I want to be a guest speaker at a university Currently, I am doing a PhD and I want be a guest speaker at one of my previous universities. The topic of my PhD is very similar to what I studied at the university and therefore, I want to give a small talk to the students there. How can I ask the authorities at my previous university to do the same?
Find out who organizes the relevant seminar. Often there is a website listing talks and a contact person. If you can't find this information, just pick someone from the faculty who is likely to be interested in your talk.
Either email that person yourself, or get someone (e.g. your advisor) to introduce you. Introduce yourself; say you are interested in visiting the department and/or giving a talk; indicate what you would like to talk about; and specify what dates work for you. The following email is a good start.



  Dear So-and-so,
  
  I am visiting Madeuptown the week of Sextember 47th and would be interested in visiting the University of Madeuptown underwater basket weaving department. 
  I am currently a PhD student in underwater basket weaving at Fictitious University advised by John Namedrop and I have been working on the use of new wicker weaving materials, which may be of interest to the students in your department.
  Please let me know if you would be interested in arranging something.
  
  Best,
  
  Das


I have been the organizer of a computer science group seminar (in the US) and I regularly received such emails. Indeed many of them were much more direct than this. 

Most likely, you'll be invited to speak. However, it is possible that they are unable to schedule something suitable at the time you specify. (If you don't get a positive response, then leave it at that. I.e., don't pester them.)

It's worth a try if you are interested in giving a talk. The worst thing that could happen is that your email is ignored.

If you are lucky, they may offer to cover some of your expenses. However, don't bet on it if you are "self-inviting."