I have fellowships for postdocs and students . If you are interested in the type of things I am doing (within some error margin) please send me an email (with an appropriate subject that reflects your intention to apply). If studying at Hebrew University interests you, I also supervise our international grad school.

Research Interest

Combinatorics. Currently, combinatorial constructions for manifolds and spaces, the topology and algebra of subspace arrangements, models of intersection theory in their various disguises (including skeletal rigidity), Hodge theory and Lefschetz theorems, moduli spaces of combinatorial objects (such as polytopes). Also, metric geometry, in its various disguises.

Books

I currently work with Geva Yashfe on a long-overdue update to Stanley's book on combinatorial commutative algebra. Updates will be found here.

Current Students

2016-- Or Raz (formerly MSc, now PhD)
2017-- Geva Yashfe (formerly MSc, now PhD)
2017-- Lukas Kuehne (PhD)
2018-- Johanna Steinmeyer (PhD)

Selected Publications

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

I prove that face rings of general triangulated spheres and manifolds satisfy the Hard Lefschetz theorem for suitable (and in particular generic) Artinian reductions, generalizing the classical version for rationally smooth projective toric varieties greatly beyond the setting of Kähler structures. This has several applications to combinatorial topology, in particular:

- 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.

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.

The resolution of the semistable reduction conjecture and log-smoothness over valuation rings

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.

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.

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

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.

(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.

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.

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.)

Notes

Smaller notes are collected here.

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

Karim Adiprasito

Contact

Open Positions

Research Interest

Books

Current Students