There have been many open challenges questions in this forum.
For instance,
- Research and open challenges in Programming Language Theory
- What are current open problems in compiler theory?
- Outstanding conjectures and open problems in (algorithmic) game theory?
- Open problems on the frontiers of TCS
- Open problems related to Graph isomorphism
What are some of research avenues, areas available within communication complexity? What are some of most interesting open problems?
Some famous open problems AFAIK are status of log-rank conjecture, status of $\mathsf{IP}^{}\notin\mathsf{PH}^{cc}$, status of examples of natural problems in $\mathsf{NP}_k^{cc}$,$\mathsf{coNP}_k^{cc}$.
What are some of lesser known open problems or areas that are not yet mature within communication complexity or areas that have made new progress but much more needs to be done?
Are there any problems relating other areas such as circuit complexity, machine learning etc?
AFAIK communication complexity is more about finding bounds that about algorithms. Are there any problems with number theoretic, algebraic, geometric flavor or more generally in an algorithmic flavor?
More generally what would be some areas a newly entered person should focus energy to select a problem to work on (comm complexity field does not have many hearsay research topics as other traditional areas in mainstream theory CS with direct real world applications)?