BIO
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 where I was hosted by Professor Anandavardhanan, with whom I organized a graduate seminar in Nevanlinna Theory and diophantine analysis. Since 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 spring 2011, I spent an extended period visiting UC Berkeley visiting (then) Miller Fellow Raman Sanyal. In winter 2011/2012, I spent 3 months at the Einstein Institute of Mathematics @ the Hebrew University of Jerusalem, hosted by Gil Kalai. 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.
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).
Research
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
- A Note on the Simplex Cosimplex Problem
2011, 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.