阮希石

  阮希石畢業於臺灣大學數學系 (1966年學士, 1968年碩士), 以及美國布蘭代斯大學數學研究所(1974年博士)。之後曾 任教於美國加州柏克萊大學 (1974-1976), 台灣清華大學 (1976-1988) 及美國克拉克大學 (1988-1989), 曾受聘於美國 普林斯敦高等研究院(1981-1982), 日本京都大學數理研究所 (1989-1990), 德國馬克斯布朗克數學所(1990-1993) 為客座 研究教授。於1993年7月受聘為中研院數學所研究員。

  阮希石的研究領域為代數幾何, 複數幾何, 數學物理及 表現論, 近年來的工作著重於與物理相關之幾何問題。   

 

 



 

FIELDS OF RESEARCH

Algebraic Geometry:    Complex variety with trivial canonical bundle, Orbifold geometry, Toric geometry.

Mathematical Physics :   2-dimensional solvable statistical mechanics, Quantum integrable systems, String theory.



                                                   PUBLICATIONS (2004 -   )

1. On hypersurface quotient singularity of dimension 4, Intern. J. Math. & Math. Sciences , 48 (2004) 2547-2581; math.AG/0011151, (with L. Chiang).

2. On branching indices of affine A-D-E diagrams : a geometrical characterization by Kleinian singularities, math.AG/0405580.

3. Intermediate Jacobian and some arithmetic properties of Kummer-surface-type CY 3-folds, math.AG/0407193.

4. Bethe Ansatz equations of XXZ model and q-Sturm-Liouville problems, math-ph/0407033 (with Mourad E. H. Ismail and S. S. Lin ).

5. Chiral Potts rapidity curve descended from six-vertex model and symmetry group of rapidities, J. Phys. A: Math. Gen. 38 (2005) 7483-7499; cond-mat/0410011 .

6. Structure of certain Chebyshev-type polynomials in Onsager's algebra representation, Journal of Computational and Applied Mathematics 202 (2007) 88-104; math-ph/0501045.

7. The Onsager Algebra Symmetry of τ(j) -matrices in the Superintegrable Chiral Potts Model, J. Stat. Mech. (2005) P09007; cond-mat/0505698.

8. Bethe ansatz and symmetry in superintegrable chiral Potts model and root-of-unity six-vertex model, in Nankai Tracts in Mathematics Vol. 10, Differential Geometry and Physics, ed, Mo-Lin Go and Weiping Zhang (World Scientific, Singapore 2006), pp 399-409; cond-mat/0511543.

9. The Q-operator for root-of-unity symmetry in six vertex model, J. Phys. A: Math. Gen. 39 (2006) 12303-12325 ; cond-mat/0602375.

10. Fusion operators in the generalized τ(2) -model and root-of-unity symmetry of the XXZ spin chain of higher spin, J. Phys. A: Math. Theor. 40 (2007) 1481-1511; cond-mat/0607258.

11. The Q-operator and functional relations of the eight-vertex model at root-of-unity η = 2mK/N for odd N, J. Phys. A: Math. Theor. 40 (2007) 11019-11044; cond-mat/0611316 .

12. On Q-operators of XXZ spin chain of higher spin, cond-mat/ 0702271.

13. The transfer matrix of superintegrable chiral Potts model as the Q-operator of root-of-unity XXZ chain with cyclic representation of Uq(sl2), J. Stat. Mech. (2007) P09021; arXiv: 0705.2856.

14. On the equivalent theory of the generalized τ(2) -model and the chiral Potts model with two alternating vertical rapidities, arXiv: 0710.2764.

15. Bethe equation of τ(2)-model and eigenvalues of finite-size transfer matrix of chiral Potts model with alternating rapidities, J. Stat. Mech. (2008) P10001; arXiv.0805.1585.

16. On τ(2)-model in chiral Potts model and cyclic representation of quantum group Uq(sl2), J. Phys. A: Math. Theor. 42 (2009) 072003; arXiv:0806.0216.

17. Duality and symmetry in chiral Potts model, J. Stat. Mech. (2009) P08012; arXiv:0905.1924.

18. Eigenvectors of an arbitrary Onsager sector in superintegrable  τ(2)-model  and chiral Potts model, arXiv:1003.3621

19. Toric representation and positive cone of Picard group and deformation space in mirror symmetry of Calabi-Yau hypersurfaces in toric varieties,
 arXiv:1106.2323

20. Quantum group theory in τ(2)-model, duality of τ(2)-model and XXZ-model with cyclic Uq(sl2)-representation for q^n =1, and chiral Potts  model,
arXiv:1206.4356

21. Fermat surface and group theory in symmetry of rapidity family in chiral Potts model,  arXiv:1301.5260



                                                   PUBLICATIONS (1994- 2003 )

1. Topological properties of Calabi-Yau manifolds, in: R. Penner and S. T. Yau (Ed.), Perspectives in Mathematical Physics, Intern. Press, 1994, pp. 251-264.

2.      On c1 = 0 resolution of quotient singularity, Intern. J. Math. 5 (4) (1994) 523-536.

3. A stochastic approach to Euler Poincare number of loop space of a developable orbifold, J. Geom. Phys. 16 (1995) 71- 98, (with R. Leandre ) .

3.      Minimal resolutions of Gorenstein orbifolds in dimension three, Topology 35 (1996) 489-508.

5. A note on ODEs from mirror symmetry, in: Functional analysis on the eve of the 21st century-in honor of the 80th birthday of I.M. Gelfand, Vol. II , Birkhäser Boston, Inc.; 1996, pp. 301- 32 ; hepth/9407192 , ( with A. Klemm,  B. H. Lian  and  S. T. Yau  ).

6. Toric geometry in mirror geometry, Academia Sinica Math.  preprint R940907-1, 1994.

7. Picard groups of hypersurfaces in toric varieties, Publ. RIMS Kyoto Univ. 32 (1996) 797-834;  arXiv:1106.1615

8. Mirror symmetry of elliptic curves and Ising model, J. Geom. Phys. 20 (1996) 273-296; hep-th/9509051.

9. Weighted projective surfaces of splitting-type, MIAS 95-5, 1995,  ( with Y. Goto ) .

10. Bethe Ansatz for Heisenberg XXX model, MIAS 95-6 , 1995, cond-mat /9509183 ( with     S.S. Lin ).

11. Orbifold Euler characteristic, in:  B. Greene  and  S. T. Yau  (Ed.),  Mirror Symmetry II, Amer. Math. Soc., Intern.  Press, 1997,  pp. 129-140 .

12. Yang-Baxter relation for sl(2)-Yangian, MIAS 96-2, 1996, ( with M. L. Ge and  S.S. Lin ).

13. Crepant resolution and fibred CY manifolds, MIAS 97-1, 1997.

14. Calabi-Yau fourfolds for M- and F-Theory compactifications,  Nucl. Phys.B 518, 3 (1998) 515-574;   hep-th/ 9701023,  ( with A. Klemm,  B. H. Lian and  S. T. Yau  ).

15. Gauge symmetry and integrable models, MSRI Preprint 1997-88, hep-th/ 9709073, ( with     O. K. Pashaev and S. S. Lin ).

16. Schwarz's function and chiral Potts model,  Helv. Phys. Acta 71 (1998) 242-261.

17. The algebraic structure of the Onsager algebra, Czech. J. Phys., Vol. 50 No. 1 (2000) 37-44;  cond-mat/0002418, (with E. Date) .

18. The structure of quotients of the Onsager algebra by closed ideals,  J. Phys. A: Math. Gen. 33 (2000) 3275-3296;  math.QA/9911018, (with E. Date) .

19. Heisenberg and modular invariance of N=2 conformal field theory,  Intern. J. Mod. Phys. A, 15 (2000) 3065-3094; hep-th/9902198.

20. Algebraic geometry approach to the Bethe equation for Hofstadter type models , J. Phys. A: Math. Gen. 35 (2002) 5907-5933; cond-mat/9912473,  (with S. S. Lin) .

21. Orbifolds and finite group representations, Intern. J. Math. & Math. Sciences  26:11 (2001) 649-669, math.AG/0007072, (with L. Chiang).

22. Algebraic geometry and physics, in: T.Sunada, P. W. Sy and Y. Lo (Ed.), Proceedings of the Third Asian Math. Conf. 2000, World Scientific 2002, pp. 455- 482 ; math-ph/0011038.

23. The superintegrable chiral Potts quantum chain and generalized Chebyshev polynomials, in: S.Pakuliak and G.von Gehlen (Ed.),  Integrable Structure of Exactly Solvable Two-Dimensional Models of Quantum Field Theory, Kluwer Academic Publ., Dordrecht 2001, pp. 155-172; hep-th/0104144,  (with G. von Gehlen).

24. Baxter's T-Q relation and Bethe Ansatz of discrete quantum pendulum and sine-Gordon model, hep-th/0105140, ( with S. S. Lin ).

25. Algebraic geometry in discrete quantum integrable model and Baxter's T-Q relation, Czech. J. Phys., 52 (2002) 737-820; hep-th/0107047, (with S. S. Lin ).

26. Characters of representations of affine sl(2|1) algebra and N=2 superconformal algebra, preprint, 2002, (with V. Kac and M. Wakimoto).

27. Algebraic geometry and Hofstadter model, Intern. J. Mod. Phys. B, 16 (2002) 2097- 2106; math-ph/0206014,  (with S. S. Lin ).

28. Crepant resolutions of C^n/A_1(n) and flops of n-folds for n= 4, 5 , in: N. Yui and J. D. Lewis (Ed.), Calabi-Yau varieties and mirror symmetry, Fields Institute Comm. 38 , 2003, Amer. Math. Soc. Pp. 27-41 ; math.AG/0208052, (with L. Chiang).

29. Quantum reduction for affine superalgebras, Comm. Math. Phys. 241 (2003) 307-342; math-ph/0302015, (with V. Kac and M. Wakimoto).

30. Rational curves in rigid Calabi-Yau three-folds, math.AG/0312399.

 

                                                  PUBLICATIONS (1984- 1993 )

1.      Homogeneous vector bundles over projective varieties,  Chinese J. Math. 12 (1984) 137-169.

2.  On Ricci flat 3-fold ,  Acta Mathematica Sinica, New Series, Beijing 3 (3) (1987) 256-288,  ( with S. T. Yau  ).

3.      On the generalization of Kummer surfaces,  J. Diff. Geom. 30 (1989) 523 - 537.

4.  On the Calabi-Yau orbifolds in weighted projective spaces,  Intern. J. Math. 1 (2) (1990) 211-232.

5.      Excitation spectrum and phase structure of chiral Potts model,  Phys. Lett. A 150 (1990) 347-354, (with B. M. McCoy ).

6.  Geometric singularities and spectra of Landau-Ginzburg models, Comm. Math. Phys. 142 (1991) 245-259, ( with B. R. Greene and S. T. Yau ).

7.      The mirror of Calabi-Yau orbifold,  Intern. J. Math. 2  (4)  (1991) 439-455.

8.  A characterization of "rapidity" curve in chiral Potts model, Comm. Math. Phys. 145  (1992) 605-634.

9.      Modular invariance of manifolds with SU(n) holonomy, Intern. J. Math. 3 (3) (1992) 359-395.

10.  Vafa's formula and equivariant K-theory,  J. Geom. Phys. 10 (1993) 287-294, (with K. Ono ).

11.  Onsager's algebra, loop algebra and chiral Potts model, Preprint Max-Planck-Inst. Fur Math., Bonn, MPI 91-70, 1991.

12.   Abelian integral in chiral Potts model, Preprint Max-Planck-Inst. fur Math., Bonn, MPI 92-27, 1992.

13.  Mirror symmetry and Arnold's duality, Preprint Max-Planck-Inst. fur Math., Bonn,  MPI 92-86,1992; Zap. Nauchn. Sem. POMI  234 (1996) 153-176,        (English): J. Math. Sci. (New York), 94 (1999), 1237-1253.

14. The geometry of Calabi-Yau orbifolds, in: S.T Yau (Ed.), Essays on mirror manifolds, Intern. Press, Hong Kong, 1992, pp. 342-348.

15.  New constructions of mirror manifolds: probing moduli space far from Fermat points. In: S.T Yau (Ed.), Essays on mirror manifolds, Intern. Press, Hong       Kong, 1992, pp. 408-448 (with B. R. Greene and M. R. L. Plesser).

16.  Topological coupling of Calabi-Yau orbifold,  J. Group Theory in Physics 1 (1993) 83- 103.

17.  A new example of N=2 supersymmetric Laudau-Ginzburg theories: the two ring case,  Mod. Phys. Lett. A 8 (31) (1993) 2989-2999; math-ph/0209061        ( with  A. M. Perelomov ).

 

                                                  PUBLICATIONS (1974- 1983 )

1.      Degeneration of K3 and abelian surfaces,  Ph.D. thesis of  Brandeis University , 1974.

2.  A remark on Grauert's theorem,  Bull. Inst. Math. Academia Sinica 5 (1977)  167-170.

3. Subvarieties of codimension 2,  Bull. Inst. Math. Academia Sinica 6 (1978) 189-196.

5.      The configuration of singular varieties,  Bull. Inst. Math. Academia Sinica 7 (1979) 419-429.

5.      The fundamental group of residual space of a curve in an algebraic surface, Chinese J. Math. 7 (1979) 163-175.

6.  Branched covering spaces,  Chinese J. Math. 7 (1979) 177-206.

7.      Picard-Lefschetz transformation on homotopy group,  Chinese J. Math. 9 (1981) 81-90.

8.      Subbundles of the tangent bundle of the complex projective spaces, Bull. Inst. Math. Academia Sinica 9 (1981) 1-28.

9.      On elliptic fiber surfaces,  Chinese J. Math. 11 (1983) 129-142.

10. On flat fiber surfaces,  Chinese J. Math. 11 (1983) 157-180.