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# Mousumi Mandal

INdAM-COFUND Incoming fellow between 2012-01-10 and 2013-02-15

### Project information

#### Title

Castelnuovo-Mumford Regularity: Homological and Computational Aspects

#### Partner organisation

Dipartimento di Matematica

Università di Genova

Via Dodecaneso 35

16136 Genova, Italy

#### Abstract

The research training project is concentrated around an important topic in the international research: the study of the Castelnuovo-Mumford regularity of a finitely generated module over a Noetherian commutative ring.

The Castelnuovo-Mumford regularity is a kind of universal bound for important invariants of graded k-algebras such as the maximum degree of the syzygies and the maximum non-vanishing degree of the local cohomology modules.

A central problem in Commutative Algebra and Algebraic Geometry is the Eisenbud-Goto conjecture on regularity, multiplicity and codimension of domains. We can try to prove the conjecture for specific classes of graded domains. In particular we are interested to characterize the initial ideals (or the Groebner bases) of prime ideals. This problem has a strong interest in our project because there is a clear connection between the Castelnuovo-Mumford regularity of an ideal and those of its initial ideal. We try to find suitable extension of Eisenbud-Goto Conjecture for local domains.

We try to see whether the regularity of the associated graded ring can be bounded by a polynomial function (possibly linear) of the multiplicity and the codimension.

Another problem is to find some uniform bounds for all the coefficients of the Hilbert polynomial using regularity bounds. We try to give some optimal uniform bounds for all the coefficients of the Hilbert polynomial corresponding to m-primary

ideals.

### Fellow information

#### Research interests

- Castelnuovo-Mumford Regularity
- Hilbert Coefficients and Blowup Algebras
- Combinatorial Commutative Algebra

#### Previous positions, awards

*September 2011-January 2012*: Visiting Fellow, Tata Institute of Fundamental

Research, Mumbai, India*April 2011-August 2011*: Research Associate, IIT Bombay, Mumbai, India*January 2006-March 2011*: Research Scholar, IIT Bombay, Mumbai, India

#### Publications, preprints, other works

- M. Mandal, J. K. Verma,
*On the Chern number of I-admissible filtrations of*, to appear in Journal of Commutative Algebra (2012)

ideals - M. Mandal, S. Masuti, J. K. Verma,
*Normal Hilbert polynomials: a survey*, to appear in Proceedings of the International

Conference in Commutative Algebra and Algebraic Geometry (2012) - S. Goto, J. Hong, M. Mandal,
*The positivity of the first coefficients of normal*, Proc. Amer. Math. Soc. 139 (2011), 2399-2406

Hilbert polynomials - S. Goto, M. Mandal, J. K. Verma,
*Negativity of the Chern number of parameter ideals*, Proceedings of the International Conference in

Algebra at Aligarh Muslim University - M. Mandal, B. Singh, J. K. Verma,
*On some conjectures about the Chern numbers of*, Journal of Algebra. 325 (2011), 147-162

filtrations - M. Mandal, J. K. Verma,
*On the Chern number of an ideal*, Proc. Amer. Math. Soc. 138 (2010), 1995-1999

#### Conferences, schools, other events

*Chern number formulas and consequences*

Poster presented at*Midwest Commutative Algebra and Geometry*

Conference

Purdue University, Indiana, US, May 2011*The positivity of the first coefficients of normal*

Hilbert polynomials

Talk delivered at*32nd Symposium on Commutative Algebra and the 6th*

Japan-Vietnam Joint Seminar on Commutative Algebra

Japan, December 2010*The positivity of the first coefficients of normal*

Hilbert polynomials

Talk delivered at*International Conference on Commutative Algebra and*

Algebraic Geometry

IISc. Bangalore, India, December 2010*On the Chern number of an ideal*

Talk delivered at*CIMPA School and International Conference in*

Commutative Algebra and its application to Algebraic Geometry and

Combinatorics

Galatasaray University, Istanbul,

Turkey, September 2010*International Congress of Mathematics*

Hyderabad, India, August 2010*School and Workshop on Local Rings and Local Study*

of Algebraic Varieties

ICTP, Trieste, Italy, June 2010