TALK

Title: What is Algebraic Combinatorics and What Should It Be

Abstract: I will survey the history and trends of this relatively new field, and speculate about the future, and discuss what it SHOULD be, but also what it should NOT be.

What is (current) Algebraic Combinatorics: Pretentious, Pompous, Snotty, Overly Abstract

What Should it Be: Inclusive, Open-Minded, and Include ALL kinds of algebra, including "high school algebra" so efficiently implemented in Computer Algebra Systems

1. Gian-Carlo Rota's charming 1994 talk

2. Dr. Z.'s Opinion 5

3. Richard Stanley's Essay

4. The New JACO (under the leadership of Ilias Kotsireas), [Lots of great articles, e.g. one by Bill Chen and Amy Fu]

5. ALCO: Algebraic Combinatorics gone astray:

   Goulden-Jackson issue of ALCO

6. Kauers-Z masterpiece rejected by ALCO for the Goluden-Jackson issue

7. Opinion 178 of Dr. Z.

8. An example of "Elementary" (but interesting!) Algebraic Combinatorics

9. Symmetric Chain Decomposition of the Boolean Lattice:
   • Lubell's short proof of Sperner's Lemma

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   Du:=proc(C,n) local k,i: k:=nops(C): if k=1 then {[C[1],C[1] union {n}]} else {[op(C),C[k] union {n}],[seq(C[i] union {n},i=1..k-1)]}:fi:end: SCD:=proc(n) local S,C: option remember: if n=1 then RETURN({[{},{1}]} ): else S:=SCD(n-1): {seq(op(Du(C,n)), C in S)}:fi:end:

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   Ap:=proc(L,c): [op(L),c]:end: Du1:=proc(C) local i,k:k:=nops(C):if k=1 then [[seq(Ap(C[i],0),i=1..k),Ap(C[k],1)]]: else [[seq(Ap(C[i],1),i=1..k-1)],[seq(Ap(C[i],0),i=1..k),Ap(C[k],1)]]:fi:end: SCD1:=proc(n) local S,i:option remember:if n=1 then RETURN([[[0],[1]]]): else S:=SCD1(n-1):[seq(op(Du1(S[i])),i=1..nops(S))]:fi:end:

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10. X. Wen's breakthrough

11. (Shame on you: Patrice Ossona de Mendez, editor-in-chief of the European Journal of Combinatorics)

12. The Bottom Line (as Gian-Carlo Rota would say)

    What is (current) Algebraic Combinatorics: Pretentious, Pompous, Snotty, Overly Abstract, Elitist, Human-Centric

    What Should it Be: Inclusive, Open-Minded, and Include ALL kinds of algebra, including "high school algebra" so efficiently implemented in Computer Algebra Systems