Algebra/Topology Seminar

Speaker: Karim Adiprasito (Hebrew University)

Title: T < 4E and the standard conjectures beyond positivity

Abstract: Assume X is a simplicial complex on n vertices that allows for 
an embedding into R^2d? How many d-dimensional simplices can it have?

This is a rather fundamental question. For d=1, it goes back to 
Descartes and Euler, who established that planar simple graphs have at 
most three edges for every vertex. The case d>1 remained elusive, and 
notoriously resisted modern topological and combinatorial techniques. I 
will discuss how this question is related to a deep problem in algebraic 
geometry, Grothendieck's hard Lefschetz conjecture, and indicate a new 
method to prove this conjecture for a case that was previously out of 
reach: beyond projectivity of the underlying variety. This has several 
interesting implications:

- We prove that for a simplicial complex that PL embeds into R^2d, the 
number of d-dimensional simplices exceeds the number of 
(d-1)-dimensional simplices by a factor of at most d+2. This generalizes 
a result going back to Descartes and Euler, and resolves the 
Gruenbaum-Kalai-Sarkaria conjecture.

- A consequence of this is a high-dimensional version of the celebrated 
crossing number inequality of  Ajtai, Chvatal, Leighton, Newborn and 
Szemeredi: For a PL map of a simplicial complex X into R^2d, the number 
of pairwise intersections of d-simplices is at least

         f_d^(d+2)(\varDelta)/(d+3)^(d+2)f_{d-1}^{d+1}(\varDelta)

         provided f_d(\varDelta)> (d+3)f_{d-1}(\varDelta).

- We 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.

- We verify a conjecture of Kuehnel, proving tight lower bounds on the 
complexity of a triangulated manifold in terms of its Betti number.

I intend to assume almost no background, and give a gentle introduction 
to the theory.