MATHEMATICSRough plan of my Lecture • • • • What is Mathematics ? Information about various Exams. Careers in Mathematics. Philosophy of Mathematics. What is Mathematics? • Mathematics is a study of Sets with Structures. There are three basic structures. Three Basic Structures of Mathematics • Algebraic Structures . .Three Basic Structures of Mathematics • Algebraic Structures → • An algebraic structure consists of one or more sets closed under one or more operations satisfying some axioms. Three Basic Structures of Mathematics • Algebraic Structures • Topological Structures . The branch of mathematics that studies topological spaces in their own right is called topology. connectedness. . and continuity in every mathematical space where it is possible to do so.Three Basic Structures of Mathematics • Algebraic Structures • Topological Structures → • Topological structures allow the formal definition of concepts such as convergence. Three Basic Structures of Mathematics • Algebraic Structures • Topological Structures • Ordered Structures . Three Basic Structures of Mathematics • Algebraic Structures • Topological Structures • Ordered Structures → • In Mathematical space: the relations are defined to relate or compare different elements. These relations together with the set under consideration gives rise to ordered structure in Mathematics. . Building Of Mathematics A L G E B R A T O P O L O G Y O R D E R E D . Building Of Mathematics A L G E B R A T O P O L O G Y O R D E R E D . Building Of Mathematics A L G E B R A T O P O L O G Y O R D E R E D Logic . . • It makes possible to assign meaning to statements in Mathematics. • The rules of logic allows us to distinguish between valid and invalid statements in mathematics.What is Logic? • Logic is the basis of Mathematical reasoning. Rules of Logic . . How the empty set is subset of every set? . We prove So to prove φ is subset of S. We prove .To prove A is subset of B. the statement (*) is true.Consider the statement q (*) It is the Statement of the form where p is always false. Since p implies q is always true when p is false (regardless of truth value of q). . Definition The definition of definition consists of certain axioms. It is studied in the subject Logic. . Two properties of Definition Definition is always Fundamental. Definition is if and only if type of statement. . 1. Definition is always Fundamental Consider the following argument Where is the Mistake? . Definition is always Fundamental Consider the following argument Where is the Mistake? .1. Triangle is equilateral All its sides are equal.1. . Definition is an If and only If statement. Every Definition in Mathematics is ultimately a Set.Set is the most fundamental concept of Mathematics. . definition of function. sequence. (How?) .e.g. group is a set. NBHM MA/M. Interview 11. GATE Entrance Examination 9. Scholarship Examination 5. Scholarship Examination 8. NET Exam .D.D. JEST Entrance Examination 6. ISI Examination 12. IIT JAM Entrance Examination 2. IMSC Ph. IISER Ph.D.Sc. NBHM Ph.Various Entrance Exams/Interviews at Graduate and Post-Graduate level 1. INAT Entrance Examination 7. IISC Bangalore (I-Math) Entrance Examination 4. TIFR Entrance Examination 3. Interview 10. SET Exam 13. Basic Books For these Exams Analysis i) Real Analysis Author: N. L. Carothers Publisher: Cambridge University Press Real Analysis Author: H.L. Royden Publisher: Prentice Hall of India Analysis on Manifolds Author: J. R. Munkres Publisher: Addison Wesley Publishing Company. Functions of One Complex Variable Author: John B. Conway Publisher: Narosa Publishing House ii) iii) iv) Algebra • i) An Introduction to Algebra Author- Donald Lewis Publisher- Harper and Row, New York. • ii) A First Course in Abstract Algebra Author- John B. Fraleigh Publisher- Narosa publishing house. • iii) Linear Algrbra Author- Seymour Lipschutz Publisher- Schaum Series Topology • i) Topology Author. Simmons Publisher.F.Seymour Lipschutz Publisher.Schaum Series .G.Prentice Hall of India • ii) An Introduction to Topology & Modern Analysis Author. Munkres Publisher. • iii) Topology Author.James R.Tata McGraw-Hill. IIT JAM Entrance Examination .1. curl and Laplacian. Uniform Continuity. Linear differential equations of second order with constant coefficients. Algebra : Groups. Dimension. absolute and conditional convergence. Convergent sequences and series. Taylor 's theorem. Basis. Vector Calculus : Gradient. rank. Maxima and minima of functions of a single variable. quotient rings and fields. subgroups and normal subgroups. Differential Equations : Ordinary differential equations of the first order of the form y'=f(x. . divergence. completeness of R. inverse. Euler-Cauchy equation. Real Analysis : Open and closed sets. Power series. Integral Calculus : Integration. Finite Dimensional Vector Spaces over Real and Complex Numbers. group homomorphisms and basic concepts of quotient groups. Linear Transformations. Surface areas and volumes. Fundamental theorem of calculus. Lagrange's Theorem for finite groups. Green's. Double and Triple. Stokes' and Gauss' theorems and their applications. Method of variation of parameters. rings. Uniform convergence. integrals. Eigenvalues and eigenvectors. limit points.y). Mean value theorem. maxima and minima. Functions of two and three variables. determinant. Linear Algebra : Systems of linear equations. ideals.SYLLABUS • • • • • • • Sequences. Matrices. Partial derivatives. Series and Differential Calculus : Sequences of real numbers. 2) Differential Equations Author. Meerut. .Gupta.John Wiley & Sons. Malik.Tom M.How to Prepare? • Basic Books on Algebra and Analysis (except : Functions of One Complex Variable) 1) Calculus Volume II Author.Pragati Publications. Mittal Publisher. Apostol Publisher. TIFR Entrance Examination (Tata Institute of Fundamental Research. Mumbai) • Website: http://univ.res.in/gs2010/index.tifr.2.html . .Question Paper Pattern • There are questions on Algebra. Topology and simple questions on Combinatorics and on other topics. • Questions are of only True/False type! • There is negative marking. Analysis. K.How to Prepare? • Basic Books on Algebra and Analysis and Topology • Combinatorics Author. .Schaum Series. Balakrishnan Publisher.V. IISC Bangalore (I-Math) Entrance Examination (Indian Institute of Science.) . Ph.) 2. Programme (After M. Programme (After B. Bangalore) • At IISC Bangalore: 1.Sc. Integrated Ph.3. D.D.Sc. or IISc Entrance Test. or GATE. . or JEST. or UGC-NET for JRF. or NBHM.How to get into IISC? • By Qualifying in one of the following Tests CSIR-UGC NET for JRF. or DBT JRF or ICMR JRF. How to prepare for IISC Entrance? • Basic Books on Algebra and Analysis and Topology • Topics from Applied Mathematics chosen from the syllabus . nbhm.Sc.4.in/msc.dae. Department of Atomic Energy.gov. Mumbai) • Website: http://www. Scholarship Examination (National Board for Higher Mathematics.html . NBHM MA/M. 000 for two years of post graduation.44. 1.Sc. In Mathematics. .Sc. Scholarship? • NBHM (National Board of Higher Mathematics) gives scholarship to the selected students doing MA/M. Around Ten students are selected from all over India on the basis of their performance on written test and interview.What is NBHM MA/M. At present this scholarship is Rs. 6000 per month which amounts to Rs. B.Who can apply for this Scholarship? • Student appearing/appeared in T. I year or in M.Y.Sc. .Sc.Sc. II year. M. Analysis 3. Algebra 2.How To Prepare? • There are three sections in the question paper: 1. . Topology/Geometry (optional) Student can choose Topology or Geometry from third section. Great Book!!! . Concise.Donald Lewis Publisher.Books • 1. Ring Theory.Harper and Row. Field Theory. You just need to go through all the statements and results in this book. Algebra • i) Introduction to Algebra Author. New York. This collection will help you to solve any question on these topics. (This is a MUST MUST have book). Clear collection of all the results in Group Theory. Linear Algebra (Vector Spaces). By doing so you will have Complete. Royden Publisher: Prentice Hall of India Prepare only first Nine chapters.Churchil and Brown Publisher.McGraw Hill Prepare only the first Six Chapters of this book. This book is written in simple language and is very useful.L. . ii) Complex Variables and Applications Authors.2. Analysis • i) Real Analysis Author: H. l. Khanna Publisher: Jai Prakash Nath & Co. Chand Prepare only first Fourteen Chapters. . Geometry (optional) • i) Book: Solid Geometry Author: M.L. ii) Book: Elements of Coordinate Geometry Author: S..3. Prepare only first Eight chapters of this book. Loney Publisher: S. 3.F. Topology (optional) • i) Book: An Introduction to Topology & Modern Analysis Author: G. Lipschutz Publisher: Schaum Series • . Simmons Publisher:Tata McGraw-Hill Prepare only chapters Three to Six ii) Book: General Topology Author: S. 5.veccal.ernet. JEST Entrance Examination Joint Entrance Screening Test • Website: http://jest09.in/ . programmes in these institutions. programs in physics (including various inter-disciplinary research areas in physics) and Theoretical Computer Science. Programme in Physics (including various inter-disciplinary research areas in Physics)/ Theoretical Computer Science. Applications are invited from motivated students.D. at any of the 21 premier participating institutions. / Integrated M. with consistently good academic record.What is JEST? • A number of reputed institutes in the country have come together to conduct a Joint Entrance Screening Test (JEST) for enrolment of students in Ph. D. D.D. Programmes in Physics/ Theoretical Computer Science in these institutions in the country.Sc. . JEST is conducted for selecting candidates to be interviewed for admission to Ph. JEST score forms an important component in the selection of candidates for the Ph.-Ph.D. for appearing in JEST leading to enrolment in a Ph. SNBNCBS. IISER PUNE. 13. Kolkata . Nainital 2. 21. IPR. Bangalore 3. Mumbai UGC-DAE CSR. Indore HRI. Ahmedabad 10. IISC. Indore SINP. Bangalore 11. Bhubaneswar IMSC. Chennai. 12. 19. 18. Kalpakkam IOP. Bangalore 4.Institutes which allow admission on qualifying JEST 1. 17. Allahabad IGCAR. Pune 6. Kolkata VECC. JNCASR. Gandhinagar RRCAT. 15. Mohalii 5. ARIES. Kolkata BARC. IIA. NCRA. 14. IISER MOHALI. Pune 9. Pune 7. PRL. 20. 16. IUCAA. Bangalore 8. RRI. / M. Talented final year B. JNCASR.Ph. in Computer Science and related disciplines. Programme (Theoretical Computer Science) M. and should be interested in the mathematical aspects of computer science. will also be considered at IISc. / B. Programme (PHYSICS) M. in Physics or M.A./ M. or B.Tech (in Engg Physics/Applied Physics/Post B. Applied Physics.D.Sc.E. with at least 55% aggregate. Hons only) will also be considered at IIA. / M.Sc. Sc. Ph. IISER Mohali.Sc.Tech.E.Tech.Sc. Minimum Qualifications & Selection Procedures Ph.E.Eligibility • • • • Students who expect to complete their final examination by August 2009 are also eligible to apply.Tech. NCRA.Sc. Candidates with M.D. Talented B. RRI. Engineering Physics.D. in Mathematics and B. / B. graduates will also be considered for integrated M.Sc. in related disciplines. At IPR candidates should have a Masters degree in Physics.Sc. Graduates will also be considered for pre-selection at IUCAA.. programme. IUCAA.E. etc.Tech.C. IMSc. IISER Pune and SNBNCBS. / M. / M. • • . Graduates with a B. 3.Ph.Tech. / B.1. At HRI. degree will be considered for the integrated Post-B. each institute will call a limited number of candidates for its further selection procedure depending on its requirements. programme in subject areas mentioned above. programme in subject areas mentioned above. Programme at IIA 6. At IMSc.E.D.Sc. (Physics) / B. candidates with a Bachelor's degree will also be considered for the integrated M.Sc.D. 4. (Physics/Mathematics) / B.Sc. Details of the programme can be found on the IIA webpages.Sc.E.D. graduates with B.-Ph.E. Ph. degree will also be considered for admission to M. Tech.D. / B.Sc. graduates with M.Tech. graduates with B. At IIA. degree will be considered for the integrated M. Using the JEST results. 5.-Ph. degree will be considered for the integrated M.E. .D. 2.D. (Physics / Applied Physics) / B.Sc. graduates with B. programme. At IIA. At SNBNCBS.Tech . programme in Physics.Sc. as part of an integrated Ph. Integrated M.Tech-Ph. Mere qualifying in JEST does not entitle one to get a Research Fellowship. / B. / B. programme in subject areas mentioned above. All candidates selected after interview will receive Research Fellowship from the respective institutes. (by Research) programme in Physics and Theoretical Computer Science.Tech.Sc. / B. Walker Publisher.I.MIR Publishers Moscow/CBS Publisher India 4.MIR Publishers Moscow/CBS Publisher India .E. Fundamentals of Physics Author. Irodov Publisher. Irodov Publisher.E. Problems in General Physics Author. Fundamental Laws of Mechanichs Author.I.How to Prepare? • Basic Books on Analysis and Topology 1. 2.John Wiley & Sons. Fundamental Laws of Electromagnetism Author.I.MIR Publishers Moscow/CBS Publisher India 3. Rescnick.Halliday.E. Irodov Publisher. 6. INAT Entrance Examination IUCAA-NCRA Admission Test . D. . Pune OR National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research (NCRA-TIFR). Pune.What is INAT Test? • IUCCA-NCRA Admisson Test is being conducted to select candidate for research scholarship to join Ph. programme either at Inter University for Astronomy and Astrophysics (IUCAA). Eligibility and Preparation • Same as that of JEST test. . dae.in/phd. NBHM Ph.html .7.gov. Scholarship Examination • Website: http://www.nbhm.D. in Mathematics.D.D. At present this scholarship is Rs. 22. .000 per month.What is NBHM Ph. Scholarship? • NBHM (National Board of Higher Mathematics) gives scholarship to the selected students for doing Ph. Around Ten students are selected from all over India on the basis of their performance on written test and interview. The scholarship is given maximum up to 5 years. .B.Sc. II year or those who have completed M.Sc.Sc.Who can apply for this Scholarship? • Student appearing/appeared in T.Sc.Y. I year M.. M. Pattern of the Exam • There are five sections. Algebra 2. • Answer as many questions as possible. Miscellaneous. • Question Types: Objective and Short answer. containing ten questions each 1. Topology 4. • The assessment of the paper will be based on the best FOUR sections. Analysis 3. Applied Mathematics 5. . How to prepare? • Basic Books on Algebra and Analysis and Topology • Selected topics from Applied Mathematics . 8. GATE Entrance Examination • Graduate Aptitude Test in Engineering . in selected subjects in selected institutes including IIT’s. • All GATE qualified students who have taken admissions for Ph.D. IISC. IISER etc.Tech.What is GATE? • Clearing GATE (Graduate Aptitude Test in Engineering) examination in Mathematics opens door to do Ph./M.D. in many National Institutes such as IIT.Tech get scholarship from MHRD (Ministry of Human Resource and Development). • GATE qualified candidates (in Mathematics) can also apply for M. . Eligibility • M. . II year appeared or passed.Sc. Cayley-Hamilton Theroem. maximum modulus principle. minimal polynomial. Skew-Hermitian and unitary matrices. Lebesgue integral. line. compactness. functions of several variables. systems of linear equations. minima. Ordinary Differential Equations: First order ordinary differential equations. metric spaces. existence and uniqueness theorems. Lebesgue measure. method of Laplace transforms for solving ordinary differential equations. Riemann integration. Complex Analysis: Analytic functions. Stokes and Gauss. completeness. systems of linear first order ordinary differential equations. Real Analysis: Sequences and series of functions. dominated convergence theorem. Taylor and Laurent’s series. maxima. Liouville’s theorem. linear ordinary differential equations of higher order with constant coefficients. Fourier series. Finite dimensional inner product spaces. linear second order ordinary differential equations with variable coefficients. rank. multiple integrals. Fatou’s lemma. Legendre and Bessel functions and their orthogonality. theorems of Green. conformal mappings. surface and volume integrals. power series.Syllabus • Linear Algebra: Finite dimensional vector spaces. Gram-Schmidt orthonormalization process. • • • . complex integration: Cauchy’s integral theorem and formula. diagonalisation. uniform convergence. Weierstrass approximation theorem. series solutions. measurable functions. Hermitian. Linear transformations and their matrix representations. residue theorem and applications for evaluating real integrals. selfadjoint operators. bilinear transformations. eigen values and eigen vectors. method of undetermined parameters. interpolation: error of polynomial interpolation. wave and diffusion equations in two variables. Gauss Legendre quadrature. Lagrange’s equations for holonomic systems. numerical integration: Trapezoidal and Simpson rules. • • • . Hahn-Banach extension theorem. Cauchy. secant method. open mapping and closed graph theorems. LU decomposition). Euclidean domains. Hamiltonian equations. Fourier series and Fourier transform and Laplace transform methods of solutions for the above equations. automorphisms. Runge-Kutta methods. numerical differentiation. bounded linear operators.Syllabus • • • Algebra: Normal subgroups and homomorphism theorems. Hilbert spaces. method of characteristics. Prime ideals and maximal ideals in commutative rings. finite fields. product topology. Sylow’s theorems and their applications. Topology: Basic concepts of topology. least square polynomial approximation. principle of uniform boundedness. Principle ideal domains and unique factorization domains. Functional Analysis: Banach spaces. matrix eigenvalue problems: power method. second order linear equations in two variables and their classification. fixed point iteration. numerical solution of systems of linear equations: direct methods (Gauss elimination. countability and separation axioms. Group actions. Newton interpolations. connectedness. numerical solution of ordinary differential equations: initial value problems: Taylor series methods. orthonormal bases. Lagrange. Dirichlet and Neumann problems. Fields. Newton-Raphson method. Euler’s method. compactness. Urysohn’s Lemma. iterative methods (Jacobi and Gauss-Seidel). Numerical Analysis: Numerical solution of algebraic and transcendental equations: bisection. Mechanics: Virtual work. Partial Differential Equations: Linear and quasilinear first order partial differential equations. solutions of Laplace. Riesz representation theorem. standard parametric tests based on normal. basic feasible solution. Random variables. Sampling distributions. UMVU estimators. Interval estimation. Hungarian method for solving assignment problems. u -u method for solving transportation problems. Calculus of Variation and Integral Equations: Variation problems with fixed boundaries. t. maximum likelihood estimators. Testing of hypotheses. standard probability distributions and their properties. simplex method. Balanced and unbalanced transportation problems. Dual problem and duality theorems. conditional expectation. linear integral equations of Fredholm and Volterra type. moments. their iterative solutions. convex sets and their properties. joint and conditional distributions. sufficient conditions for extremum. Linear programming: Linear programming problem and its formulation. dual simplex method and its application in post optimality analysis. F – distributions. big-M and two phase methods. conditional probability. independence.Syllabus • Probability and Statistics: Probability space. infeasible and unbounded LPP’s. Bayes theorem. Weak and strong law of large numbers. central limit theorem. X2 . alternate optima. Linear regression. graphical method. • • . expectation. 3 2/ 1/ mark will be deducted for each wrong answer.20 : Will carry one mark each (sub-total 20 marks). The answer to the second question of the last two pairs will depend on the answer to the first question of the pair.60.Question Paper Pattern Patterns of Question papers Q. Q.50 : Will carry two marks each (sub-total 60 marks) Question pairs (Q.58 and Q. 3 There will be negative marks only for wrong answer to the first question of the linked answer question pair i. . Q. then the answer to the second question in the pair will not be evaluated. Q. for Q.57 and Q.60) will be linked answer questions.21 to Q.57.1 to Q.58) and (Q. There is no negative marking for Q. If the first question in the linked pair is wrongly answered or is un-attempted. 2/3 mark will be deducted for each wrong answer.59.e. Each question will carry two marks Negative Marks for wrong Answer mark will be deducted for each wrong answer.59. Malik. 2) For functional analysis: Book: An introduction to Topology and Modern Analysis Author. Mittal Publisher. Meerut.Tata McGraw Hill .F.G. Simmons Publisher.Gupta.Pragati Publications.How to prepare? • Basic Books on Algebra and Analysis and Topology 1) Differential Equations Author. Prentice Hall of India 4) For Partial Differential Equations: Book: Partial Differential Equations Author.W. E. Williams Publisher. Science and Engineering. Publisher.Clarendon Press Oxford . Author.John H.Books 3) For Numerical Analysis: Book: Numerical Methods for Mathematics. Mathews. Kapoor Publisher- .K.V. 6) For Linear Programming Problems: Book: Operations Research Author.Books 5) For Linear Programming Theory: I used internet resources for preparing theory of various methods of LPP. C. Joag Publisher.N. P.Tata McGraw Hill 8) For Calculus of Variation and Integral Equations Book: AuthorPublisher- .Books 7) For Mechanics: Book: Classical Mechanics Author.C. Rana. Books 9) For Probability and Statistics Book: AuthorPublisher- . 9. Interview (Indian Institute of Science Education and research) .D. IISER Ph. These institutes are located in Bhopal. Each IISER is an autonomous institution awarding its own Masters and Doctoral degrees. Kolkata and Thiruvanantapuram. The IISERs represent a unique initiative in India where teaching and education are totally integrated with state-of-the-art research nurturing both curiosity and creativity in an intellectually vibrant atmosphere of research. Pune. • . Mohali. has established five Indian Institutes of Science Education and Research (IISER).About IISER • The Government of India. through the Ministry of Human Resource Development (MHRD). D. GATE (Minimum percentile 95) 3. NBHM Ph. UGC CSIR NET with JRF 2. Interview.D.Eligibility • To appear for interview candidate should have cleared any one of these exams: 1. Scholarship Examination .IISER Ph. iiserbhopal. IISER Thiruvanantapuram: Website: www. IISER Kolkata: Website: www.in 5.in 4.iiserkol.ac.in .ac.iiserpune.in 3.ac. IISER Mohali: Website: www.ac.in 2.For more information. visit the IISER’s websites: 1.ac.iisermohali.iisertvm. IISER Pune: Website: www. IISER Bhopal: Website: www. res.D. Chennai • Website: www.in . IMSC Ph. Interview Institute of Mathematical Sciences.10.imsc. Ph. Programme (After B.IMSC Institute of Mathematical Sciences Chennai At IMSC Chennai: 1. Programme (After M.) .D.Sc.) 2. Integrated Ph. D.Sc. . • Applicants for the integrated PhD programme should have completed a Bachelors degree in Mathematics or Statistics by the time they actually join the programme.Eligibility • Applicants for the PhD programme should have completed a Masters degree in Mathematics or Statistics by the time they actually join the programme. This screening is typically based on the applicant's performance in the PhD Scholarship screening test of the NBHM. . • There is NO separate entrance test of IMSC.D.D and integrated Ph. Candidates are selected for the interview by screening applications.How to get into IMSC? • Admission to both of these programmes (Ph.) is based on the candidate's performance in an interview. 000. housing and medical facilities will be provided.000 in subsequent years (subject to satisfactory performance). 20. 22.000 for the first two years and Rs. .IMSC Scholarships • A monthly stipend of Rs. 24. together with an annual contingency grant of Rs. in . ISI Examination INDIAN STATISTICAL INSTITUTE Kolkata • Website: www.11.ac.isical. Mahalanobis. C. founded by Professor Prasanta Chandra Mahalanobis.its general applicability and its dependence on other disciplines for its own development.C. In keeping with this long tradition. the Institute also initiated and promoted the interaction of statistics with natural and social sciences to unfold the role of statistics as a key technology which explicated the twin aspects of statistics . Within a few years the Institute's achievements in research that included innovative projects on sample surveys of agricultural crops and socio-economic after-effects of the Bengal famine (1943-44) as well as pathbreaking research publications of Professor R. grew out of the Statistical Laboratory set up by him in the Presidency College in Kolkata. brought recognition in India and abroad. Bose on experimental designs in the Annals of Eugenics (1939). Under the leadership of Professor P. . The Institute is now considered as one of the foremost centres in the world for training and research in statistics and related sciences.About ISI • The Indian Statistical Institute (I. the Institute has been engaged in developing statistical theory and methods and their practical applications in various branches of science and technology. In 1932 the Institute was registered as a non-profit making learned society for the advancement of statistics in India.).S.I. Functional Analysis. • 1) Topics for MI (Forenoon examination) : Real Analysis. Linear Algebra. Ordinary Differential Equations and General Topology. and RM-2 of 2 hours duration each in the forenoon and in the afternoon. Elementary Number Theory and Combinatorics. Lebesgue Integration.ISI Exam for JRF/Scholarship • There will be two tests RM-1. • Candidates will be judged based on their performance in both the tests. . Complex Analysis. • 2) Topics for MII (Afternoon examination) : Algebra. Gupta.Pragati Publications.Tata McGraw Hill . Simmons Publisher.How to prepare? • Basic Books on Algebra and Analysis and Topology 1) Differential Equations Author. Mittal Publisher.G. Meerut. 2) For functional analysis: Book: An introduction to Topology and Modern Analysis Author.F. Malik. V. Balakrishnan Publisher. Zuckerman Publisher. K.I. H.Wiley Eastern Limited 4) For Combinatorics: Book: Combinatorics Author.Books 3) For Elementary Number Theory: Book: An Introduction to Number Theory Author. Niven.Schaum Series . CSIR NET/JRF Exam . SET Exam 13.12. SET Exam 13. . CSIR NET/JRF Exam • You may join my community on Orkut for these two exams.12. Toughest Exam/Interview . Toughest Exam/Interview • TIFR Interview (after clearing written test) . Toughest Exam/Interview • TIFR Interview (after clearing written test) Easiest Exam . Toughest Exam/Interview • TIFR Interview (after clearing written test) Easiest Exam SET Exam . Research .Careers in Mathematics: 1. Teaching 2. Who should do the teaching? . I love lecturing. H... Hardy . and have lectured a great deal to extremely able classes.. -G.I hate 'teaching'. Who is good teacher of Mathematics? . but supreme beauty — a beauty cold and austere. yet sublimely pure. the sense of being more than Man. rightly viewed. The true spirit of delight. without appeal to any part of our weaker nature. and capable of a stern perfection such as only the greatest art can show. without the gorgeous trappings of painting or music. the exaltation.” .Beauty of Mathematics “Mathematics. which is the touchstone of the highest excellence. is to be found in mathematics as surely as poetry.Bertrand Russell . like that of sculpture. possesses not only truth. .The good teacher is one who can convey this beauty to the students. 3. an improvement in student ability to read mathematics. an increase in mathematics communication among students. 2.What is an Effective teaching in Mathematics? It results in 1. an increase in the understanding of mathematics by the students. . . 5. and 7.4. a greater use of mathematics vocabulary by students. 6. an increased student interest in the class because of the variety in ‘teachers’ throughout the course. a more ‘student-centred’ classroom . a positive student attitude toward the class and mathematics. Research Who should do the research? .2. Keep other options open. Philosophy of Mathematics On the one hand, philosophy of mathematics is concerned with problems that are closely related to central problems of metaphysics At first blush, mathematics appears to study abstract entities. This makes one wonder what the nature of mathematical entities consists in and how we can have knowledge of mathematical entities. If these problems are regarded as intractable, then one might try to see if mathematical objects can somehow belong to the concrete world after all. On the other hand, it has turned out that to some extent it is possible to bring mathematical methods to bear on philosophical questions concerning mathematics. The setting in which this has been done is that of mathematical logic when it is broadly conceived as comprising proof theory, model theory, set theory, and computability theory as subfields. Thus the twentieth century has witnessed the mathematical investigation of the consequences of what are at bottom philosophical theories concerning the nature of mathematics. they are said to contribute to the philosophy of mathematics. .When professional mathematicians are concerned with the foundations of their subject. When professional philosophers investigate philosophical questions concerning mathematics. and the more interaction there is between philosophers and mathematical logicians working on questions pertaining to the nature of mathematics. Of course the distinction between the philosophy of mathematics and the foundations of mathematics is vague. they are said to be engaged in foundational research. the better. The Beauty of Mathematics is the BEAUTY OF ITS CREATOR who made it perfect as He is the only one who is PERFECT. LET’S PRAISE HIM. LET’S ADORE HIM. AND LET’S LISTEN AND DO WHAT HE TELLS US TO DO. -Unknown . Hardy . H. and that the theorems which we prove. and which we describe grandiloquently as our 'creations' are simply our notes of our observations.Locus of Mathematical Reality I believe that mathematical reality lies outside us. -G. that our function is to discover or observe it.