Professor Shrawan Kumar

You may download a PDF version of my curriculum vitae here.

Here is a brief summary of my works in PDF format.

Contact Information:

Department of Mathematics
University of North Carolina at Chapel Hill
Chapel Hill, NC 27599-3250, USA
Telephone: 919-962-9615, Telefax: 919-962-2568
e-mail: shrawan [at] email.unc.edu

Personal:


Born in Ghazipur, U.P., INDIA.

Academic Qualifications:

Positions:

Books Authored:

Publications:

    

  1. A remark on universal connections

    Math. Ann. 260, 453-462 (1982). [pdf]

  2. A G-minimal model for principal G-bundles

    Annales de L'Institut Fourier, Grenoble 32, 205-219 (1982). [pdf]

  3. Geometry of Schubert cells and cohomology of Kac-Moody Lie algebras

    J. Diff. Geometry 20, 389-431 (1984). [pdf]

  4. Rational homotopy theory of flag varieties associated to Kac-Moody groups

    In: "Infinite dimensional groups with applications" (ed. by V. Kac). MSRI (Springer-Verlag) publications Vol. 4, 233-273 (1985). [pdf]

  5. Cohomology algebra of Schubert varieties associated to Kac-Moody groups

    In: "Proceedings of the 1984 Vancouver Conference in Algebraic Geometry" (ed. by J. Carrell et. al.). CMS Conference Proceedings Vol. 6, 277-299 (1986). [pdf]

  6. A generalization of the Conner conjecture and topology of Stein spaces dominated by C^n

    Topology 25, 483-493 (1986). [pdf]

  7. The nil Hecke ring and cohomology of G/P for a Kac-Moody group G (with B. Kostant)

    Proc. Natl. Acad. Sci. USA 83, 1543-1545 (1986). [pdf]

  8. A homology vanishing theorem for Kac-Moody algebras with coefficients in the category O

    J. Algebra 102, 444-462 (1986). [pdf]

  9. Non-representability of cohomology classes by bi-invariant forms (Gauge and Kac-Moody groups)

    Commu. Math. Physics 106, 177-181 (1986). [pdf]

  10. The nil Hecke ring and cohomology of G/P for a Kac-Moody group G (with B. Kostant)

    Advances in Math. 62, 187-237 (1986). [pdf]

  11. T-equivariant K-theory of generalized flag varieties (with B. Kostant)

    Proc. Natl. Acad. Sci. USA 84, 4351-4354 (1987). [pdf]

  12. Extension of the category O^g and a vanishing theorem for the Ext functor for Kac-Moody algebras

    J. Algebra 108, 472-491 (1987). [pdf]

  13. Demazure character formula in arbitrary Kac-Moody setting

    Invent. Math. 89, 395-423 (1987). [pdf]

  14. Proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture

    Invent. Math. 93, 117-130 (1988). [pdf]

  15. Existence of certain components in the tensor product of two integrable highest weight modules for Kac-Moody algebras

    In: "Infinite dimensional Lie algebras and groups" (ed. by V.G. Kac). Advanced series in Mathematical Physics Vol. 7, World Scientific, 25-38 (1989). [pdf]

  16. A refinement of the PRV conjecture

    Invent. Math. 97, 305-311 (1989). [pdf]

  17. Bernstein-Gelfand-Gelfand resolution for arbitrary Kac-Moody algebras

    Math. Annalen 286, 709-729 (1990). [pdf]

  18. A new geometric invariant associated to the space of flat connections (with K. Guruprasad)

    Compositio Math. 73, 199-222 (1990). [pdf]

  19. T-equivariant K-theory of generalized flag varieties (with B. Kostant)

    J. Diff. Geometry 32, 549-603 (1990). [pdf]

  20. Proof of Wahl's conjecture on surjectivity of the Gaussian map for flag varieties

    In: " Algebraic groups" (ed. by S. Ramanan). Proceedings of the Hyderabad conference. Manoj Prakashan, 275-278 (1991). [pdf]

  21. A geometric realization of minimal rak k-type of Harish-Chandra modules for complex s.s. groups (with B. Kostant)

    In: "Kazhdan-Lusztig theory and related topics" (ed. by V. Deodhar). Contemporary Mathematics vol. 139 published by AMS, 249-264 (1992). [pdf]

  22. Proof of Wahl's conjecture on surjectivity of the Gaussian map for flag varieties

    American J. of Math. 114, 1201-1220 (1992). [pdf]

  23. Cohomology of quantum groups at roots of unity (with V. Ginzburg)

    Duke Math. J. 69, 179-198 (1993). [pdf]

  24. Finiteness of local fundamental groups for quotients of affine varieties under reductive groups

    Comment. Math. Helvetici 68, 209-215 (1993). [pdf]

  25. Equivariant cohomology with generalized coefficients (with M. Vergne)

    Ast\'erisque 215, 109-204 (1993). [pdf]

  26. Toward proof of Lusztig's conjecture concerning negative level representations of affine Lie algebras

    J. Algebra 164, 515-527 (1994). [pdf]

  27. Symmetric and exterior powers of homogeneous vector bundles

    Math. Annalen 299, 293-298 (1994). [pdf]

  28. Representations of quantum groups at roots of unity

    In: "Quantum topology" (ed. by D.N. Yetter). World Scientific, Singapore, 187-224 (1994). [pdf]

  29. Infinite Grassmannians and moduli spaces of G-bundles (with M. S. Narasimhan and A. Ramanathan)

    Math. Annalen 300, 41-75 (1994). [pdf]

  30. The nil Hecke ring and singularity of Schubert varieties

    In: "Lie theory and geometry (in honor of B. Kostant)" (ed. by J.-L. Brylinski et. al.). Progress in Math. vol. 123, Birkhauser, 497-507 (1994). [pdf]

  31. The nil Hecke ring and singularity of Schubert varieties

    Inventiones Math. 123, 471-506 (1996). [pdf]

  32. Shapovalov determinant for restricted and quantized restricted enveloping algebras (with G. Letzter)

    Pacific J. of Math. 179, 123-161 (1997). [pdf]

  33. Infinite Grassmannians and moduli spaces of G-bundles

    In: "Vector Bundles on Curves- New Directions," (ed. by M.S. Narasimhan). Springer Lecture Notes in Math. vol. 1649, 1-49 (1997). [pdf]

  34. Picard group of the moduli spaces of G-bundles (with M.S. Narasimhan)

    Math. Annalen 308, 155-173 (1997). [pdf]

  35. Fusion product of positive level representations and Lie algebra homology

    In: "Geometry and physics" (ed. by J.E. Andersen et. al.). Lecture Notes in Pure and Applied Mathematics vol. 184, Marcel Dekker, Inc., 253-259 (1997). [pdf]

  36. Positivity of the cup product in cohomology of flag varieties associated to Kac-Moody groups (with M.V. Nori)

    Inter. Math. Res. Not. 14, 757-763 (1998). [pdf]

  37. Frobenius splitting of cotangent bundles of flag varieties (with N. Lauritzen and J. Thomsen)

    Inventiones Math. 136, 603--621 (1999). [pdf]

  38. Homology of certain truncated Lie algebras

    In: "Recent Developments in Quantum Affine Algebras and Related Topics," (ed. by K.C. Misra et. al.). Contemporary Mathematics vol. 248, 309--325 (1999), published by AMS. [pdf]

  39. Frobenius splitting in characteristic zero and the quantum Frobenius map (with P. Littelmann)

    J. of Pure and applied Algebra 152, 201--216 (2000). [pdf]

  40. Frobenius splitting of Hilbert schemes of points on surfaces (with J.F. Thomsen)

    Math. Annalen 319, 797--808 (2001). [pdf]

  41. Algebraization of Frobenius splitting via quantum groups (with P. Littelmann)

    Annals of Mathematics 155, 491--551 (2002). [pdf]

  42. A conjectural generalization of the n! result to arbitrary groups (with J.F. Thomsen)

    Transformation Groups 8, 69--94 (2003). [pdf]

  43. Equivariant analogue of Grothendieck's theorem for vector bundles on \Bbb P^1

    In: "Perspectives in Geometry and Representation Theory (A tribute to C.S. Seshadri)," Hindustan Book Agency, 500--501 (2003). [pdf]

  44. The generalized triangle inequalities for rank 3 symmetric spaces of noncompact type (with B. Leeb and J. Millson)

    Contemporary Mathematics vol. 332, published by AMS, 171--195 (2003). [pdf]

  45. A new realization of the cohomology of Springer fibers (with J.F. Thomsen)

    In: "Proceedings of the International Conference on Algebraic Groups and Arithmetic" (Volume dedicated to M.S. Raghunathan on his sixtieth birthday), Narosa Publishing House, 119--125 (2004). [pdf]

  46. Induction functor in noncommutative equivariant cohomology and Dirac cohomology

    J. Algebra 291, 187--207 (2005). [pdf]

  47. Explicit determination of the Picard group of moduli spaces of semi-stable G-bundles on curves (with A. Boysal)

    Math. Annalen 332, 823--842 (2005). [pdf]

  48. Extensions of algebraic groups (with K-H. Neeb)

    In:"Studies in Lie Theory" (Volume dedicated to A. Joseph on his sixtieth birthday), Progress in Mathematics vol. 243, Birkhauser, 365--376 (2006). [pdf]

  49. Eigenvalue problem and a new product in cohomology of flag varieties (with P. Belkale)

    Inventiones Math. 166, 185--228 (2006). [pdf]

  50. On Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras

    J. of Am. Math. Soc. 21, 797--808 (2008). [pdf]

  51. The eigencone and saturation for Spin(8) (with M. Kapovich and J. Millson)

    Pure and Applied Math. Quarterly 5, 755--780 (2009). [pdf]

  52. Descent of line bundles to GIT quotients of flag varieties by maximal torus

    Transformation Groups 13, 757--771 (2008). [pdf]

  53. Special isogenies and tensor product multiplicities (with J. Stembridge)

    International Mathematics Research Notices, vol. 2007 (no. 20), 1--13 (2007). [pdf]

  54. Eigencone, saturation and Horn problems for symplectic and odd othogonal groups (with P. Belkale)

    J. Alg. Geom. 19, 199--242 (2010). [pdf]

  55. Characters of simplylaced nonconnected groups versus characters of nonsimplylaced connected groups (with G. Lusztig and D. Prasad)

    Contemporary Math. vol. 478, 99--101 (2009). [pdf]

  56. On positivity in T-equivariant K-theory of flag varieties (with W. Graham)

    International Mathematical Research Notices, vol. 2008, 1--43 (August, 2008). [pdf]

  57. A conjectural presentation of fusion algebras (with A. Boysal)

    In: "Algebraic Analysis and Around" (in honor of M. Kashiwara's sixtieth birthday). Advanced Studies in Pure Mathematics vol. 54, 95--107 (2009). [pdf]

  58. A generalization of Cachazo-Douglas-Seiberg-Witten conjecture for symmetric spaces

    Math. Annalen 346, 67--84 (2010). [pdf]

  59. Finiteness of the number of compatibly split subvarieties (with V. Mehta)

    International Mathematics Research Notices, vol. 2009 (no. 19), 3595--3597 (2009). [pdf]

  60. Tensor Product Decomposition

    In: Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010. [pdf]

  61. A generalization of Fulton's conjecture for arbitrary groups (with P. Belkale and N. Ressayre)

    Math. Annalen 354, 401-425 (2012). [pdf]

  62. Geometry of orbits of permanents and determinants

    Comment. Math. Helv. 88, 759--788 (2013). [pdf]

  63. An algebro-geometric realization of equivariant cohomology of some Springer fibers (with C. Procesi)

    J. Algebra 368, 70--74 (2012). [pdf]

  64. An approach towards the Kollar-Peskine problem via the Instanton Moduli Space

    Proceedings of Symposia in Pure Mathematics 86, 217-225 (2012). [pdf]

  65. Richardson varieties have Kawamata log terminal singularities (with K. Schwede)

    International Mathematics Research Notices, rns241, (2012), 23 pages [pdf]

  66. Positivity in T-Equivariant K-Theory of flag varieties associated to Kac-Moody groups

    J. Eur. Math. Soc. 19, 2469--2519 (2017). [pdf]

  67. A study of the representations supported by the orbit closure of the determinant

    Compositio Math 151, 292-312 (2015). [pdf]

  68. Dimension of zero weight space: an algebro-geometric approach (with D. Prasad)

    J. Algebra 403, 324--344 (2014). [pdf]

  69. A study of saturated tensor cone for symmetrizable Kac-Moody algebras (with M. Brown)

    Math. Annalen 360, 901-936 (2014). [pdf]

  70. A survey of the additive eigenvalue problem (with an appendix by M. Kapovich)

    Transformation Groups 19, 1051-1148 (2014). [pdf]

  71. Hitchin's conjecture for simply-laced Lie algebras implies that for any simple Lie algebra (with N. Bushek)

    Differential Geometry and its Applications 35, 210--223 (2014). [pdf]

  72. The multiplicative eigenvalue problem and deformed quantum cohomology (with P. Belkale)

    Advances in Mathematics 288, 1309--1359 (2016). [pdf]

  73. Connections between conjectures of Alon-Tarsi, Hadamard-Howe, and integrals over the special unitary group (with J.M. Landsberg)

    Discrete Mathematics 338, 1232--1238 (2015). [pdf]

  74. Additive Eigenvalue Problem

    EMS Newsletter (2015), 8 pages. [pdf]

  75. Components of V(ρ) ⊗ V(ρ) (with R. Chirivi and A. Maffei)

    Transformation Groups, 22, 645--650 (2017). [pdf]

  76. Representation ring of Levi subgroups versus cohomology ring of flag varieties

    Math. Annalen 366, 395--415 (2016). [pdf]

  77. Positivity in T-equivariant K-theory of flag varieties associated to Kac-Moody groups II (with S. Baldwin)

    Representation Theory (An electronic journal of the American Mathematical Society) 21, 35--60 (2017). [pdf]

  78. A complete set of intertwiners for arbitrary tensor product representations via current algebras

    Transformation Groups 24, 115--125 (2019). [pdf]

  79. Conformal blocks for Galois covers of algebraic curves (with Jiuzu Hong)

    Compositio Math. 159, 2191--2259 (2023) [pdf]

  80. On the faces of the tensor cone of symmetrizable Kac-Moody Lie algebras (with Nicolas Ressayre), Preprint (2024), 45 pages.

    [pdf]

  81. Representation ring of Levi subgroups versus cohomology ring of flag varieties II

    J. Algebra 556, 340--362 (2020). [pdf]

  82. Elliptic classes of Schubert varieties (with Richard Rimanyi and Andrzej Weber)

    Math. Annalen 378, 703--728 (2020). [pdf]

  83. Root components for tensor product of affine Kac-Moody Lie algebra modules (with Samuel Jeralds)

    Representation Theory (An electronic journal of the American Mathematical Society) 26, 825--858 (2022). [pdf]

  84. Subalgebra generated by ad-locally nilpotent elements of Borcherds Generalized Kac-Moody Lie algebras, Journal of Algebra 594, 307--312 (2022).

    [pdf]

  85. Category C_k of multi-loop algebra representations versus modular representations: Questions of G. Lusztig, Preprint (2022), 14 pages.

    [pdf]

  86. Representation ring of Levi subgroups versus cohomology ring of flag varieties III (with Jiale Xie), Journal of Pure and Applied Algebra 228, (2024).

    [pdf]

  87. Twisted conformal blocks and their dimension (with Jiuzu Hong), Math. Zeitschrift, 29 pages (2024).

    [pdf]

  88. Conjectural positivity of Chern-Schwartz-MacPherson classes for Richardson cells, International Mathematics Research Notices (2023), 12 pages.

    [pdf]

  89. Components of V(ρ) ⊗ V(ρ) and dominant weight polyhedra for affine Kac-Moody Lie algebras (with Sam Jeralds), Transformation Groups (2023), 22 pages.

    [pdf]

  90. Lie algebra cohomology of the positive part of twisted affine Lie algebras (with Jiuzu Hong), Proceedings of Symposia in Pure Mathematics, Volume 108, 301--336 (2024).

    [pdf]

  91. Positivity in T-Equivariant K-theory of partial flag varieties associated to Kac-Moody groups (with Joseph Compton), Preprint (2023), 25 pages.

    [pdf]

  92. Nonexistence of regular maps between homogeneous projective varieties, Preprint (2023), 6 pages.

    [pdf]

  93. Errata for the book ‘Conformal Blocks, Generalized Theta Functions and the Verlinde Formula’ by Shrawan Kumar, 5 pages (2024).

    [pdf]

  94. Erratum to the paper `Conformal blocks for Galois covers of algebraic curves' published in Compositio Mathematica, volume 159 (2023), 2291--2259 (with Jiuzu Hong), 2 pages (2024).

    [pdf]

Lecture Notes for Some Talks:

  1. Eigenvalue problem for Hermitian matrices and its generalization to arbitrary reductive groups

    (Talk given at AMS Sectional meeting held at Davidson College, March 2007) [pdf]

  2. Geometry of Schubert varieties and Demazure character formula

    (Talks given at the Hausdorff Research Institute for Mathematics, Bonn, Germany, April 2011) [pdf]

  3. Borel-Weil-Bott theorem and geometry of Schubert varieties

    (Lectures given at the MRC Meeting, Snowbird, Utah, June 2012) [pdf]

  4. Equivariant K-theory of flag varieties

    (Talks given at Jacobs University, Bremen (Germany), August 2012) [pdf]