among other things, that every semialgebraic set defined over the integers appears as the realization space of some polytope. This was an amazing achievement.

Our contribution: Well, there was a conjecture in his paper that the same should be true for open semialgebraic sets, and simplicial polytopes. In other words, the general position version of his theorem. This turned out to be much harder, but we managed and thereby solved his conjecture.

• Hodge theory for combinatorial geometries
  Annals of Mathematics, with June Huh and Eric Katz (publication version)
  

The characteristic polynomial of a matroid is a fundamental and mysterious invariant of matroids with many problems surrounding it. Among the most resilient problems is a conjecture of Rota, Heron and Welsh proposing that the coefficients of the characteristic polynomial are log-concave. We prove this conjecture by relating it to, and then establishing a, Hodge theory on certain Chow rings associated to general matroids. Ultimately, once you realize that the proof is essentially an adaption of a technique developped and/or used earlier several times (by Elias-Williamson, Karu and others) and original credit for developping the program that now is combinatorial Hodge Theory should go to the amazing work of Peter McMullen, it becomes a tad less exciting, but it was certainly very surprising back then.

(See also Matt Baker's blog entry about this work. June and I won the New Horizons Prize for this paper in 2018).

• Filtered geometric lattices and Lefschetz Section Theorems over the tropical semiring
  2014, with Anders Bjoerner (arxiv:1401.7301)

This proves a conjecture of Mikhalkin and Ziegler, and prove a Lefschetz section theorem for  "smooth" (that is, locally matroidal) tropical varieties. The paper went through several iterations, where we noticed smaller new results but also improved presentation, as this draws from several different areas and people were not familiar with, for instance, stratified Morse theory. At some point, I also noticed it contained a mistake: that the Lefschetz theorem is true with respect to integral tropical Hodge groups. In the end, the proof and theorem with respect to Mikhalkin's Hodge theory works only in characteristic 0, and I added a counterexample in positive characteristic. The paper is, to best of my knowledge, correct in this form, though it is also somewhat obsolete and I have not submitted it again: because of the work on the Rota conjecture, we now know that the hard Lefschetz theorem holds as well (this implication was figured out by Amini and Piquerez). Still, there are several results in this paper that may be useful to the reader, and the methods too: one of the applications is to prove that complete intersections in polyhedral geometry have nice topology, see the note below (published here).

• Relative Stanley--Reisner theory and Upper Bound Theorems for Minkowski sums
  Publications mathématiques de l'IHÉS, with Raman Sanyal (arxiv:1405.7368)

Here I extend and formalize some of the machinery of Stanley-Reisner complexes to relative complexes, especially manifolds, and apply this to solve the upper bound problem for Minkowski sums. This was quite useful for several later developments in the area, and contains several results beyond just Minkowski sums. I also later discovered that one could also prove an analogous lower bound using much the same methods (prodded by Yue Ren who asked me about it). You can find it here, and Yue´s followup here.

• The Hirsch conjecture holds for normal flag complexes
  2013, Math. of Operations Research, with B. Benedetti (arxiv.org:1303.3598)

We use the basic fact that locally convex sets of small intrinsic diameter in CAT(1) spaces are convex to prove the following result: Every flag and normal simplicial complex satisfies the nonrevisiting path conjecture, and in particular the diameter bound conjectured by Hirsch for all polyhedra. Furthermore, this paper contains a combinatorial proof of mine of the same result that previously appeared on G. Kalai's Blog (though I found the geometric proof before the  combinatorial one.) It is also rather cool as a standalone result.

 

Smaller notes are collected here.

• Topology of complete intersections of polyhedral hypersurfaces (shows that the stable complete intersection of tropical hypersurfaces is Cohen-Macaulay).