# Shi Shyr Roan

Shi-shyr Roan received a B.S. and a M.S. from the Mathematics Department of National Taiwan University in 1966 and 1968 respectively. He received a Ph.D. in mathematics from Brandeis University, U.S.A. in 1974. Over the years, Roan taught at various universities: U. C. Berkeley, U.S.A. (1974-1976), National Tsing Hua University, Taiwan (1976-1988), University of Virginia, Charlottesville, U.S.A. (1997-1998) , Clark University, Worcester, U.S.A. (1988-1989). He was a member of the Institute for Advanced Study, Princeton, U.S.A. (1981-1982), and of the Mathematical Sciences Research Institute, Berkeley, U.S.A. (1997-1998). As a visiting professor, he conducted researches at Research Institute for Mathematical Sciences, Kyoto University, Japan (1989-1990), and at Max-Planck-Institut fur Mathematik, Bonn, Germany (1990-1993). He joined the Institute of Mathematics, Academia Sinica, Taiwan in July, 1993, and retired in July, 2008. He has resided in Albany, California, U.S.A. since then.

Commentary by Shi Shyr Roan : In my professional career, I have never at any time held a post-doctoral position. In fact, I devoted my time and energy mainly on teaching and education in the first fourteen years. For my research career, I am most indebted to Ichiro Satake and Friedrich Hirzebruch, who had invited me to conduct research in Japan and Germany. The research environments and experience of those years (1989-1993) have been the most pivotal and productive period of my research development.

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** 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 Physic*s, Intern. Press, 1994, pp. 251-264.

2. On c_{1 }= 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 ) .

4. 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 symmetr*y, 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.

4. 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.

**MAJOR RESEARCH CONTRIBUTIONS **

1. Degeneration of K3 surfaces: Ref. Pubs. (1974-i983) 1

2. Mirror symmetry of Calabi-Yau 3-folds: Ref. Pubs. (1984-1993) 7

3. Crepant resolution of Gorenstein 3-orbifolds : Ref. Pubs. (1984-1993) 3, Pubs. (1994-2003) 2, Pubs. (1994-2003) 4

4. Onsager's algebra and loop algebra: Ref. Pubs. (1984-1993) 11, Pubs. (1994-2003) 18

5. Kramers-Wannier duality in Chiral Potts model: Ref. Pubs. (2004- ) 17