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, you can also check our international
grad school.
A brief CV
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". 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. Since 2016, you
can also find me at the University of Leipzig in the spring semesters.
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.
Selected
Publications
Combinatorial 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)
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 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)
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.
Filtered
geometric lattices and Lefschetz Section Theorems over the tropical
semiring 2014, with Anders Bjoerner (arxiv:1401.7301)
The
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
Akizuki–Kodaira–Nakano.
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
matroids.
Relative Stanley--Reisner theory
and Upper Bound Theorems for Minkowski sums 2014, with Raman Sanyal (arxiv:1405.7368)
In
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.
Kalai's Blog.
Characterization 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. Padrol
(arxiv:1301.2960)
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
seventies.
Drafts
and Notes
May
get polished and published at some point
A
Note on the Simplex Cosimplex Problem 2011, European J of Combinatorics PDF
Kalai,
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 anyway.