Explore the fascinating world of face numbers in simplicial polytopes and triangulations of spheres in this 47-minute lecture by Isabella Novik. Delve into the upper bound problem, examining the maximum number of i-dimensional faces in various geometric structures. Investigate the properties of cyclic polytopes, simplicial complexes, and the Upper Bound Theorems. Compare the abundance of spheres to polytopes and analyze the proportion of neighborly polytopes. Examine centrally symmetric polytopes and spheres, exploring their unique properties and limitations. Discover recent developments in the field, including the existence of certain structures and open problems. Gain insights into this captivating area of mathematics through a comprehensive survey of known results and ongoing research questions.
Isabella Novik - Face Numbers - The Upper Bound Side of the Story
International Mathematical Union via YouTube
Overview
Syllabus
Intro
I. Basics on Polytopes
Motivation --- the Upper Bound Problem
Properties of cyclic polytopes
A digression: simplicial complexes and simplicials W
The Upper Bound Theorems
Comments
II. There are many more spheres than polytopes
What proportion of d-polytopes are li l-neighborly
Summary so far
III. Cs polytopes and cs spheres
The Upper Bound Problem for cs polytopes and sp.
How neighborly can a cs polytope be?
Cs-2-neighborliness of cs polytopes W
From non-neighborliness to f-numbers W
Does it exist?
It does exist!
Summary and Open problems
Taught by
International Mathematical Union
