Admissions Advice Online

Hello,
Your statement of purpose reads more like a research paper and not a statement of purpose. Remember, your statement of purpose should provide some background about you as a person, your likes, some of your research and a LOT about the program you are applying to and how it can help you achieve your stated goals. All I see here is a recap of your past research and interests and very little about the program and why you are specifically applying to this program. -selectiveadmissions.com
 

Just leave it be, it doesn't matter. They want to know in order to know who their competition is, nothing else, so it doesn't matter if it's not the most accurate or you changed your mind about some school.

Hello,
All programs take a range of students. While many students do have some research, all schools accept a good number of students that have absolutely no research. This is where you fit in. The key for you here is to take whatever experiences you do have and link them to why you are applying to the program and how this specific program will help you get from point A to B. When the adcom sees that their program is NEEDED to help you achieve your career goals, they are more likely to admit you. -selectiveadmissions.com

My opinion is no. Fix it for other applications, but this is unlikely to make a difference. I think asking to resubmit will draw even more attention to it and could potentially hurt you because it might indicate that you are focussing too much on details that do not matter.
But again, this is just my opinion based on what I think is important in my field. Get other opinions too

Generally, 1,000 words or less is the standard length of a well written SOP unless otherwise stated in application. I would advise you to always use single space unless otherwise requested as no one wants to really sit and read a 4 page SOP. In most cases you will enter your essays into an online system where the admissions office can format the SOP to print/display the way they want it....single/double spaced. - Admissions Advice Online

As an international student, your GRE scores will be even more important. Unless you come from an undergraduate institution that is well recognized, the GRE is the only way that schools can gauge the quality of your degree. The reality is that most of the schools that you listed receive far more qualified applicants than they have spaces for. As such, virtually none will lower their acceptance criteria. You can either study for a year while you intern to eventually retake the exam while building experience(Recommended) or select other programs that better align with your stats. - Admissions Advice Online 

Never. You should never talk about anything that will not positively highlight your application in your SOP. Almost all programs will review each applicant on his/her merits. That is to say, did they make the most with what was available to him/her? If you are lacking in one area, they will expect to see strength in other. If your application is assembled correctly, they will notice this. There is no reason to draw attention to a possible weak area of your application, especially when you attempt to explain it away with an excuse that might be considered offensive no matter how valid it is. - Admissions Advice Online

Hello,
 
As always, I recommend that applicants NEVER mention anything negative in their SOP. Remember, your SOP should always come from a standpoint of strength. Also, as a 40 year old applicant, you will not be expected to have a extra curricular activities  portfolio as robust as let's say a traditional undergraduate. However, there will be no excuse for a complete lack of involvement. Just highlight the experiences that you do have and frame them as best as you can. -Admissions Advice Online

This is called "senioritis." Schools can, and have many times rescinded an offer for admissions because of this.You are best not to even say anything. As long as your graduation and degree is posted, adcoms will not get involved. The office secretary will just post as normal. However, if you send an email and bring attention to this, then an admissions officer will read this and that might raise some problems. - Admissions Advice Online  

This is called "senioritis." Schools can, and have many times rescinded an offer for admissions because of this.You are best not to even say anything. As long as your graduation and degree is posted, adcoms will not get involved. The office secretary will just post as normal. However, if you send an email and bring attention to this, then an admissions officer will read this and that might raise some problems. - Admissions Advice Online  

Without a doubt, the best way to get into these programs is research. If you complete undergrad research, you will be very valuable to most top programs. Your GPA and GRE scores are in line with most top programs and the math courses that you have already taken show that you can handle the quant nature of the program. You also have varied experience. You just need to mix everything together for a powerful SOP and you should have no problem with admissions. -Admissions Advice Online