Problem 1
A critical point where the Hessian is nonsingular; its index counts negative Hessian directions.
Faculty solution guide
Handles, Cell Attachments and Morse Inequalities
Accept equivalent mathematically correct reasoning. Require explicit definitions, theorem hypotheses, proof structure, verification, and limitations.
Problem 2
Identify the first invalid inference, missing hypothesis, quantifier error, unjustified limit interchange, or unproved existence claim; then supply a valid repair.
Problem 3
A complete response explicitly states the definitions, domain, hypotheses, exact target, and theorem dependencies for Handles, Cell Attachments and Morse Inequalities.
Problem 4
The argument must address relating critical points to homology and manifold construction with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in shape reconstruction, optimization landscapes, dynamical systems, relativity, and global analysis, verifies conditions, and names at least one limitation.
Problem 6
Accept a second valid proof, a counterexample showing necessity of a hypothesis, an independent estimate, or a reproducible finite check that is correctly described as supporting evidence.