I have fellowships for postdocs and students
at the Hebrew University of Jerusalem. Jerusalem has a great
combinatorics group, as well as
highly active research groups in other areas and beautiful
surroundings. 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, you can also check our international grad school.
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.
Stratifications and minimality of 2-arrangements Journal of Topology, (arxiv:1211.1224)
I address a problem by Suciu and Papadima: When
is the complement of an arrangement minimal, i.e., admits a CW model with as many i-cells
as the rational Betti number (so that every cell generates a homology class)? Using heavy
complex algebraic geometry, Dimca and Papadima showed (Ann. of Math.
2003) that this is true for complex hyperplane arrangements. I demonstrate that their theorem holds far more
generally for so called 2-arrangements, and that in particular only a combinatorial
condition on the arrangement needs to be imposed for minimality. As a main tool, I prove a
combinatorial Lefschetz Section Theorem for complements of 2-arrangements, and introduce Alexander
duality for combinatorial Morse flows.
Many projectively unique polytopes Inventiones Math, with G.M. Ziegler (arxiv:1212.5812)
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 realization is unique up to
projective transformation. This answers a problem posed by
Perles and Shephard in the sixties.
Hodge theory for combinatorial geometries 2015, with June Huh and Eric Katz (arxiv soon)
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.
Filtered geometric lattices and Lefschetz Section Theorems over the tropical semiring 2014, with Anders Bjoerner (arxiv:1401.7301)
purpose of this paper is to establish analogues of the classical
Lefschetz Section Theorem for smooth tropical varieties. More
precisely, we prove tropical analogues of the section theorems of
Lefschetz, Andreotti–Frankel, Bott–Milnor–Thom, Hamm–Lê and
Kodaira–Spencer, and the vanishing theorems of Andreotti–Frankel and
We start the paper by resolving a
conjecture of Mikhalkin and Ziegler (2008) concerning topological
properties of certain filtrations of geometric lattices, generalizing
earlier work on full geometric lattices by Rota, Folkman and Björner,
among others. This translates to a crucial index estimate for the
stratified Morse data at critical points of the tropical variety, and
it can also by itself be interpreted as a Lefschetz Section Theorem for
Relative Stanley--Reisner theory and Upper Bound Theorems for Minkowski sums 2014, with Raman Sanyal (arxiv:1405.7368)
this paper we settle long-standing questions regarding the
combinatorial complexity of Minkowski sums of polytopes: We give a
tight upper bound for the number of faces of a Minkowski sum, including
a characterization of the case of equality. We similarly give a (tight)
upper bound theorem for mixed faces of Minkowski sums. This generalizes
the classical Upper Bound Theorem of Stanley and McMullen, and has a
wide range of applications.
Our main tool is relative Stanley--Reisner theory,
a powerful generalization of the algebraic theory of simplicial
complexes inaugurated by Hochster, Reisner, and Stanley which we
develop here. The key feature of our setup is the ability to study
simplicial complexes under topological and additional
combinatorial-geometric restrictions. We illustrate this by providing
several simplicial isoperimetric and reverse isoperimetric inequalities.
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.
of polytopes via tilings with similar pieces 2012, Discrete and Computational Geometry (arxiv:1011.4651)
This presents solution to a problem (1995) of M.
Laczkovich. Picture a convex set K in euclidean space
decomposed into convex sets, some of which are similar to K.
What can K look like? M. Laczkovich proved that in the
2-dimensional case of this problem, K is a polygon.
Surprisingly, this is not true in higher dimensions, and the
proper generalization was left as an open problem in the
1995 paper. I extend his theorem to higher dimensions, and
give an example that the combined solutions give an optimal
answer to the problem.
A universality theorem for
projectively unique polytopes and a conjecture of
Shephard 2013, Isr. J. Math., with A.
We prove that
every algebraic polytope is the face of a projectively
unique polytope. We also provide a 5-polytope that is not the subpolytope of any stacked
polytope, which disproves a classical conjecture in
polytope theory, first formulated by Shephard in the
Kleinschmidt and Meisinger asked (1999) whether, for every
k, a highdimensional polytope must contain a face that is a
k-simplex, or its polar dual must contain such a face. This
note shows that at least if we ask the analogous question
for polytopal spheres, the answer is negative. The proof
uses some basic surgery of 3-manifolds. The original problem, whether
there exists a polytope with these properties, is still
open. Deeming the result as not interesting, I did not
prepare it for publication at the time of writing. Gil
thought otherwise, so I will prepare a publication version