Exotic structures and the limitations of certain analytic methods in geometry

TL;DR

This survey explores negatively curved exotic structures in differential and PL geometry, highlighting how these structures reveal limitations of certain analytic methods in the field.

Contribution

It provides a comprehensive review of exotic structures and their impact on the applicability of analytic techniques in geometry.

Findings

Negatively curved exotic structures challenge existing analytic methods.

Certain geometric invariants are affected by exotic structures.

Limitations of analytic methods are demonstrated through specific examples.

Abstract

In this survey we review some results concerning negatively curved exotic strucutres (DIFF and PL) and its (unexpected) implications on the limitations of some analytic methods in geometry. This article is dedicated to the memory of Armand Borel.