For a Mathematics Postgraduate, who is interested in doing his PhD in areas related to Algebra and Analysis, what research field would you suggest. Preferably, the domain should be promising and relevant as per the current research interests of the Mathematics community in particular and the society in general, both theoretically and practically?
This is a good question. In addition to the helpful answer already given by @ Romeo P.G , there is a bit more to add.
Rich sources of algebra in contemporary mathematics can be found in the following areas:
Homology Computation using Pyramids (HCP) leads to the study of generators of homology groups. For an introduction to homology theory, see Section 2, starting on page 4 in a 2006 Vienna University of Technology report:
Conference Paper Computing Homology Group Generators of Images Using Irregula...
A very readable introduction to computing generators of homology groups is given in Section 4.2, starting on page 7. Notice that HCP has a number of practical applications such as the computer vision and the study of digital images (sse, e.g., Section 5, starting on page 12 in the HCP paper.
Membrane Topology focuses on the study of f homological quantum field theories. For the details, see
https://arxiv.org/pdf/math/0612218.pdf
Dear Venkatrajam
Regarding Algebra, apart from very familiar research areas like Commutative , Lie Algebra, Module Theory etc one can look into the topic like Automata theory , Language theory. Especially problems like recognizability of a language by an automaton, finite basis problem for languages are specially interesting.
This is a good question. In addition to the helpful answer already given by @ Romeo P.G , there is a bit more to add.
Rich sources of algebra in contemporary mathematics can be found in the following areas:
Homology Computation using Pyramids (HCP) leads to the study of generators of homology groups. For an introduction to homology theory, see Section 2, starting on page 4 in a 2006 Vienna University of Technology report:
Conference Paper Computing Homology Group Generators of Images Using Irregula...
A very readable introduction to computing generators of homology groups is given in Section 4.2, starting on page 7. Notice that HCP has a number of practical applications such as the computer vision and the study of digital images (sse, e.g., Section 5, starting on page 12 in the HCP paper.
Membrane Topology focuses on the study of f homological quantum field theories. For the details, see
https://arxiv.org/pdf/math/0612218.pdf
To complement the other answers, there are (at least) two large areas where algebra and analysis interact very fruitfully: differential geometry and mathematical physics, including in particular the research area where I work, the integrable systems.
For example, looking at the introductory notes
https://www1.maths.leeds.ac.uk/~siru/ISsurv%282015%29.pdf
or, say, the introduction of a recent paper of mine
Article New integrable (3+1)-dimensional systems and contact geometry
and references therein, could give you a bit of an idea where things now stand in the field of integrable systems.
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