DEPARTMENT OF MATHEMATICS IISER BHOPAL PHD PROGRAMME MANUAL (2021) DEPARTMENT-SPECIFIC GUIDELINES Refer to the PG Manual for Institute-wide guidelines for the Ph.D. programme. This document consists of guidelines specific to the Mathematics department. 1. COURSEWORK Students are required to take three mandatory courses in the first semester as described below. These courses cover the bare minimum background required for a student to begin preparatory study in the research areas represented in the department. First Semester Second Semester 1. MTH 601: Algebra I At least two DE 2. MTH 603: Real Analysis ● A student is allowed to credit at most one 3. MTH 605: Topology I reading course (4 credits) in the entire PhD 4. One Departmental curriculum Elective (DE) ● A student can also take Ph.D. thesis credits (4/8 credits) to meet 16 credit requirement in the semester Third semester onwards, a student can take DE/Reading course/Ph.D. thesis credits, as per the advice of DPGC Convener/supervisor. 2. GUIDELINES a. Assignment of Mentor Every PhD student will be assigned a mentor upon joining the program. The mentor will provide guidance to the student towards choosing the appropriate courses and also monitor the student’s progress from time-to-time until he/she finds a thesis supervisor. b. Assignment of Thesis Supervisor 1. Each student is required to choose a potential supervisor by the end of the first year. 2. Once chosen, the student must submit the duly filled change of supervisor form to the academic office, via Maths department office. c. Comprehensive Examination The comprehensive examination consists of two components, written and oral. As per the institute’s guidelines, the PhD students must clear the comprehensive examination(s) within a period of four registered semesters from the date of admission to the programme. ● Written Examination: The written exam will be of three-hour duration. This exam will be in one of the three core areas (algebra, analysis, geometry/topology). The syllabi of the same are detailed in a separate section later in the manual. The written exams will be conducted during the semester breaks (preferably, in December and July) after the completion of their first semester. The DPGC convener should nominate the chairperson of the question paper committee for each of the three subjects and each of the chairpersons should then form a committee of two to four faculty members from the department. Members of each such committee shall decide among themselves about the pass marks which they deem equivalent to a CPI of 7 on 10, and they should then grade the answer sheets. The student must submit Form A to the Department office by April 30 or October 30 to appear for the written comprehensive examination in July or December, respectively. This should be done in consultation with a potential thesis supervisor or DPGC (in case a potential supervisor has not been identified). As per institute’s guidelines, a student would get a maximum of two chances to clear the written comprehensive examination. ● Oral Examination: This exam cannot be taken unless the student has passed the written section and it is to be conducted by a committee consisting of the following members 1. Potential thesis supervisor. 2. At least one external expert, preferably, from within the institute. In case, the external member is from outside the institute, a suitable justification must be provided by the department. 3. Minimum two experts related to the topic. 4. One nominee of the HOD. 5. In case of repeated oral examinations, the HOD/ DPGC convener must attend the oral examination. This examination should consist of a presentation by the student followed by the interview by the committee. There will be no weightage of the written comprehensive exam in this round, and a student must independently clear the oral exam. Once again, as per institute’s guidelines, a student would get a maximum of two chances to clear the oral comprehensive examination. The student must submit Form B to the Department office at least 4 weeks prior to the oral exam along with the synopsis prepared in consultation with the potential thesis supervisor. The oral exam can be scheduled anytime subject to availability of the committee. d. Doctoral Advisory committee The thesis supervisor(s), in consultation with the DPGC, will form a Doctoral Advisory Committee (DAC), consisting of the supervisor(s) and at least two other faculty members from the department. The DAC will oversee the progress of the candidate towards completing the requirements for Ph.D. after the comprehensive exam has been successfully cleared. e. Annual Progress Seminar and Report Following successful completion of comprehensive exam requirements, each student is required to give an annual seminar based on the research progress made in the previous year and submit the report of progress towards the thesis. This report must be submitted to the DAC with a copy to Convener, DPGC. The timelines of these seminars and the submission of reports is announced by DPGC on a yearly basis. f. Graduate Seminars Each student has to clear two graduate seminars, as per the rules given in the Institute PG manual. In this regard, Form C must be submitted at least one month before the graduate seminar date. FORMS: Form A (Written Comprehensive Examination), Form B (Oral Comprehensive Examination with Synopsis), Form C (Graduate Seminar with DAC) are available at the department website. Other forms (such as the reports of various exams etc) are available on the Academic Office webpage. Reading Courses Students can register for a four-credit Reading Course (MTH 698) on a research topic with a prospective thesis supervisor prior to clearing the Ph.D. candidacy requirements. A student is allowed to credit at most one reading course in the entire PhD curriculum. ● This will be a semester long course with only one instructor. ● The goal of the course is two fold: - to promote a focused self-study, and - to provide the instructor and student the opportunity to work with each other. ● The student should register for the reading course during the pre-registration. The student should submit a hard copy of the title of the reading course with the instructor's signature, to the Department office, by the pre-registration deadline. ● The student should submit a hardcopy of the course contents for the reading course to the department office within a week after the classes begin. ● The evaluation in the course will be done by either the instructor or by a committee comprising three faculty members (including the instructor) based on the performance of the student in the material covered during the course, seminars/the end semester examination. ● In the case that the Reading course is evaluated solely by one instructor, the evaluation will also involve an end-semester examination. ● The instructor is expected to intimate the student about the evaluation policy in the beginning of the course. PhD Comprehensive Examination Syllabi ● Algebra ● Group Theory ● Groups: Groups, subgroups, cyclic groups, quotient groups, Lagrange’s theorem and some applications, isomorphism theorems. ● Group actions: Definition and examples, Cauchy’s theorem, class equation, Sylow’s theorems and its applications to finite groups. ● Other topics: Direct product, semi-direct product, derived series, upper and lower central series, basic results on solvable and nilpotent groups, structure theorem for finitely generated abelian groups, elementary divisors, invariant factors of a finite abelian group, free groups. ● Rings and Modules ● Rings: Rings, subrings, ideals, isomorphism theorems, principal, prime and maximal ideals, characterization of prime and maximal ideals, Chinese remainder theorem. ● Integral domains: Euclidean domain, Principal ideal domain, Unique factorization domain, primes and irreducible elements, Gauss’s lemma, Eisenstein’s criterion. ● Modules: Modules, submodules, quotient modules and isomorphism theorems, modules over PID, Jordan decomposition, rational canonical form, finitely generated modules, short exact sequence, short five lemma, tensor product and its properties. ● Fields and Galois theory ● Fields: Fields, subfields, characteristic of a field, finite and algebraic extensions, splitting fields, normal extensions, algebraic closure, separable extensions, cyclotomic fields, finite fields. ● Galois Theory: Primitive element theorem, fundamental theorem of Galois theory, computation of Galois groups. Recommended Books 1. Dummit and Foote, Abstract Algebra, Wiley Publications, 2nd edition. 2. Hungerford, Algebra, Springer Publications. 3. Serge Lang, Algebra, Addison Welsey, 3rd Edition. 4. Jacobson, Basic Algebra, parts-I and II, Dover Publications Inc.; 2nd edition. 5. Birkhoff and McLane, Algebra, Chelsea Publishing Co. 6. D.J.S. Robinson, A course in the theory of groups, Springer, 2nd edition. ● Analysis ● Sigma-algebras, Lebesgue measure, measurable functions, Lebesgue integration, modes of convergence, Egoroff’s theorem, Fatou’s lemma, Lebesgue’s monotone convergence theorem, Lebesgue’s dominated convergence theorem, Lusin’s theorem. ● Product measure, Fubini’s theorem, convolution, integration in polar coordinates. ● Signed measures and differentiation, complex measures, total variation, absolute continuity, the Lebesgue-Radon-Nikodym theorem and its applications. ● Lebesgue’s differentiation theorem, functions of bounded variation, fundamental theorem of calculus for Lebesgue integrals. ● Lp spaces, Holder’s inequality, Minkowski