?

1
???
? ? ?????? 2007-12-26
2
????

• ????
• ????
• ????
• ????

3
??????

• ???(1562-1633),??????
• ?????????
• ?Geometry????
• ???????????6?
• ???(1811-1882),?????
• ??????????
• ?Algebra????
• ????13??????
• ???????9?

4
?????????

• Algebra (???)?????
• ??????????
• ?????(?19?????)????
• ???????

5
??????????

• ??????2???????
• 13?????????????
• 16????3?4???????
• 18??? ??????
• 18???Carl Friedrich Gauss
• (1777-1855) ???
• ??????

6
???????

• 4???????200??,?
• ??????????????
• ?????,????????
• Lagrange???????????
• Niels H. Abel(1802-1829)??
• ?5?????????????
• ?????????

7
????????

• Evariste Galois(1811-1832)
• 17?????????????
• ???????Galois?????
• ???.??,5????????
• ???????????????
• ????(??????????),?????
• ?Galois??????????????????

8
Galois???

• 1829Galois???Cauchy??,???
• 1830?Fourier??,??Fourier??
• 1831??Poisson???????,??????
• 1832-5-30?Galois??1???
• ?2?????????
• 1846 Liouville????Galois???
• 1870 Jordan???????Galois??
• ??Galois?????????

9
Hermann Weyl ???

• Galois?????????????
• ??????,???????
• ????????????????
• ????????????????
• ????,???????????
• ????????????

10
????
11
??????

• 1843Hamilton???????
• 1846Cayley????????
• 1871Dedekind????
• 1872Klein?????????
• 1873Lie??Lie?
• 1894Cartan?????Lie??
• 1896Frobenius????????
• 1904Schur???????

12
??????

• 1905Wedderburn??????
• 1911Steinitz????
• 1921Noether????
• 1931Van der Waerden??????
• 1942Lefschetz??????
• 1946Weil?????????
• 1956Cartan-Eilenberg??????
• ??,????????
• ?????

13

AMS?????????

• 06??? Order, lattices, ordered algebraic
  structures
• 08??? General algebraic systems
  
• 12??? Field theory and polynomials
• 13??? Commutative rings and algebras
• 14??? Algebraic geometry
• 15??? Linear and multilinear algebra matrix
  theory
• 16??? Associative rings and algebras
• 17??? Nonassociative rings and algebras
• 18??? Category theory homological algebra
• 19??? K-theory
• 20??? Group theory and generalizations
• 22??? Topological groups, Lie groups
• 43??? Abstract harmonic analysis
• 55??? Algebraic topology
• 81??? Quantum theory
• 15/95

14

AMS?????????

• ????
• ????
• Lie??
• ????????
• K-??
• ??
• ?????

15
ArXiv?????????

• Algebraic Geometry (math.AG)
• Algebraic Topology (math.AT)
• Category Theory (math.CT)
• Commutative Algebra (math.AC)
• Group Theory (math.GR)
• K-Theory and Homology (math.KT)
• Mathematical Physics (math.MP)
• Operator Algebras (math.OA)
• Quantum Algebra (math.QA)
• Representation Theory (math.RT)
• Rings and Algebras (math.RA)
• 11/32
• 
  

16
ArXiv?????????
??? (math.CT) ???? (math.AC) ?? (math.GR)
K-????? (math.KT) ????? (math.QA) ???
(math.RT) ???? (math.RA)

17
????????
???? ?????
Kac-Moody??
?????K-?? ???????
??????? ????
???Hopf??

18
????????

• ???????????????????????????????????????????
  ???????????????

19
?????

• ???????????????????????
• ????????,?????????????
• ?????????????(??)???Lie??
• ?????????5?????????????Lie
• (?????)??????(Hopf?Kac-Moody?Poisson?
• Clifford???????????Koszul?Calabi-Yau)??,??

20
?????????????

• ??????????????????,?????
• ???????????
• ??????????????????????????
• ???????,???????????????
• ????????????????,?????????
• ???????????????????????
• ??????????????????????????
• ???,?????????????????

21
??????????

• ????
• ??????????? ?????
• ????????,?????
• ???????,??????
• ??????????????
• ?????????????????????
• ??

22
??

• Hopf?????????????,??????
• ????????????????????????
• ????
• 1980s Drinfeld???????????Hopf,?
• ???Yang-Baxter?????????????

23
???????

• ????Lie??????????
• ????Hopf??
• ????????Kac-Moody??
• ??Hopf??,???????
• ???Hopf???Frobenius??
• ???Hopf??H??Hopf???????H???
• ???3????????????,?????
• ??0?????G???????????G???
• Kaplanski????0???????Hopf??H??????
• ?????H???

24
????????

• 1970s Auslander??Brauer???
• ?????????????????????1990s
• Ringel ????Hall????Green????????
• ??Hall???????????????????
• Van den Bergh ?Hall??????????Kac-Moody
• ????????????????????

25
??????? ???????

• ??????????????????? ??Lie
  ?????Dynkin????Gabriel
  ?Ringel???????????????????????????????????Dynkin??
  ?????????????????????????

26
???
????
????
? ?
???
Hall??
CY??
?? ??

Poisson
????
???
?? ??
??
YM??
Koszul
??? ??
Lie??
Hopf ??
27
????
? ? ? ?
Serre??
????
?? CY YM Hopf Poisson
?
CY??
???
????
Hall??

28
Calabi-Yau??????

29
??????
??????

• ???????????????????????
• ?,?Borcherds??????Moonshine?????
• ?????????
• ?????????
• ????????
• ?????????

30
??????? ??????? ??????? ??????? ??????? ???????
31

• ????!
• ??????!