Department of Mathematics

University of Illinois at Urbana-Champaign

1409 W. Green Street (MC-382)

Urbana, Illinois 61801

** Office: 331 Altgeld Hall Phone: (217) 333-3168
Fax: (217) 333-9576 e-mail: **

- M.A., Calcuta University (Pure Mathematics), 1970
- Research Assistant, Tata Institute of Fundamental Research, Bombay, India. 1971-75.
- M.S., Ohio State University (Pure Mathematics), 1978
- Ph.D., The University of Michigan, 1981.

Advisor: Prof. M. Hochster; Thesis Title: "Multiplicities over local rings"

Topic: Intersection Multiplicity on Local Rings.

1981-86 | Assistant Professor, University of Pennsylvania. |

1983-84 | Visiting Fellow, Institute for Advanced Study, Princeton. |

1986 | Assistant Professor, Department of Mathematics, University of Illinois. |

1987 | Post Doctoral Fellow, M.S.R.I., Berkeley (January-August). |

1988 | Associate Professor, Department of Mathematics, University of Illinois. |

1993 | Fellow, Danish Research Council (April-June). |

1993 | Visiting Professor-University of Utah (Fall). |

1997 | Professor, Department of Mathematics, University of Illinois. |

- Intersection Mulitplicity and related problems
- Canonical element conjecture and related problems
- Characteristic
*p*( > 0 ) techniques and its applications - Several homological questions and conjectures

Undergraduate: | Calculus in several variables, Linear Algebra |

Graduate: | Introductory level courses in commutative algebra and algebraic geometry |

Topic Courses: | 1. Local Cohomology 2. Smooth and Etale extensions These two topics courses are offered at regular intervals. |

- Sichang Lee, 1995
- Guided Claudia Miller in 1996-97, during the final year of her dissertation
- Jinjia Li, 2006
- Bart Snapp, 2007
- Jason McCullough, 2009
- Jesse Beder, 2012

- Sumner B. Myers Prize for the best Mathematics Department thesis,University of Michigan, 1981.
- Visiting member, Institute for Advanced Study 1983-1984.

- Multiplicities over local rings, Thesis, 1981, University of Michigan.
- Weak linking and Multiplicities, Journal of Pure and applied Algebra, Vol. 27, Number 2, February 1983, 111-130.
- Generalized Intersection Multiplicity of Modules, Trans. Amer. Math. Soc., Vol. 276, No. 2, April 1983, 657-669.
- Frobenius and multiplicities, Journal of Algebra, vol. 85, No. 2, December 1983, 424-448.
- Symbolic powers, intersection multiplicity and asymptotic behaviour of Tor, Journal of the London Math. Soc. (2) 28 (1983), 261-281.
- Generalized Intersection Multiplicities of Modules II, Proceeedings of AMS, Vol. 93, No. 2, February 1985, 203-204.
- (With M. Hochster & J. Mclaughlin) Modules of finite projective dimension with negative intersection multiplicities, Invent. Math. 79 (1985), 253-291.
- A note on a question of Bass, Trans. Amer. Math. Soc., Vol. 290, No 2, August 1985, 691-700.
- On the Canonical Element Conjecture, Trans. Amer. Math. Soc., Vol. 299, #2, Feb. 1987, 803-811.
- A Special Case of Positivity, Proceedings of the AMS, Volume 103, #2, June 1988, 344-346.
- Ext and Frobenius, Journal of Algebra, vol. 127, #1, Nov. 15, 1989, 163-177.
- Syzygies and Homological conjectures, Commutative Algebra, Hochster, Huneke, Sally Editors MSRI Publications, Springer-Verlag, 1989, 139-156.
- Modules of finite projective dimension with negative intersection multiplicities 11 (with M. Hochster and J. E. McLaughlin), preprint.
- A Note on Chow Groups and Intersection Multiplicity of Modules, The Journal of Algebra, Vol. 161, #1, October, 1993, 186-198.
- A Note on Chow Groups and Intersection Multiplicity of Modules 11, Journal of Algebra, Vol. 171, 370-382 (1995).
- On the Canonical Element Conjecture II, Math Zeit, Vol. 216, 379-388 (1994). [download pdf file]
- Dualizing Complex and the Canonical Element Conjecture, Journal of the London Math. Society, (2) 50 (1994) 477-487.
- A characterization of System of Parameters (with P. Roberts), Proc. of the AMS, Vol. 124, No. 3, March 1996, 671-675.
- Ext and Frobenius II, Journal of Algebra, vol. 186, 724-735 (1996). [download pdf file]
- The Canonical Element Conjecture--A brief survey and some new results, Commutative Algebra, Bruns. Herzog. Hochster. Vetter, International Conference, Vechta 1994, Vechtaer Universitiitschriften, 51-53.
- Dualizing Complex and the Canonical Element Conjecture II, Journal of the London Math Society, (2), 56(1997), 49-63. [download pdf file]
- A note on the Monomial Conjecture, Transactions of the AMS, vol. 350, #7, July 1998, p. 2871-2878. [download pdf file]
- A Theorem on Smoothness-Bass-Quillen, chow groups and Intersection Multiplicity of Serre, Transaction of the AMS, vol. 352, #4, 1635-1645. [download pdf file]
- A characterization of Gorenstein rings in characteristic
*p*( > 0 ), Proc. of the AMS, vol. 126, #6, June 1998, p. 1637-1643, 2000. [download pdf file] - Splitting of Local Cohomology of Syzygies of the residue field, J. of Algebra, 244, 168-185 (2001). [download pdf file]
- On Modules of finite projective dimension over complete intersection, Proc. of the AMS, Vol. 131, No. 1, 2002, pp. 113-116. [download pdf file]
- On Negativity of Higher Euler Characteristics, American Journal of Mathematics, 126 (2004), 1341-1354. [download pdf file]
- Intersection Multiplicity of Modules in the Positive Characteristics, Journal of Algebra, 280 (2004), 394-411. [download pdf file]
- A Special Case of Positivity II, Proceedings of the AMS, Vol.133, No.7, 1891-1896, 2005. [download pdf file]
- On Efficient Generation of Pull-back of T
_{pn}(-1), Illinois J. of Math, Vol. 51, No. 1, Spring 2007, 57-65. [download pdf file] - Intersection multiplicity of Serre on regular schemes, J. of Algebra, No. 319, 2008, 1530-1554. [download pdf file]
- (With Phil Griffith), Intersection multiplicities, canonical element conjecture and the Syzygy problem, Michigan Math. J. 57 (2008), 227-247 [download pdf file]
- A connection between two sets of conjectures, The Mathematics Student, Special Centenary Volume (2007) pp. 71-80, Indian Mathematical Society. [download pdf file]
- Intersection Multiplicity on Blow-ups, Journal of Algebraic Geometry, 22 (2013), 85-103.[download pdf file]
- The monomial conjecture and order ideals, Journal of Algebra, 383 (2013), 232-241. [download pdf file]
- On a consequence of the order ideal conjecture, Journal of Pure and Applied Algebra, 219 (2015), 482-487. [download pdf file]
- On modules of finite projective dimension, Nagoya Math. Journal, 219 (2015), 87-111. [download pdf file]
- The monomial conjecture and order ideals II, Journal of Algebra, 454 (2016), 123-138. [download pdf file]