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.

I was born 1988 in Aachen/Aix-la-Chapelle, Germany. From 2007 to 2010, I studied Mathematics at TU Dortmund, where I received my Diploma in Mathematics with Tudor Zamfirescu. From 2008 to 2009, I spent an academic year at the Institute of Mathematics @ IIT Bombay.

Starting Fall 2010, I was a doctoral student at FU Berlin supported by a DFG scholarship via the Research Training Group MDS under the advision of Guenter Ziegler.

In May 2013, I defended my thesis entitled "Methods from Differential Geometry in Polytope Theory", which was awarded the Ernst Reuter prize of the Free University.

After being an EPDI fellow at IHES, a Minerva fellow at Hebrew University and a member at the Institute for Advanced Study, I joined the faculty of the Hebrew University in 2015 (tenured 2016)

In 2016, I won a grant by the European Research Council (ERC) and the Israel Science foundation (ISF), which support my research projects. In addition, I was awarded the European Prize in Combinatorics in 2015, the Klachky prize of the Hebrew University in 2017, was named a Knut & Alice Wallenberg fellow (at KTH) in 2018, and was awarded the 2019 New Horizons Prize in Mathematics. Starting 2020, I am found at the University of Copenhagen most of the time.

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.

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

2016-- Or Raz (formerly MSc, now PhD)

2017-- Geva Yashfe (formerly MSc, now PhD)

2017-- Lukas Kuehne (PhD)

2018-- Johanna Steinmeyer (PhD*)
*

- 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.

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.

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)

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

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 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

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)

(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)

- 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.)

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