Annual Report for the year 2000-2001 |
The Department of Mathematics comprises of 25 faculty members with expertise in various areas of Mathematics, Statistics and Theoretical Computer Science. The faculty members strive to maintain high standards in teaching and research. During the year under review, three of the faculty members were receipients of the following awards: Best Young Mathematician award by the Indian Society of Industrial and Applied Mathematics, C.L.Chandana Award for Distinguished and Outstanding contributions to Mathematics Teaching and Research in India for the year 2000 and IIT, Bombay Excellence in Teaching award for the year 2000, respectively.
There is an active collaboration between the Department and various leading Indian and Foreign Institutes. Some of the Faculty members are also interacting with the Industry. The Department hosted an International conference, and many Seminars were organised during the year. Many faculty members were also involved in organising Workshops, CEP Programmes, participating in national/international conferences, and writing of books and monographs. They also carried out significant honorary work like editorial and referee jobs for journals, reviewing of projects sponsored by national agencies, examining of theses and serving on selection committees of various Universities and Institutes.
Ph. D. : 7 M. Sc. (Mathematics) : 18 M. Sc. (ASI) : 25 Degrees awarded Ph. D. : 03 M. Sc. (Mathematics) : 10 M. Sc. (ASI) : 13
1.A. R. Shastri, Department of Mathematics, IIT, Bombay Topology of Surfaces, 2.Ketan Mulmule, Department of Computer Science, IIT, Bombay Algebra and Geometry in Complexity 3.U. A. Yajnik, Department of Physics, IIT, Bombay Fundamental Interactions and Geometry 4.S. S. Abhyankar, Purdue University, USA Roots of Algebraic Geometry 5.R. R. Simha, Department of Mathematics, University of Mumbai Perimeter of Ellipse and Beyond 6.B. V. Limaye, Department of Mathematics, IIT, Bombay Ancient Indian Mathematics 7.M. S. Raghunathan, TIFR, Mumbai Geometry and Arithmetic in SL(2, R)R and D Activities
The current research interest in the Department covers a wide range of fields:
Algebra, Combinatorics, Algebraic Geometry, Differential Geometry, Algebraic Topology, Functional Analysis, Harmonic Analysis, Set-Valued Analysis, Approximation Theory, Nonlinear Analysis, Partial Differential Equations, Nonlinear waves, Numerical Analysis, Tribology, Mechanics of Solids, Fluid Mechanics, Magnetohydrodynamics, Biomechanics, Non-Newtonian Fluid Flow, Statistical Inference, Geostatistics, Reliability Theory, Industrial Mathematics and Statistics, Combinatorial Optimization, Algorithms, Computational Biology and Biostatistics, Bio-informatics.
More specifically, the faculty members are currently pursuing research in the following areas.
Hilbert polynomials, Cohen-Macaulay property of fiber cones, local cohomology, mixed multiplicities.
Subintegrality, invertible modules, Hilbert functions and polynomials of local and graded rings. Investigation of subintegrality and invertible modules; the relationship between Hilbert coefficients and depths of form rings.
Grassmannians and Schubert varieties, Determinantal and pfaffian rings and their Hilbert functions, MDS Codes and Codes associated to higher dimensional varieties, Counting the number of points of varieties over finite fields.
Harmonic manifolds, Classification of noncompact Harmonic manifolds with minimal horo-spheres.
Eigenvalue Problems for large full matrices, Refinement and Acceleration of approximate eigenevalues, eigenvectors and spectral subspaces of Integral and Differential Operators.
Vector valued functions, Analysis of vector-valued Mean Periodic Functions and their Applications.
Stability and well-posedness in Optimization and Approximation.
Development of a common platform for parameter estimation for Data Reconciliation , Neural Network and Electrical Power System, based on nonlinear analysis and contol and optimization theory.
Development of efficient, robust and reliable numerical approximations to real world problems.
Finite Element Approximations to Partial Integro-Differential Equations, Evolutionary Variational Inequalities, Viscoelastic Fluid Flow Problems, Alternate Mixed Methods and Numerical Methods for Particle Size Distributions in Emulsion Polymeriz- ation Process.
Hyperbolic Systems Of Conservation Laws. Analysis of the transport equations governing the asymptotic behaviour of highly oscillatory solutions of the Lundquist eqations in Magnetohydro- dynamics.
Asymptotic solutions of nonlinear hyperbolic systems and stability.
Search for approximate analytical solutions using nonlinear geometrical optics approach.
Magnetic Fluids, Blood Flow.
Combinatorial Optimization, Algorithms for VLSI CAD, Computer Aided Geometric Design.
Algebraic and Enumerative Combinatorics.
Artificial Neural Networks for Molecular Recognition, Nonparametric Statistical and AI Methods for Protein Engineering, Artificial Immune Systems.
Development of Nonparametric Regression and Knowledge Based Algorithm for ab-initio Prediction of Protein Structure.
Development of ANN based AI Software for epitope/paratope
designing. Understanding Autoimmunity by Artificial Immune
System Modelling.
Projects and Consultancies
Combinatorial and Computational Methods in Algebraic Geometry, AICTE, Delhi.
Effect of Megnetic field on blood flow with blood as a two-phase
Design, estimation and software development of single and double
Failure Data Analysis, Mahindra and Mahindra Ltd., Tractor Division, Mumbai.
----------------------------------------------- Sponsored Projects Ongoing : 06 Sanctioned outlay : 50.89 lakhs Faculty involved : 08 Consulting jobs : 04 Income generated : 1,66,850 -----------------------------------------------Collaboration
There has been an active interaction between the Department and
a number of leading institutes such as : Institut fur Mathematik
der Universitat Wien Vienna, Austria, Federal Univ. Parana,
Brazil, Queen's University, Kingston, Canada, University of
Tsukuba, Japan, Institut de Mathematiques de Luminy, Marseille,
France, Univ. Jean Monnet, St Etienne, France, Seoul National
University, Korea, Institute for Information Transmission
Problems Moscow, Russia, Colorado School of Mines, Golden, USA,
I. B. M. Research Center,USA, Chennai Mathematical Institute,
Institute of Mathematical Sciences, Jamia Millia Islamia
University, New Delhi, Rajasthan University, Jaipur Tata
Institute of Fundamental Research, Mumbai, VRCE, Nagpur, NSTI,
New Delhi.
Visitors
Mrinal Ghosh, IISc, Bangalore, Gilles Lachaud, Institut de Mathematiques de Luminy, France, C. Rabut, INSA, Toulouse, France, G. S. Ladde, University of Texas, Arlington, USA, Graeme Fairweather, Colorado School of Mines, USA, Peter Zvengrowski, University of Calgary, Canada, Vaidyanathan Sundarapandian, IIT, Kanpur Ganesh Sundaram, Bell Labs, New Jersey, USA.
Balwant Singh
Member of the organizing committee for a Winter School and
International Conference on Cohomological Methods in Commutative
Algebra held at Mumbai from Nov 27 to Dec 16, 2000.
A. K. Pani
Resource Person in the `International Workshop on Analysis and
Applications', held in Chiangmai University, Chiangmai from 15th
to 17th May'2000.
Member of the Organising Committee for the International
Conference on `Nonlinear Systems : Modeling, Simulation and
Applications', held in NES College, Nanded during 26th Dec. to
29th Dec. 2000.
I. K. Rana
organised the 7th Workshop in Mathematics for final year BA/BSc
students form Maharashtra region. The Workshop was sponsored by
DST.
S. B. Patkar
coordinated a 5-day CEP course on Java 2 Standard Edition
conducted for Zee Interactive Learning Systems.
S. V. Sabnis
organised an `International Conference on Statistics,
Combinatorics and Related Areas' jointly with the Forum for
Interdisciplinary Mathematics, USA during 19 - 21 December 2000.
conducted a two-day in-house CEP workshop on 'Industrial
Statistics' at the premises of Sterlite Copper company during
22-23 July 2000.
J. K. Verma
was convener of Winter School and International conference on
`Cohomological Methods in Commutative Algebra' which was funded
by NBHM and International Mathematical Union
Awards and Honours
A. K. Pani
Best Young Mathematician Award by the Indian
Society of Industrial and Applied Mathematics for
outstanding contributions in Numerical Analysis,
Partial Differential Equations and Industrial
Mathematics.
I. K. Rana
C.L.Chandana - 2000 Mathematics Award for Distinguished and
Outstanding contributions to Mathematics Teaching
and Research.
D. V. Pai
Best Teacher Award for Excellence in Teaching for the year
2000 by IIT Bombay.
Publications
P. Chaturani (with J. Toravi, A. M. Patwardhan, S. Saxena
Bharatiya and S. K. Pathak)
In-vivo measurement of pressure gradient profiles of blood flow
in human thracic aorta, 27th National Conference on FMFP, 2000,
pp. 369-378.
(with J. Toravi and S. Saxena Bharatiya)
Two-layered Two-phase magnetic fluid model for blood flow
through a circular tube hemodialyser, Proc. Math. Anal. Appl.,
2000, pp. 185-189.
Presentation/Participation in Conferences/Workshops
Balwant Singh
1. participated in the Third National Discussion Meeting on
Commutative Algebra and Algebraic Geometry held at Bangalore
from Oct 16 to 21, 2000.
2. participated and lectured in a Winter School and Inter-
national Conference on Cohomological Methods in Commutative
Algebra held at Mumbai from Nov 27 to Dec 16, 2000.
P. Chaturani
1. delivered an Invited lecture on pulsatile flow of blood with
external excitations in the 45th Congress of ISTAM, Dec. 2000
2. participated in project monitoring workshop of Extramural
Research Schemes of Mathematical Sciences organised by CSIR, New
Delhi on 27th Feb 2001.
S. R. Ghorapade
1. Rings of invariants and constant term identities, Workshop
on Computational and Combinatorial Aspects of Commutative
Algebra, Universitat GH Essen, Essen, Germany, May 2000
2. Hyperplane sections of Grassmannians, with applications,
Conference on Algebra and Algebraic Geometry with Applications
(in honour of Prof. Shreeram Abhyankar's 70-th Birthday), Purdue
University, West Lafayette, USA, July 2000.
3. Poincare polynomials and the number of solutions of
equations over finite fields, International Conference on Number
Theory and Discrete Mathematics (in Honor of Srinivasa
Ramanujan), Panjab University, Chandigarh, October 2000.
4. Hilbert series of ladder determinantal rings, Third National
Discussion Meeting on Commutative Algebra and Algebraic
Geometry, Indian Institute of Science, Bangalore, October 2000.
5. Teaching Calculus using Intac: Some Experiments and
Experiences, CASTME-UNESCO-HBCSE International Conference on
Science, Technology and Mathematics Education for Human
Development, Goa, February 2001.
6. Polynomial congruences, Geometry and Topology, Conference on
Number Theory, The Institute of Science, Mumbai, February 2001.
7. School on Vanishing Theorems and Effective Methods in
Algebraic Geometry, directed by J.-P. Demailly and R. Lazarsfeld,
The Abdus Salam International Centre for Theoretical Physics,
Trieste, Italy, April--May 2000.
M. C. Joshi
1. organized a special session on " Industrial Mathematics " at
the World Congress of Nonlinear Analysts, University of Catania,
Italy- July 2000.
2. gave one hour invited lecture at the World Congress of
Nonlinear Analysts at Catania , Italy on various aspects of the
sponsored project on Industrial Mathematics currently in
progress at IIT Bombay.
K. D. Joshi
participated in a Workshop at Babasaheb Ambedkar Marathwada
University, Aurangabad in December 2000
S. B. Patkar
1. presented a research paper (coauthored with H.Narayanan) on
Improving Graph Partitions using Submodular Functions at
International Symposium on Mathematical Programming, Atlanta,
Georgia, Aug. 2000.
2. presented a research paper (coauthored with H.Narayanan)
entitled Fast On-line/Off-line Algorithms for Optimal
Reinforcement of a Network and its Connections with Principal
Partition at the Annual International Conf. on Foundations of
Software Technology and Theoretical Computer Science, (FST
TCS-20), New Delhi, December 2000.
S. V. Sabnis
gave an invited talk entitled Identification of Successful
Business Practices in Indian Exporting Firms at xxth Annual
Conference of Indian Society for Probability and Statistics and
National Seminar on Industries Oriented Statistical Reserach,
Pt. Ravishankar Shukla University, Raipur, Feb 19-21, 2001
A. R. Shastri
attended the International Instructional Conference on Low
Dimensional Topology, at Allahabad, organised by Harischandra
Research Institute and Department of mathematics, University of
Allahabad, during 19th Dec. to 30 Dec. 2000. He also gave
lectures on the use of the PL category and on Plumbing in
manifolds.
A. K. Pani
1. organised a session on Computational Mathematics in the
International Conference on Nonlinear Systems : Modeling,
Simulation and Applications, held in NES College, Nanded during
26th Dec. to 29th Dec. 2000.
2. organised a special session on New Challenges in Mathematical
Education in the International Conference on Nonlinear Systems:
Modeling, Simulation and Applications, held in NES Science
College, Nanded during December 26--29, 2000.
3. chaired a session in the International Conference on
Analysis and Applications, held in Chiangmai University,
Chiangmai from 17th to 18th May, 2000.
4. chaired a whole day session on Partial Differential
Equations, in the International Conference on Nonlinear
Systems : Modeling, Simulation and Applications, held in NES
College, Nanded during 26th Dec. to 29th Dec., 2000.
5. delivered two and half hours lecture on Finite Element
Methods for Elliptic Problems in the International Workshop on
Analysis and Application, held in Chiangmai University during
15th to 17th May, 2000
6. delivered 40 minutes invited lecture on An Alternate Mixed
Finite Element Method for Evolution Equations in the
International Conference on `Analysis and Applications', held in
Chiangmai University during 18th to 19th May, 2000.
7. delivered an invited talk entitled An H^1-Galerkin Mixed
Method for Heat Equation, in the International Conference on
`Applied Nonlinear Analysis 2000', in Pukyong National
University, Pusan (South Korea), during 15-16th June, 2000
8. delivered an invited talk entitled Oldroyd Model in
Viscoelastic Fluids: Some Theoretical and Computational Issues
in the annual meeting of Korean Applied Mathematics Forum, held
in Songee Mountain (June 26-28th, 2000).
9. delivered one hour Invited Talk entitled Particle Size
Distribution in Emulsion Polymerizations : Some Computational
Issues in the session on Applied Analysis and Industrial
Mathematics in the Third World Congress on Nonlinear Analysts
2000, held in Univ. Catania (Italy), during 19-26th July, 2000.
10.delivered one hour invited talk entitled Numerical Simulation
of Particle Size Distribution in Emulsion Polymerization
Process, in the International Conference on ` Nonlinear Systems
: Modeling, Simulation and Applications ', held in NES College,
Nanded during 26th Dec. to 29th Dec., 2000.
11.delivered an invited talk entitled Particle Size Distribution
in Emulsion Polymerisation - An Industrial Case Study and Its
Imact in Mathematical Research, in the International Conference
on Industrial and Applied Mathematics in the Indian
Subcontinent, held at Amritsar during 22nd to 25th January, 2001.
I. K. Rana
chaired a session in the `Workshop in Mathematics Education' at
ICICI Center for Mathematics,St.Stephen's College, University of
Delhi in Jan,2001
J. K. Verma
1. delivered an invited talk entitled On two theorem of
Macaulay at the 3rd National Discussion meeting in Commutative
Algebra and Algebraic Geometry held at I.I.S.c, Bangalore in
October, 2000
2. delivered 4 lectures on `Cohen-Macaulay rings' in Winter
School on Cohomological Methods in Commutative Algebra ,
organized by IIT, Bombay at the Homi Bhabha Centre for Science
Education, Mumbai, November 27-December 16, 2000.
3. delivered 13 lectures on `Fundamental theorems of
commutative algebra' in NBHM Mathematics Nurture Programme, TIFR,
Mumbai, June-July 2000.
P. Vellaisamy
particaipated in the Spring Meeting of the Mathematical Society
of Japan, held at Keio University, Tokyo, during March 26- 30,
2001.
Seminars/Colloquium/Invited Lectures
S. R. Ghorapade
delivered the following lectures:
1. On and around Reed-Muller codes, Institut de Mathematiques
de Luminy, Marseille, France, May 2000.
2. Old and new inequalities for the number of points of
varieties over finite fields, Institut de Mathematiques de
Jussieu, Universite Pierre et Marie Curie (Paris 6),
Chevaleret, Paris, May 2000.
3. Codes, matroids and hypergraphs with applications to Algebraic
Geometry, Mathematisches Kolloquium, Christian-Alberchts
Universitaet Kiel, Kiel, Germany, May 2000.
4. Lefschetz Theorems and the number of points of singular
varieties over finite fields, Oberseminar Algebra, Geometrie,
Zahlentheorie, Universitaet GH Essen, Essen, Germany, May 2000.
5. Error correcting codes, matroids and linear sections of
Grassmannians, Mathematischen Kolloquium, Universitaet
Osnabruck, Osnabruck, Germany, May 2000.
6. Fermat's Last Theorem and its impact on Mathematics,
California State University, Los Angeles, USA, July 2000.
7. Grassmann varieties, matroids and error correcting codes,
University of Southern California, Los Angeles, USA, July 2000.
8. Review of Classical Commutative Algebra, A series of two
lectures, Winter School on Cohomological Methods in Commutative
Algebra (supported by NBHM and IMU), Homi Bhabha Centre for
Science Education, Mumbai, Novermber 2000.
9. Topics in Algebraic Number Theory, A series of ten lectures,
Christian Albrechts Universitaet Kiel, Kiel, Germany,
December 2000.
K. D. Joshi
delivered invited talk at Karnatak University, Dharwad in
December 2000.
R. R. Joshi
1. delivered an invited Seminar on "Artificial Intelligence and
Computational Methods in Molecular Immunology" at CBME, IIT
Delhi. (June, 2000)
2. delivered an invited lecture series on "Neural Computing and
Applications in Molecular Biology/Bioinformatics" at
Bioinformatics Center, Pune (16-18 Feb, 2001).
B. V. Limaye
delivered an invited talk on "Spectral Approximation and
Computation" at Karnataka University, Dharwad on 12th Jan
2001.
A. K. Pani
1. delivered a series of Lectures on "Parabolic Integro-
Differential Equations and Numerical Methods" during June,
2000 in the Department of Mathematics, Seoul National
University, Seoul (South Korea).
2. delivered a colloquium talk entitled An alternate mixed method
for some evolution equations, Department of Mathematics
(14th June'2000), Yonsei University, Seoul (South Korea).
3. delivered a colloquium talk entitled An H^1- mixed method
for parabolic problems, Department of Mathematics (19th
June 2000), Korean Advanced Institute of Science and
Technology, Taejon (South Korea).
4. delivered a colloquium talk entiled A Posteriori Estimate:
A Step Towards Adaptive Method, (June 23rd'2000), Department
of Mathematical Education, Seoul National University, Seoul
(South Korea).
5. delivered a seminar talk entitled An alternate mixed finite
element method for parabolic equations, Department of
Mathematics, IIT Delhi (19th January, 2001).
S. B. Patkar
1. delivered a colloquium talk entitled Applications of
Submodular Functions at University of Iowa, USA, Aug. 2000
2. delivered a 6-hour lecture series on "Network Simplex Method
and its Applications", in : UGC Refresher Course on
Optimization, Mumbai University
S. V. Sabnis
delivered an invited talk entitled Time Series Modelling,
NITIE, 14th March 2001.
A. R. Shastri
1. delivered talks on `Why PL'? and Plumbing,,International
Instructional Conference on Low dimensional Topology,
University of Allahabad, 19th Dec. to 30 Dec. 2000.
2. delivered an invited lecture entitled Complex Numbers, SDM
college Honavar, North Karnataka, Dec 3rd 2000.
3. delivered a series of three lectures entiled Paper folding:
a non technical introduction to PL topology, Departement
of Mathematics, Karnatak University Dharwar, Dec 4th-6th 2001.
M. K. Srinivasan
delivered 4 lectures on "Linear Programming Duality", UGC
refresher course on Optimization at Mumbai University,
Feb. 2001.
J. K. Verma
delivered a lecture entitled Sums of squares at Ram Narain
Ruia College, Mumbai, September 30, 2000.
P. Vellaisamy
delivered invited lectures in:
(i) Regional conference of Japan Annual Meeting of
Mathematical Society held in July 2000 at Osaka
Prefecture University, Osaka.
(ii) National Meeting on the Applications of Statistics to
Enviornmental Science, (Dec. 2000), Keio University, Tokyo.
(iii) International Meeting on Comprehensive Research in the
Theory of Statistics, ( February, 2001), Osaka University,
Osaka.
(iv) The Institute of Statistical Mathematics (ISM), (in March,
2001) , Tokyo.
Honorary Work
Balwant Singh
participated in several selection committee
meetings at IIT's and other institutions.
P. Chaturani
worked as a member of ISTAM best paper award committee
S. R. Ghorapade
is nominated as a Co-opted Member (in the `Eminent Persons'
category) on the Board of Studies in Mathematics, University of
Mumbai, August 2000.
worked as referee for the Journal of the Indian Academy of
Sciences
M. C. Joshi
worked as a referee of Ph.D. Thesis , participated in the Board
of Study meetings and Selection Commiitee Meetings in various
universities.
R. R. Joshi
worked as a member, National Task Force for Bioinformatics,
refereed papers submitted to the international journals
"Computers and O.R." and "Neural Processing Letters".
A. K. Pani
was invited to join as a UGC nominee in the Advisory Committee
for the UGC-DRS project (stage II) of the Department of
Mathematics in the Berhampur University.
participated in the discussion on the course structures on
Numerical Analysis and Scientific Computing which would be a
part of the applied mathematics curriculum of Chiangmai
University Undergraduate and Masters programme in Mathematics;
refereed papers for SIAM J. Numer. Anal., Korean J. Comput and
Appl. Math., IMA J. Numer. Anal., Numer. Meth. PDE, Communi-
cations in Applied Analysis, J. Computational and Applied
Mathematics;
worked as an Associate Editor of Differential Equations
and Dynamical Systems: An International Journal for Theory and
Applications;
evaluated Ph. D. Thesis : One from Goa Uni.,
One from Aligarh Muslim Univ;
particiapted in evaluation of on a candidate for the position of
a Professor in the Department of Mathematics in the Pohang
Institute of Technology, Pusan, Korea.
D. V. Pai
participated in several selection committee meetings at IIT's
and other institutions,
conferred with an Honarary Appointment to the Research Board of
Advisors of the American Biographical Institute, U. S. A.,
refereed three Research Projects for Department of Science and
Technology, Government of India,
refereed Ph. D. these: one for University of Lucknow, one for
University of Rewa and one for I. I. T.,Kanpur.,
refereed two research articles for Indian Academy of Sciences.
B. V. Limaye
participated in several selection committee meetings at IIT's
and other institutions.
S. B. Patkar
refereed papers for Discrete Applied Mathematics.
J. Prakash
reviewed papers for Zentralblatt fur Mathematik.
A. Ranjan
refereed one paper.
V. D. Sharma
appointed Reviewer for Zentrablatt Math (Germany)
reviewed papers for Math Rev. (USA), examined one M.Phil Thesis
from the University of Hyderabad
M. K. Srinivasan
reviewed papers for Maths. Review
J. K. Verma
worked as Convener, "NBHM Nurture programme" for winners of
Indian National Mathematical Olympiad Batch 0f 1997.
refereed for papers in "Communications in Algebra". Writer for
"Mathematical Reviews".
P. Vellaisamy
reviewed a paper for Journal of Indian Statistical Association.
Faculty and their Specializations
1. Balwant Singh Commutative Algebra Algebraic Geometry. 2. Chaturani P. Fluid Mechanics, Magnetohydrodynamics, Biomechanics, Magnetic Fluids, Blood Flow, Non-Newtonian Fluid Flow 3. S. R. Ghorpade Algebraic Geometry, Combinatorics 4. Joshi K. D. Topology, Discrete Mathematics 5. Mohan C. Joshi Nonlinar Analysis, Control Theory 6. R. R. Joshi Computational Biology and Biostatistics, Bioinformatics. 7. Rekha P. Kulkarni Numerical Functional Analysis, Spline Theory 8. Balmohan V. Limaye Functional Analysis, Numerical Analysis, Spectral Approximation 9. Devidas V. Pai Functional Analysis, Approximation Theory, Set-valued Analysis 10. Amiya K. Pani Numerical Analysis Partial Differential Equations, Industrial Mathematics 11. K. S. Parihar Mechanics of Solids 12. Sachin B. Patkar Combinatorial Optimization Algorithms 13.J. Prakash Tribology 14. Prem Narain Numerical Analysis 15. Inder K. Rana Harmonic Analysis 16. Akhil Ranjan Differential Geometry. 17. S. V. Sabnis Reliability Theory Industrial Statistics 18. V.D.Sharma Quasilinear Systems of PDEs Nonlinear Waves 19. A. R. Shastri Algebraic Geometry Algebraic Topology 20. Sivaramakrishnan N. Complex Analysis 21. G. K. Srinivasan Partial Differential Equations. 22. M. K. Srinivasan Combinatorics 23. A. Subramanyam Statistical Inference Geostatistics 24. P. Vellaisamy Statistical Inference 25. J. K. Verma Commutative Algebra