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 .
Best Answer
I think you will like "Berkeley Problems in Mathematics" by de Souza and Silva.
(Also a link to Amazon so you can look at the reviews there.)