My Research

Algebraic Geometry and Algebraic Number Theory: Motives, motivic cohomology, multiple polylogarithms, various multiple zeta functions and their special values, multiple harmonic sums, and arithmetic of function fields over finite fields.

 

 

Publications:

 

1.      Witten volume formulas for semi-simple Lie algebras. Integers 11A (2011) #A22.

2.      Joint with X. Zhou, Reducibility of Signed Cyclic Sums of Mordell-Tornheim Zeta and L-Values . To appear in J. Ramanujan Math. Soc. 2011.

3.      Alternating Euler sums and special values of Witten multiple zeta function attached to so(5). J. Aust. Math. Soc. 89(2011), pp. 419-430. Preprint MPIM2009-42, 2009.

4.      Mod p structure of alternating and non-alternating multiple harmonic sums. J. Théor. Nombres Bordeaux 23 (1)(2011), pp. 259-268.

5.      Finiteness of p-divisible sets of multiple harmonic sums. To appear in Annales des Sciences Mathématiques du Québec.

·           Examples of finite p-divisible sets of MZV series.

·          Finiteness conjecture of J(s|p).

·         Finiteness conjecture of J(s,t|p), J(r,s,t|p) and J(s,t,s,t|p).

·         The conjecture of J(s|2)={0}.

·         Finiteness of J(s|3).

·         Finiteness of J(s|5).  

  1. Joint with R. Tauraso, Congruences of alternating multiple harmonic sums. Preprint MPIM2009-67, 2009. J. Comb. and Number Theory 2 (2)(2010), pp. 129-159.
  2. Joint with X. Zhou, Witten multiple zeta values attached to sl(4). Preprint MPIM2009-41, 2009. Tokyo J. Math. 34 (2011), pp. 135-152.
  3. A note on colored Tornheim's double series. Preprint MPIM2009-66, 2009. Integers 10 (2010) A59, pp. 879-882.
  4. Multi-polylogs at twelfth roots of unity and special values of Witten multiple zeta function attfached to the exceptional Lie algebra g2. J. of Alg. and Its Applications, 9 (2010), pp. 327-337.
  5. Standard relations of multiple polylogarithm values at roots of unity. Documenta Mathematica 15 (2010), pp. 1-34.
  6. On a conjecture of Borwein, Bradley and Broadhurst.  J. Reine Angew. Math. 639 (2010), pp. 223–233.
  7. Renormalization of multiple q-zeta values, Acta Mathematica Sinica, English Series, 24 (2008), pp. 1593-1616. 
  8. An exotic shuffle relation for multiple zeta values. Arch. der Math.  91 (5)(2008), pp. 409-415
  9. Multiple polylogarithm values at roots of unity. C. R. Acad. Sci. Paris, Ser. I, 346 (2008) pp. 1029-1032. DOI: 10.1016/j.crma.2008.09.011
  10. Wolstenholme type theorem for multiple harmonic sums. International J. of Number Theory, 4 (2008), pp. 73-106.
  11. Multiple q-zeta functions and multiple q-polylogarithms. Ramanujan J. Math. 14 (2007), pp. 189-221.
  12. Bernoulli numbers, Wolstenholme's Theorem, and p5 variations of Lucas' Theorem. J. of Number Theory 123 (2007), pp. 18-26. 
  13. Goncharov's relations in Bloch's higher Chow group CH3(F,5). J. Number Theory, 124 (2007), pp. 1-25.
  14. Analytic continuation of multiple polylogarithms. Analysis Mathematica, 33 (2007), pp. 301-323.
  15. Variations of mixed Hodge structures of multiple polylogarithms. Canad. J. Math. 56 (6) (2004), pp. 1308-1338.
  16. Motivic cohomology of pairs of simplices. Proc. of the London Math. Soc. 88 (3) (2004), pp. 313-354.
  17. Multiple polylogarithms: analytic continuation, monodromy, and variations of mixed Hodge structures. In: ``Contemporary Trends in Algebraic Geometry and Algebraic Topology'': Proceedings of Conference in Memory of K.T. Chen and W.-L. Chow, Nankai Institute of Mathematics, edited by S.S. Chern, L. Fu and R. Hain, Nankai Tracts in Mathematics, Vol. 5, pp. 167-193,  World Scientific, 2002. 
  18. Motivic complexes of weight three and pairs of simplices in projective 3-space. Adv. Math. 161 (2001), pp. 141-208.
  19. Joint with A.B. Goncharov, Grassmannian trilogarithms. Compositio Math. 127 (2001), pp. 83-108.
  20. Joint with F. Xu: "Euler systems" in global function fields. Israel J. Math. 124 (2001), pp. 367-379.
  21. Remarks on a Hopf algebra for defining motivic cohomology. Duke Math. J. 103 (2000), pp. 445-458.
  22. Hopf algebra structure of generalized scissors congruence groups. PhD Thesis, Brown University, 1999
  23. Analytic continuation of multiple zeta functions. Proc. Amer. Math. Soc. 128 (2000), pp. 1275--1283.
  24. Joint with C. Li: Class number growth in a family of global function fields. J. of Algebra 200 (1998), pp. 141--154.
  25. Joint with C. Li: Iwasawa theory of Zpd-extensions over global function fields. Expos. Math. 15 (4) (1997), pp. 315--337.
  26. On root numbers connected with special values of L-functions over Fq(T). J. Number Theory 62 (2) (1997), pp. 307--321. 
  27. Joint With F. Xu: Maximal independent systems of units in global function fields. Acta Arith. 78 (1) (1996), pp. 1-10
  28. Genus fields, Conducors and relative integral bases for abelian function fields. Acta Mathematica Scientia, 16 (1) (1996), pp. 31-37.
  29. Class number relation between type (l,l,...,l) function fields over Fq(T) and their subfields. Science in China (Series A). 38 (6) (1995), pp. 674-682.
  30. On decomposition law in Carlitz-Kummer function fields. Chinese Science Bulletin. 39 (7) (1994), pp. 538-540.

 

Preprints:

 

1.      Joint with M. Conrad: Integral structures of multiple polylogarithms at roots of unity. Preprint, 2009.

 

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