Packet 4: Bonus 7

This theorem can be used to demonstrate that Morse functions are dense in C-infinity, since arbitrarily small perturbations can be made to remove critical values. For 10 points each:
[10h] Name this theorem from differential topology which asserts that almost every point in the image of a smooth map is a regular value.
ANSWER: Sard’s theorem [or Sard’s lemma or Morse–Sard theorem]
[10m] An equivalent statement of Sard’s theorem is that the set of critical values, or the set of values that are not regular, has this property. Sets with this property, like the Cantor set, have complements whose membership holds “almost everywhere.”
ANSWER: (Lebesgue) measure zero [or Lebesgue zero or null set; prompt on (Lebesgue) measurable; reject “empty”]
[10e] Sard’s theorem can be used to prove Brouwer’s theorem about these points, which, for single-variable functions, lie on the line y equals x. Contraction mappings have these points by a theorem of Banach.
ANSWER: fixed points [or invariant points]
<Harvard, Other Science> | D. Prelims 4 - Toronto B + Harvard + Michigan + Minnesota

HeardPPBE %M %H %
2110.9581%24%5%

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Conversion

TeamOpponentPart 1Part 2Part 3TotalParts
British ColumbiaColumbia A0101020ME
BruinChicago B001010E
CambridgeUCF001010E
Columbia BNYU B001010E
GeorgetownOhio State0000
Georgia Tech ASwarthmore0000
Illinois BGeorgia Tech C001010E
Johns HopkinsBrown001010E
MITWaterloo0101020ME
Maryland ACornell001010E
NYU AVanderbilt001010E
Northwestern AStanford A001010E
Northwestern BIllinois A001010E
Penn StateGeorgia Tech B0000
PittsburghIndiana001010E
RutgersVirginia0000
TexasWashU0101020ME
Toronto AUCLA0101020ME
UC Berkeley ANorth Carolina10101030HME
UC Berkeley BStanford B001010E
Virginia TechMaryland B001010E

Summary

TournamentEditionMatchHeardPPBE %M %H %
Main Site2026-04-17✓2110.9581%24%5%