How to email a professor requesting to supervise a PhD thesis? I am interested in applying to PhD in Mathematics in University of Luxembourg. In the webpage it is written that &quot;We recommend that you contact potential thesis supervisors by email to discuss your areas of general research interest and specific ideas for a thesis topic. You may consider sending a description of the research project and your CV. Once you have identified a supervisor, he/she will initiate the application process and complete the application for admission.&quot;
Therefore, I need to contact potential thesis supervisors by emails to discuss  areas of general research interest and specific ideas for a thesis topic. That is, first I need to convince a professor to supervise me or I am well prepared to work under the professor.
Therefore, I am thinking that I will send an email to a professor. But I am facing some troubles.
I want to apply to PhD in Mathematics on Algebraic Topology. One thing that I have understood is to be selected I need to show interest on that particular subject. I want to convince the professor that I am well prepared to work under him and eager to join his research group. But, the problem this how to show the interest?
I have planned something and I want to execute my plans. I am sharing with you. One of the ideas is to read some papers of the professor and do comment on that. But I think it's not so the good idea because it may be cause of a bad impression. But I may be wrong. Now I am writing here, what I am thinking.
I am thinking the followings.

Since, I am particularly interested in Algebraic Topology, I will attach in the email a detailed  learning roadmaps (mentioning the references) of Algebraic Topology in which way I learned the subject.
I am thinking that I will attach (or give a google drive link) solutions of the exercises of some books, the exercises of those books are not easy to solve (even for researchers). I am preparing the hand written  solutions on the IPad.
Also, I will attached a write-up for the ideas for future research on the particular topic, in which I would like to help from some papers of the professor and mention those in my write-up. That's mean I will read some papers of the professor and made a write-up where I will include my ideas as well as some parts (or results or may be some ideas) of the Professor's paper.

Please advise me. Am I thinking in a right way? Please help me. It will also be helpful for me if you add your opinions and advise in details.
Thanking in advanced.
I don't think you need to send a &quot;learning roadmap&quot; or solved exercises. A learning roadmap doesn't really convey whether you have learned the subject. The University of Luxembourg requires that you have a masters degree before you start your PhD. If you have a masters in math, then you've probably solved a few exercises in the process.
Having ideas for future research projects is a very good thing, especially if they are ideas relating to the work of a potential advisor. My advice is to keep the initial email short and to the point -- you can add more details once you get a response. Something like:
&quot;Dear Prof. [So and So],
I am interested in applying for a PhD position at the University of Luxembourg. I obtained my Masters from [the Institute of Such and Such]. I would be interested in working on a project relating to your work on [informal group laws]. I have a few ideas for potential projects in this direction. Please let me know if you would be willing to discuss these.
Thank you in advance for your time,
user454229&quot;
Of course, don't just copy this and insert the relevant details. You want to write the email in your own voice. This is just to give you an idea of how to keep it short and to the point. And good luck!