Application for research program is indeed throughout the year. However, it's not just submitting your application and hoping that the said uni will accept you, these are the things you need to do before that, in the case of UM, you need to first find a supervisor
1. Whose research field interests you.
2. Who has vacancy and is willing to accept you.
2. Who has sufficient grant that can support you for the entirety of your masters/phd research project.
4. Talk to him/her about his/her current project(s), or propose one
After that, write a proposal, get your proposal approved by your supervisor/co-supervisors, only then you can submit your application.

Good Luck 

Great views ! How long they usually reply you ?

Oh thanks for your reply. Are you studying PhD locally now ? What is the research quality in Malaysia nowadays ? I am not comparing it to other university in western country , but is it comfortable to do research locally ( those grants , fundings etc, notice that I did not include sciences subject like physics chemistry and bio which requires a lot of experiment apparatus and data analysis )

I am interested in pure mathematics phd . Personally I am not a wealthy person ( neither poor ) , so I am quite concerned whether i can study overseas , because maximally i can only pay for the flight tickets , toefl , gre test,admission Fees and one month expenses . Plus , I got a first class honors so I might probably straight away go for phd , I don't know this will affect my enrollment to graduate programmes or not ? ( since I know quite a lot of european country requires a master )

First of all , I appreciate your reply and I learnt a lot from you. Indeed , pure mathematics is something theoretical. Personally , I wish I can study in a cheapest way and get a job as a lecturer in Malaysia. I am very interested to pursue knowledge and hoping that I can make something new that's why I am here for PHD. Yet , I do need some financial assistance so that I can study without any concerns. I managed to make a mini thesis when I was in year 4 and my lecturer said I am quite suitable for a jump to PHD. I did find a supervisor which is pretty good in local university.

THE MAIN REASON I want to study overseas because I might learn new things from those modern countries. But , do you think it is actually almost the same if I studied in local U since nowadays we can access a lot of knowledge from internet , plus pure mathematics do not require any experiment apparatus ( maximally a PC with good processing power ) . For me ,

Local U - fast(don't have to wait for admissions , visas,those might take a year etc) , cheap (I got a fellowship which the salary is near to a fresh graduate) and less problematic (my parents will be comfortable with me)

Overseas - can learn from grand-masters in mathematical field . I did see a lot of scientist from Malaysia succeed in their field if they did research overseas , but we don't see a lot locally . I don't know what is the reasons though . Was it because the knowledge difference between modern countries and Malaysia?

I don't really mind studying environment though . For me , Malaysian has a good environment , what worries me was the knowledge , I am just scared I can't learn something new forever if I studied locally. As job prospects , I am not sure whether local PHD have less employability than overseas PHDs?

I truly need some suggestions on whether to take it Local U or Oversea . Anyway , thanks for your patience if you did read through my post . Regards.

Thanks for your suggestions ! I think I get what you meant , it's the supervisor that matters compared to university. By the way , which country have you been pursuing phd ?

Lastly , I have a simple question , by right i am going to apply for phd next year , while waiting for approval , and if I get  , then visa applications(if overseas ) and other sort of stuffs , these procedures take a few months to do . During these few months , what should i do ? Part time job ? Or brush up my knowledge for the research ?

As for finding a part time job, it depends. Some supervisors will ask you to start working in the lab as RA while prepping your proposal and if they didn't you might need to find a part time job because it will take some time for you to write your proposal and getting it approved by your supervisor.