Solution: MATH 15 Category Theory
Answer: SESAME CHICKEN

Written by Justin Melvin

This puzzle refers to the excellent British quiz show Only Connect. Only Connect consists of four rounds:
  1. Connections (find the category that connects 4 clues)
  2. Sequences (given a sequence of three items, find the fourth in the sequence)
  3. Connecting Wall (given 16 items, sort them into four categories of four items and identify the categories)
  4. Missing Vowels (given a category and a member of that category with vowels removed, identify that member, e.g. NLY CNNCT->ONLY CONNECT)
The first step is to partition the interactive walls into groups of four (which are confirmed when they are found) and identify the categories:
ACEKINGQUEENBOOK (of bible)
VenturaBabarFlash GordonJob
AttorneyJulienJazzJudges
BandageLouieInnuendoRuth
HardwareMufasaThe WorksNumbers
SIZEAGEOPINIONCharacters from titles that are QUESTIONS
ArianaBronzeMajorityCarmen Sandiego
Cavalieri SerStoneSecondBaby Jane
MartinIcePopularDr. Vink
ForesGoldenDissentingColin Mochrie
EYETOE (one little piggie.... nursery rhyme)____ WOOLSECRET ___
ProvidenceMarketFleeceSquirrel
SauronHomeBritchAgent
NeedleRoast beefBelliesWeapon
HurricaneNoneLocksSanta
WASHINGTON townsfamous ADAMSJEFFERSON airplane membersHOUSE (actors last names)
IndexWestSlickCameron
StartupDriverBalinCuddy
ForksSavageSpenceWilson
Black DiamondSmithCasadyForeman
EYE OF HORUSWATER ___HORNED VIPERCRAVEN
Theosodia book coverClosetUktenaScream
Spelunky itemBuffaloCerastesSwamp Thing
Pyramids of MarsSkiHornsmanCursed
The Alan Parsons Project albumGateSidewinderShocker
Raiders(Shirley) TempleLast _Scared
StablerGinger AleDitchSacred
MossCherryStrawCedars
BranchGrenadineSupperCadres
CasperLimeResortE-cards
TWOTHREE (Musketeers + D'artangan)___ FIVESPINELESS
BraceDartHighPatrick Star
TwinAramJacksonLowly Worm
PairPortTakeSpiral-bound notebook
CoupleAthoSlaughterhouseTurbo
(around) THE WORLDJUDGEMENT_ THE SUNYELLOW
Daft PunkDoomBlister InTaxi
Jules VerneVerdictHere ComesCanary
Yo-yoSentenceStaring AtBanana
Red Hot Chili PeppersFindingBrighter ThanLemon
The key insight is that three of the categories themselves in each wall form a sequence (this is the 'Round 2' column) and the extras form two 'Round 1' instances. These can be uniquely placed in the blanks.
First in sequenceSecond in sequenceThird in sequenceFourth in sequenceIndexSetRound 1 extraRound 2 extraction
ACEQUEENKINGJACK3Playing cardsBOOKC
OPINIONSIZEAGESHAPE2Adjective orderQUESTIONH
EYETOEWOOLTONGUE3MacbethSECRETN
WASHINGTONADAMSJEFFERSONMADISON5PresidentsHOUSES
RAIDERSTEMPLELASTKINGDOM5Indiana JonesSCAREDD
TWOTHREEFIVESEVEN1Prime numbersSPINELESSS
THE WORLDJUDGEMENTTHE SUNTHE MOON2TarotYELLOWH
EYE OF HORUSWATERHORNED VIPERTWISTED FLAX4Only Connect hieroglyphsCRAVENS
The 'Round 2' extraction is CHNSDSHS, which is a 'Round 4' missing vowels phrase.
The 'Round 1' sets are BOOK / QUESTION / SECRET / HOUSE (can be preceeded by 'Open') and SCARED / SPINELESS / YELLOW / CRAVEN (synonyms for each other).
The 'Round 2' answer can be re-vowelated to CHINESE DISHES, the category for the final answer.
The 'Round 1' ones, "can be preceeded by 'open' " + "synonym for scared" can be answered by
SESAME CHICKEN

Credit

Credit goes to Terry Child for the Connecting Wall code.