Karim A. Adiprasito

Subdivisions, shellability, and collapsibility of products

We prove that the second derived subdivision of any rectilinear triangulation of any convex polytope is shellable. Also, we prove that the first derived subdivision of every rectilinear triangulation of any convex 3-dimensional polytope is shellable. This complements Mary Ellen Rudins classical example of a non-shellable rectilinear triangulation of the tetrahedron. Our main tool is a new relative notion of shellability that characterizes the behavior of shellable complexes under gluing.As a corollary, we obtain a new characterization of the PL property in terms of shellability: A triangulation of a sphere or of a ball is PL if and only if it becomes shellable after sufficiently many derived subdivisions. This improves on PL approximation theorems by Whitehead, Zeeman and Glaser, and answers a question by Billera and Swartz.We also show that any contractible complex can be made collapsible by repeatedly taking products with an interval. This strengthens results by Dierker and Lickorish, and resolves a conjecture of Oliver. Finally, we give an example that this behavior extends to non-evasiveness, thereby answering a question of Welker.

Universality Theorems for Inscribed Polytopes and Delaunay Triangulations

We prove that every primary basic semi-algebraic set is homotopy equivalent to the set of inscribed realizations (up to Möbius transformation) of a polytope. If the semi-algebraic set is, moreover, open, it is, additionally, (up to homotopy) the retract of the realization space of some inscribed neighborly (and simplicial) polytope. We also show that all algebraic extensions of

Derived subdivisions make every PL sphere polytopal

{\mathbb {Q}}

The universality theorem for neighborly polytopes

Q are needed to coordinatize inscribed polytopes. These statements show that inscribed polytopes exhibit the Mnëv universality phenomenon. Via stereographic projections, these theorems have a direct translation to universality theorems for Delaunay subdivisions. In particular, the realizability problem for Delaunay triangulations is polynomially equivalent to the existential theory of the reals.

Combinatorial stratifications and minimality of 2-arrangements

Using an intuition from metric geometry, we prove that any flag normal simplicial complex satisfies the nonrevisiting path conjecture. As a consequence, the diameter of its facet-ridge graph is smaller than the number of vertices minus the dimension, as in the Hirsch conjecture. This proves the Hirsch conjecture for all flag polytopes and, more generally, for all (connected) flag homology manifolds.

A Geometric Lower Bound Theorem

We give a simple proof that some iterated derived subdivision of every PL sphere is combinatorially equivalent to the boundary of a simplicial polytope, thereby resolving a problem of Billera (personal communication).

Convexity of complements of tropical varieties, and approximations of currents

In 1967, Chillingworth proved that all convex simplicial 3-balls are collapsible. Using the classical notion of tightness, we generalize this to arbitrary manifolds: We show that all tight simplicial 3-manifolds admit some perfect discrete Morse function. We also strengthen Chillingworths theorem by proving that all convex simplicial 3-balls are non-evasive. In contrast, we show that many non-evasive 3-balls are not convex.

A note on the simplex-cosimplex problem

In this note, we prove that every open primary basic semialgebraic set is stably equivalent to the realization space of a neighborly simplicial polytope. This in particular provides the final step for Mnëv‘s proof of the universality theorem for simplicial polytopes.