Improve my email below: 

Dear Professor,

I hope this email finds you well. I just wanted to wish you a very merry Christmas and a happy holiday season.

I also wanted to let you know that I have successfully started my PhD program and everything has been going well so far.

I was also wondering how you have been doing. I hope that you have been enjoying the holiday season and that everything is going well for you.

Best wishes,
Dear Professor [Last Name],

I hope you are doing well. As the holiday season approaches, I wanted to wish you a Merry Christmas and a joyful holiday.

I also wanted to share that I have successfully started my PhD program, and everything has been going smoothly so far.

It would be great to hear how you have been doing as well. I hope you are enjoying the holiday season and that all is progressing positively.

Best wishes,
[Your Full Name]
Do you think this email looks good? Or maybe there are some ways to improve it?
From what I can tell, the email you've shared looks polite and professional overall. However, here are a few suggestions for ways you could improve it:

- Include a warmer and more personal greeting than just Dear Professor. For example, Dear [Professor's Name], I hope this note finds you well. References to shared experiences or your relationship can help strengthen the tone.
- In the first paragraph, be more specific than just wishing for a merry Christmas and happy holidays. Mention something you appreciate about the professor or your time working together. Keep the tone sincere and genuine.
- In the second paragraph, provide a bit more detail about your PhD program experience so far, rather than just saying everything has been going well. Share any challenges you've faced and what you've learned, or mention specific courses or research you're enjoying. This helps give the professor more context about your journey.
- In the third paragraph, ask an open-ended question about the professor to start a genuine two-way dialogue. For example, How has your research been progressing? or What are you most looking forward to this semester?. Show interest in their work and life, not just the holidays.
- Close with a forward-looking statement about staying in touch or meeting up, rather than just Best wishes. For example, I look forward to continuing our conversation. Please let me know if you'll be available to meet for coffee.

Overall, keep the tone warm, personal and authentic. The email is a chance to strengthen your connection, not just send a generic greeting. Feel free to modify the suggestions as you see fit. I hope this helps! Let me know if you have any other questions.