## Dr. Shiv Datt Kumar CV,Math

**CURRICULUM VITAE**

** NAME** : DR SHIV DATT KUMAR** PRESENT POSITION** : Associate Professor

Department of Mathematics

Motilal Nehru National Institute of Technology,

Allahabad (U.P.), India

Pin - 211004** Phone** - +91-532 2271806, +91-9305562338** E-MAIL** :
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** DATE OF BIRTH** : 10th August, 1965** NATIONALITY** : Indian** **

**EDUCATIONAL QUALIFICATION** :

Examination Board/University Year Division Remarks

M.A.(Maths) Allahabad University 1988 First IInd Position in Alld. Univ.

Post M.Sc.(Maths) Harish-Chandra Research 1993

Institute, Allahabad

NET(National UGC-CSIR June 1993

Eligibility Test)

Ph.D. Harish-Chandra Research 2003 Sub.-Commutative Algebra

Institute (Alld. University)** **

**RESEARCH INTERESTS** : Commutative Algebra, Algebraic Geometry, Fuzzy theory,

operation Research.** **

**RESEARCH AWARDS/FELLOWSHIPS RECEIVED**

• Mathematics Research Fellowship at the Abdus Salam, International Centre For Theo-

retical Physics, Trieste, Italy from May 18 to July 17, 2009.

• Mathematics Research Fellowship at the Abdus Salam, International Centre For Theo-

retical Physics, Trieste, Italy from May 18 to June 17, 2008.

• Visiting Scientist at the Johann Radon Institute for Computational and Applied Math-

ematics(RICAM), Linz, Austria from May 15, 2006 to June 15, 2006.

• Mathematics Research Fellowship at the Abdus Salam, International Centre For Theo-

retical Physics, Trieste, Italy from June 1, 2004 to September 2, 2004.

• Junior and Senior Research Fellowships at Harish-Chandra Research Institute, Jhusi,

Allahabad in Commutative Algebra from 14th August, 1992 to 9th May, 1996.

OTHER AWARDS : Received Best Teacher Award in Government P. G. College, Ramanagar

(Nainital) Uttrakhand in 2004.** **

**REVIEWER OF INTERNATIONAL JOURNALS** : Turkish Journal of Mathematics,

published from Ankara, Turkey

REVIEWER OF BOOKS : “Higher Engineering Mathematics′′ of Pearson Education Pub-

lication, New Delhi** **

**RESEARCH PUBLICATIONS**: REFEREED INTERNATIONAL JOURNALS

1. Uniqueness of graded primary decomposition of modules graded over finitely generated groups,

Communications in Algebra, Taylor and Francis Publication, 2010, in press (with Srini-

vas Behara).

2. Analogue of Eakin Sathye Theorem over Rees Algebra, UNESCO & International Atomic Energy

Agency, ICTP, IC/049/2009.

3. G-associated ideals with flat base change of rings and short exact sequences, UNESCO & Inter-

national Atomic Energy Agency, ICTP, IC/048/2009 (with Srinivas Behara).

4. On a variant of Bertini’s theorem and generator of ideals of a polynomial ring with monic

polynomials, International Journal of Algebra, Vol 3, 2009, no. 17, 863 - 872 (with Priti

Singh).

5. Primary Decomposition Over Rings Graded by Finite Abelian Groups, Journal of Algebra,

Elsevier Publication, 318(2), 2007, 553-561 (with Markus Perling).

6. Sections Of Zero Dimensional Ideals Over A Noetherian Ring, Archiv der Mathematik,

Birkhauser Verlag Basel, Switzerland Publication, 89, 2007, 124-130.

7. Some Important Notions of Grobner Basis and Buchberger’s Algorithm, RICAM, website of

Grobner basis, Linz, Austria, 2006.

http : //www.ricam.oeaw.ac.at/Groebner−Bases−Bibliography/gbbib files/publication 1194.

8. A Note on a graded ring analogue of Quillen’s Theorem, Expositiones Mathematicae, Else-

vier Publication, Vol 22, 2004, 29-30.

9. Lifting of generators of an ideal and Nori’s Homotopy Conjecture, Indian Journal of Pure

and Applied Mathematics, Indian National Science Academy Publication, Vol 34 (11), 2003,

1599-1610 (with Ramji Lal).

10. Some Results On Generators Of An Ideal, Journal of Pure an Applied Algebra, Elsevier

Publication, Vol. 169 (1), 2002, 29-32 (with Satya Mandal).

11. Surface Modification of Semiconductor Photo-electrode for improved Solar Cell performance,

Solar Energy Materials and Solar Cells, Elsevier Publication, Volume 91, Issue 18,

2007, 1663-1668(with M. Tripathi and K. Pandey).

12. On an analogue of a theorem of Horrocs, (with Raja Sridharan and R.K. Mishra)(to appear).

13. A note on a lifting of generators of ideals and Murthy’s conjecture as an analogue of Quillen-

Suslin theorem (to appear).

(b) IN INTERNATIONAL CONFERENCES :

1. Sections of Zero Dimensional Ideals over a Noetherian Ring at the CIMPA-UNESCO- Spain

School during August 14-21, 2006, Universidad Complutense de Madrid, Madrid, Spain.

2. Homotopy of Sections of Ideals over a Noetherian ring , School and Conference on Algebraic

K-Theory and its Applications (May 14 - June 1, 2007), Abdus Salam, International Centre For

Theoretical Physics, Trieste, Italy.

3. “Projective Modules and completion of an unimodular row to an elementary matrix′′ 74th Annual

Indian Mathematical Society Conference at Allahabad University, Allahabad, during December

27 - 30, 2008.

(c)** IN NATIONAL CONFERENCES** :

1. ”Projective Modules and completion of an unimodular row to an elementary matrix” Indian

Mathematical Society Conference at Allahabad University, Allahabad, during December 27-30,

2008.

2. “Some Remarks on Grobner basis′′ in the national conference on “Material Science in the service

of society ′′ at the CMP Degree College, Allahabad University, Allahabad during February 5-6,

2008. In this conference also worked as Joint Secretary.

3. ”On Generators of Ideals over a Noetherian Ring”, Recent Trends in Mathematics, Aligarh

Muslim University, March 24 - 26, 2001.

4. “Lifting of generators of zero dimensional ideals over Noetherian rings′′ ACADEMICA at M.B.

Govt. P. G. College, Haldwani, Nainital during March 11-12, 2005.

RESEARCH PROJECTS:

National Board of Higher Mathematics Sanctioned a Research Project entitled ”Study of Rees

Algebra, graded rings in module theoretic setup and its applications” of Rs 707850/- for three years.** **

**TEACHING EXPERIENCE** :

Position Held Institution From To Nature of Job

Asstt. Professor Motilal Nehru National June 20, 2005 Continue Teaching

Institute of Technology, Allahabad and Research

Senior Lecturer Govt. P. G. College, May 19, 2003 June 19, 2005 Teaching

Ramnagar(Nainital) and Research

Lecturer(Maths) Govt. P. G. College, Dec 29, 1998 May 18, 2003 Teaching

Ramnagar(Nainital) and Research

Lecturer(Maths) Allahbad Degree College, Nov 03, 1997 Dec 28, 1998 Teaching

Allahabad University, Allahabad and Research

Lecturer(Maths) Ewing Christian College, Sept 2, 1996 May 28, 1997 Teaching

Allahabad University, Allahabad and Research

Lecturer(Maths) Nangaland University, May 9, 1996 Sep 1,1996 Teaching

Medziphema, Nagaland and Research

PGT(Maths) Army School, Srinagar(J and K) Oct 15, 1990 Aug 9, 1992 Teaching** **

**Ph.D. GUIDANCE EXPERIENCE **: Presently supervising four Ph.D. students at MNNIT, Al-

lahabad

1. Mr Ratnesh Kumar Mishra

2. Mr Srinivas Behara(QIP- POLY)

3. Ms Priti Singh

4. Mr Pitam Singh (with Prof R. K. Singh)** **

**COURSES TAUGHT** :

• At undergraduate level : Algebra, Linear Algebra, Calculus, Real Analysis, Complex Anal-

ysis, Numerical Analysis, Discrete Mathematics, Statics and Dynamics.

• Post Graduate Level : Algebra, Topology, Operations Research, Optimization Tecniques,

Fuzzy Theory, Weblet Theory, Mathematical Modeling and Complex Analysis.

LECTURES / SEMINARS GIVEN OUTSIDE THE PERMANANT INSTITUTE:

1. ”Projective Modules and completion of an unimodular row to an elementary matrix”, Indian

Mathematical Society Conference at Allahabad University, Allahabad, during December 27-30,

2008.

2. Gave a talk on ”Homotopy of Sections of Ideals over a Noetherian ring” in the School and

Conference on Algebraic K-Theory and its Applications (May 14 - June 1, 2007) at the Abdus

Salam, International Centre For Theoretical Physics, Trieste, Italy.

3. Gave an invited talk on the research paper “Sections of zero dimensional ideals over a Noetherian

ring ” at the CIMPA-UNESCO- Spain School from on 14-21 August, 2006 at Universidad

Complutense de Madrid, Madrid, Spain.

4. Gave a talk on the topic “Projective modules and generators of an ideal over Noetherian rings”

at Allahabad University in Feb, 2005.

5. Gave a seminar on the topic “Sections of projective modules and a question of Nori” at the The

Abdus Salam, International Centre For Theoretical Physics, Trieste, Italy on 27th

August, 2004.

6. Delivered an invited talk on the topic “Projective generation of ideals” at the national seminar

organized by Jammu Mathematical Society and Deptt. of Mathematics, Jammu University from

20-22 february, 2004.

7. On ”Generators of ideals” at the University of Allahabad in June, 2001.

8. Presented a paper entitled ”Lifting of generators of ideals over a Noetherian ring” presented in

the national level conference at Aligarh Muslim University in March, 2001.

9. Talk on the topic “Homotopy sections of projective modules over a Laurent polynomial ring” at

Harish Chandra Research Institute, Allahabad in June, 2000.

10. Seminar on the topic of “Set theoretic complete intersection ideals” at Harish Chandra Research

Institute(old MRI), Allahabad in 1995.

U.G.C. SPONSORED ORIENTATION PROGRAM/ REFRESHER COURSE ATTENDED:

• Participated in National level U.G.C. sponsored Refresher Course in Mathematics from

20th January, 2005 to 9th February, 2005 in the Academic Sta College, Allahabad University,

Allahabad.

• Participated in National level U.G.C. sponsored Orientation program from 27 Sep., 2001 to

25 Oct., 2001. in the Academic Sta College, Allahabad University, Allahabad.

• Participated in National level U.G.C. sponsored Refresher Course in Mathematics from

27th February, 2001 to 30th March, 2001. in the Academic Sta College, Aligarh Muslim

University, Aligarh.** **

**EXPERIENCE OF ACADEMIC ADMINISTRATION** :

1. Worked as President, Student Activity Centre (o ciating), Motilal Nehru National In-

stitute of Technology, Allahabad from March 22, 2010 to April 29, 2010.

2. Working as Convenor, Departmental Under Graduate Committee (DUGC) in the Department

of Mathematics, Motilal Nehru National Institute of Technology, Allahabad since June, 2008.

3. Working as Coordinator, Athletics and Gymnastic Activities, Students Activity Centre, Motilal

Nehru National Institute of Technology, Allahabad since July, 2008.

4. Worked as Convenor, Departmental Post Graduate Committee(DPGC) in the Department of

Mathematics, Motilal Nehru National Institute of Technology, Allahabad from June, 2005 to

June, 2008.

5. Worked as Hostel Warden in the Govt. Post Graduate College, Ramnagar (Nainital), Uttra-

chal (India) from September, 2004 to June, 2005.

6. Worked as N.C.C. O cer (ANO) in the Govt. Post Graduate College, Ramnagar (Nainital),

Uttrachal (India) from March, 1999 to May, 2004.

7. Worked as Assistant Hostel Warden and Proctor (Science faculty) in the Govt. Post

Graduate College, Ramnagar (Nainital), Uttrachal (India) from March, 1999 to August, 2004.

8. Worked as Dean (Science faculty) in the Govt. Post Graduate College, Ramnagar(Nainital),

Uttrachal (India) from August, 2000 to May, 2004.** **

**COMPUTER COURSE** :

• One year Diploma in Computer Application ( “O” Level ) from Electronics Services and

Training Centre, Kania, Ramnagar ( Nainital ).

SUMMARY OF RESEARCH WORK :

We have worked on the following problems. The first problem is the Laurent polynomial analogue

of the corresponding theorem in polynomial rings proved by Satya Mandal.

Problem 1 : Let I be an ideal of the Laurent polynomial ring A[T, T−1] containing a doubly monic poly-

nomial. Let P be a projective A-module of rank r dim(A[T, T−1]/I)+ 2 and suppose that S : P −!

I(1) and : P[T, T−1] −! I/I2 are surjective homomorphisms such that (1) S mod (T −1, I(1)2).

Then there is a surjective homomorphism : P[T, T−1] −! I such that (1) = S and lifts .

Problem 2 : Suppose A is a commutative Noetherian ring and I is an ideal in the polynomial algebra

A[T] containing a monic polynomial. Suppose also that µ(I/I2) = r dim (A[T]/I) + 2. Assume

that I(0) is generated by r elements, say, a1, ..., ar.

(i) Can we find a set of generators f1, ..., fr of I such that fi(0) = ai, for all i = 1, · · · , r?

(ii) Can we find a set of generators of I in which one and hence all the generators are monic ?

We settle the problem 1 in a rmative. The problem 2, in general, has negative answer. We have

given examples to show this. However in certain cases result is true and interesting. In some cases

answer is complete and in some cases answer is partial. We have answered the question in the following

cases:

(1) A is semi local Noetherian ring and I(0)/I(0)2 is a free module over A/I(0) with I(0) Rad(A).

(2) A is local Cohen Macaulay ring and I(0) is a local complete intersection ideal of A.

(3) I is a zero dimensional ideal.

(4) I is a local complete intersection ideal of height r = dim(A[T]).

(5) µ(I/I2) = 1 and A is an integral domain.

(6) I(0) = I(0)2.

In some cases the Laurent polynomial analogue are also proved.

Problem 3 : We have extended the famous theorem of Quillen to the graded ring.(Expositiones

Mathematicae, Elsevier Publication, Vol. 22, 2004. 29-30)

Let S = ∞

i=0Si be a graded ring and let M be a finitely presentable S-module, which is a locally

extended, for all M2 Max (S0). Then M is a globally extended.

Problem 4 : Primary decomposition over rings graded by finite abelian groups (With Markus Perling)

(Journal of Algebra, 318(2), 2007, 553-561.)

Let R be a Noetherian ring which is graded by a finite abelian group G. In general, for G-graded

R-modules there do not exist primary decompositions which are graded themselves. This is quite

di erent from the case of gradings by torsion free group, for which graded primary decompositions

always exists.

In this paper we introduced G-primary decompositions as a natural analogue to primary decom-

position for G-graded R-modules. We show the existence of G-primary decomposition and give a few

characterizations analogously to Bourbaki’s treatment for torsion free groups.

Problem 4 (International Journal of Algebra, 2009) : We have given di erent versions of

Bertini’s theorems and proved following a variant of theorem of Bertini.

• Let A be a finitely generated K algebra with K characteristic 0, algebraically closed field. Assume

A reduced and dim(A) 2. Let f, g 2 A with that ht(f, h) 2. Then for all but a finite number

of 2 K, A/(f + h) is reduced.

We used this theorem to prove the following :

• Suppose A is a semilocal regular reduced a ne algebra over an algebraically closed field of

characterstic zero such that every maximal ideal in A[T] is complete intersection, with dim(A) =

n > 0. Suppose I is an ideal of A[T], with height(I) = n + 1. Assume co-normal module I/I2

is generated by n + 1 elements over A[T]/I. Then we can find a minimal set of generators of I

with a monic polynomial.** **

**SEMINARS / CONFERENCES ATTENDED**:** INTERNATIONAL LEVEL**

1. Participated in the “Summer School and advanced workshop on trends and developments in

Linear Algebra′′ during June 22-July 10, 2009 at the Abdus Salam, International Centre

For Theoretical Physics, Trieste, Italy.

2. Participated in the School and workshop on “Integrable systems and Scientific Computing′′ dur-

ing June 15-20, 2009 at the Abdus Salam, International Centre For Theoretical Physics,

Trieste, Italy.

3. Participated in the Indian Mathematical Society Conference at Allahabad University, Allahabad,

during December 27-30, 2008 and presented a paper entitled ”Projective Modules and completion

of an unimodular row to an elementary matrix”.

4. Participated in the School and Conference on Di erential Geometry (June 2 - June 20, 2008) at

the Abdus Salam, International Centre For Theoretical Physics, Trieste, Italy.

5. Participated in the, School and Conference on Algebraic K-Theory and its Applications (May

14 - June 1, 2007) at the Abdus Salam, International Centre For Theoretical Physics,

Trieste, Italy and presented research paper “Homotopy of Sections of Ideals over a Noetherian

ring”?.

6. Participated and gave an invited talk on the research paper on “Sections of zero dimensional

ideals over a Noetherian ring ” at the CIMPA-UNESCO- Spain School from on 14-21 August,

2006 at Universidad Complutense de Madrid, Madrid, Spain.

7. Participated in the “ Special Semester on Grobner Basis and related methods” from May 15,

2006 to June 15, 2006 at Johann Radon Institute for Computational and Applied Math-

ematics(RICAM), Linz, Austria.

8. Participated in “CIMPA-UNESCO-IRAN School on Grobner bases and its Application′′ during

9-22 July, 2005 at the Institute for Advanced Studies In Basic Sciences, Zanjan, IRAN.

9. Participated in “Advanced School and Conference on Non-Commutative Geometry′′ from 9-27

August, 2004 at the The Abdus Salam, International Centre For Theoretical Physics,

Trieste, Italy.

10. Participated in the international school on “Commutative Algebra intraction with Algebraic

Geometry and combinatorics′′ from 2nd June to 11th June, 2004 at the The Abdus Salam,

International Centre For Theoretical Physics, Trieste, Italy.

11. Participated in the International workshop on “Computational Algebraic Geometry” during 8 -

13 Dec., 2003 at H.R.I., Allahabad jointly organized by Bhaskaracharya Pritisthan, Pune and

H.R.I., Allahabad.

12. Participated in the International Conference and Instructional School on “Homological Methods

in Commutative Algebra′′ to 27th Nov. to 16th Dec., 2000 at HBCSE (TIFR), Mumbai.

13. Participated in the “International symposium on rings, modules and groups with homological

techniques and their applications′′ at I.T., B.H.U. Varanasi during 18 - 21 Dec., 1995.** **

**NATIONAL LEVEL**

14. Participated in the national seminar during 20-22 February, 2004 at Jammu University and

delivered an invited talk on topic “ Projective generation of ideals”

15. Participated in the National workshop on “Computational Algebraic Geometry” during 2 - 11th

Jan., 2003. in H.R.I., Allahabad.

16. Participated in the National level “Instructional School on Linear Algebra” during Dec 3-15,

2001. in Harish- Chandra Research Institute, Jhunsi, Allahabad.

17. Participated in the National level Conference on “ Recent trends in mathematics” during March

24-26, 2001 and presented paper entitled “Lifting of generators of an ideal” in Aligarh Muslim

University, Aligarh. .

18. Participated in the National Conference on “Commutative Algebra and and Algebraic Geometry”

during Feb 1-3, 1999 in HRI (Old Mehta Research Institute), Jhusi, Allahabad.

19. Participated in the Seminars on “Infinite Permutation Groups’ in I.I.T., Guwahati during August

6-15, 1996.

20. Visited Institute of Mathematical Sciences, Chennai, TIFR, Mumbai and H.R.I., Allahabad

several times.

21. Participated in I.S.I. winter and summer schools on non-commutative Harmonic Analysis at

I.S.I., New Delhi, and I.I.Sc., Bangalore respectively.

Last Updated (Tuesday, 04 January 2011 12:12)