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Today I would like to ask you for any references, books, pdf's etc. that comprises of a lot problems with the level advance graduate.

The syllabus is that of teaching in most under-graduate program in the topics algebra, analysis, complex analysis, general topology. I'm specifically looking for the type of books that deals with mostly problems that cross over different topics. For example, consider the following problem whose solution requires concepts from analysis and algebra:

let $R$ be a ring of all real valued continuous functions on the closed unit interval . If M is a maximal ideal of R , prove that there exists a real number $\gamma$ , $0\leq \gamma \leq 1$ such that $M=\{f(x)\in R :f(\gamma)=0\} $ .

More for instances of the kind the probelms i'm looking see this links . http://www.cmi.ac.in/admissions/sample-qp/pgmath2016.pdf http://www.cmi.ac.in/admissions/sample-qp/pgmath2015.pdf http://www.cmi.ac.in/admissions/sample-qp/pgmath2014.pdf. .

The books i mostly use are :

Alegbra: Herstein , dummit-foote , Artin ,Hoffman kunze , Sheldon Axlers

Real analysis- Rudin ,Pugh , Apostol ,

Complex analysis - Ahlfors, Conway , Bak-Newman .

Topology -Munkers ,JK Joshi , Willard , Simmons .

I already going through many Ph-D qualifying exams of various universities across the globe . But i feel this isn't enough . I know i'm asking a lot but cracking this exams means a lot to me . Any help will be greatly appreciated . Thank you all .

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  • $\begingroup$ Can you let us know what PhD qualifying exams of various universities are you going through? Otherwise this question seems to me to be a little broad. $\endgroup$ – user 170039 Jun 6 '17 at 14:18
  • $\begingroup$ Which university are you in? How did the exam go? It has been over a year now. $\endgroup$ – Subhasis Biswas Aug 15 '18 at 9:04
  • $\begingroup$ I did better than I expected in the written test . Got interview for both M.Sc applied math and pure math . Got rejected though . $\endgroup$ – Suman Kundu Sep 13 '18 at 13:39
  • $\begingroup$ Did you not try for ISI? $\endgroup$ – Subhasis Biswas Sep 24 '18 at 1:33
  • $\begingroup$ Pardon me if this is going off topic, but I am extremely curious. $\endgroup$ – Subhasis Biswas Sep 24 '18 at 1:33
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I think you will like "Berkeley Problems in Mathematics" by de Souza and Silva.

In 1977 the Mathematics Department at the University of California, Berkeley, instituted a written examination as one of the first major requirements toward the Ph.D. degree in Mathematics. Its purpose was to determine whether first-year students in the Ph.D. program had successfully mastered basic mathematics in order to continue in the program with the likelihood of success. Since its inception, the exam has become a major hurdle to overcome in the pursuit of the degree. The purpose of this book is to publicize the material and aid in the preparation for the examination during the undergraduate years. The book is a compilation of over 1,250 problems which have appeared on the preliminary exams in Berkeley over the last twenty-five years. It is an invaluable source of problems and solutions for every mathematics student who plans to enter a Ph.D. program. Students who work through this book will develop problem-solving skills in areas such as real analysis, multivariable calculus, differential equations, metric spaces, complex analysis, algebra, and linear algebra. The problems are organized by subject and ordered in an increasing level of difficulty.

(Also a link to Amazon so you can look at the reviews there.)

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