The PRIMES Experience
by The_Turtle, Jan 21, 2021, 4:49 AM
According to the poll from a previous post, it seems like you all wanted me to talk about my PRIMES-USA experience! I hope this is useful to prospective students, current students, or anyone who is curious about what high school research is like.
PRIMES-USA is a year-long, mentored mathematics research program sponsored primarily by the Massachusetts Institute of Technology’s mathematics department. Students in the program work closely with mentors to produce a research paper. You can find more detailed information about the program here.
What was my research about?
My research concerned the connection between constraint satisfaction problems and universal algebra. Informally, constraint satisfaction problems are decision problems that take the form of combinatorial puzzles; the goal is to determine whether or not there is an assignment of the variables such that each constraint is satisfied. For example, Sudoku is a constraint satisfaction problem; in Sudoku, there are 81 variables, and each constraint asserts that the nine variables the constraint applies to are distinct. Though solving a constraint satisfaction problem is generally NP-hard, certain constraint satisfaction problems can be solved in polynomial time by their linear programming relaxations (read: systems of linear inequalities). A fairly recent paper has characterized such constraint satisfaction problems as those for which the clone
My mentor conjectured that every clone containing symmetric operations of arities , , , , where is the domain of the clone, contains symmetric operations of every arity, which would imply the existence of a deterministic algorithm that can determine whether or not a constraint satisfaction problem is solved by its linear programming relaxation. The goal for my project was to prove this conjecture. While I ultimately was unable to prove the conjecture in full generality, I proved the conjecture for clones over a domain of size at most four, and I obtained strong evidence supporting the truth of the conjecture for a domain of size five.
Applying to the program
The PRIMES-USA application opens in mid-September of each year and is open to high school juniors in the United States. The application consists of three main parts: an entrance exam (which consists of around six “general” math problems and six “advanced” math problems), a personal statement, and two recommendation letters.
To an experienced olympiad contestant, the “general” math problems are mostly approachable. When I applied, the last “general” problem gave me a bit of trouble since I didn’t know the Ford-Fulkerson algorithm, but I eventually discovered it through numerous Google searches.
The “advanced” math problems require knowledge of basic abstract algebra, linear algebra, and calculus to solve. To prepare for the application, I studied sections I, II, and IV of Evan Chen’s Napkin. To my surprise, the material I learned was sufficient to solve every problem except for the second-to-last one
For the questionnaire and personal statement, one essentially explains why one wants to participate, what makes one a good candidate, and so on. I tried to finish it as early as possible so it didn’t have to be on my mind while working on the problem set. I suggest giving the question “what is your preferred area/topic of research?” some serious thought, because this is what you will be working on for an entire year.
I asked my school math team coach and Professor Loh, the director of a summer camp I attended, for recommendation letters. I originally asked Evan but I was bad and didn’t read his website, which said I couldn’t ask him for a letter since he was involved with admissions
Here are some tips to prospective applicants (these are just my opinions, so take these with a grain of salt):
Experience
As listed on the PRIMES website, the program is broadly split into three periods; the reading phase, the research phase, and the writeup phase. Though most students have weekly calls or meetings with their mentor, my mentor and I decided to have text conversations instead. The website says a dedication of around 10 hours per week is a minimum, and in my experience this is quite accurate. During the first month of the pandemic, I probably spent over double this amount, much of which was spent writing code.
I received an email from my mentor in early January, which detailed the background I needed to learn and the goals for the project. I initially found the email quite intimidating, since it contained numerous links to pretty much incomprehensible papers. However, after two months of painstakingly going through the material and asking my mentor for help when I couldn’t understand things (I’m not exactly the fastest at learning new material), I finally finished the reading phase. I mostly learned from my mentor’s constraint satisfaction problem notes, which were dense and daunting, but thankfully my mentor was patient and explained some of the concepts more thoroughly than he did in his notes document.
During the research period, I realized I didn’t need most of the background to be able to finish my mentor’s first goal: to classify the clones over a domain of size four containing a unary, binary, ternary, and quaternary symmetric operation, and then to prove that each one contained symmetric operations of every arity. I wrote a computer program to assist me with the classification, and proved, by hand, that each clone in the classification indeed contained symmetric operations of every arity. After I completed the domain of size four case, I rewrote the program twice more, making optimizations and adding functionality each time, and eventually my program was able to classify all clones over a domain of size 5 that satisfied the conjecture’s hypothesis (i.e. symmetric operations of arities 1, 2, 3, 4, and 5). However, there were too many clones to verify by hand, so I wrote additional code that allowed me to check that each one contained symmetric operations of arities up to twenty. Perhaps a brute-force proof that proves the existence of symmetric operations of all arities is possible? It’s likely doable, but it would take a long time and I don’t think it would be that instructive.
I completed the classification shortly before the interim report was due in May, so I hastily put some notes together and used it as my interim report. In retrospect, I probably should have put more effort into it so I could have received more valuable feedback. In a typical year, the interim report is used as a way to make sure that students have material to present at the annual PRIMES conference in May, but this year’s conference was moved to October. I am unsure if it will stay this way.
After completing the classification, my mentor and I used the classification to attack the general conjecture by looking at special subalgebras of the algebras corresponding to each clone. At one point I thought I had proof that any minimal counterexample to the conjecture required , but my mentor found a flaw a few weeks later and I was pretty disappointed. The method I used concerned a directed graph of subalgebras of an algebra; a similar method was able to show that every Taylor algebra (an algebra containing operations which satisfy an identity that cannot be satisfied with projection operations) contains cyclic operations of all arities greater than the size of its domain. However, summer break started before I could make any tangible progress. Since I was participating in four other summer programs, I put my PRIMES project on a hiatus, which in retrospect was a bad idea.
When summer break ended, I realized I had forgotten much of the work I had done previously, and I felt pretty unmotivated to put additional work into my project since I barely remembered how the directed graph argument worked and I didn’t write my progress
I started preparing my presentation a month before the October PRIMES conference. I spent a lot of the time writing and editing my presentation, which, save for the last slide, was entirely background material. My mentor made it clear to me that the goal for the presentation was to leave the audience with something memorable and not to present the material at a pace that half of the audience probably wouldn’t keep up with.
I gave my talk during a Saturday morning (at least in Central time) in a Zoom meeting. For the presentation, I wrote a script outline because I was afraid of forgetting some important detail that I didn’t list on the slides. In my opinion, the presentation went well. At the end, a professor asked some questions (if I recall correctly, one asked if considering an infinite domain was an interesting problem). After the PRIMES conference, I set immediately to work on writing up the paper so I could submit it to the Regeneron Science Talent Search, the application of which was due mid-November.
I made the mistake of not writing up my progress for the domain of size four case in full detail earlier. If I did, I could have applied to the Yau High School Science Award and could hae taken my time on the Regeneron Science Talent Search application. After all, I had all the material I included in the final paper in April, so there wasn’t any good reason to wait six months.
I ended up spending approximately three weeks writing the paper, and I went through approximately six revisions. I wrote the essays in three days (which is something you should NOT do). Two days before the deadline, I realized that my paper had to be in 1.5 spacing, but changing the spacing made the paper exceed the page limit of twenty pages! Out of frustration, I took out all the tables, which made the paper exactly twenty pages long. A day before the deadline, I realized that my paper was in 11pt font instead of the 12pt minimum font size, but adjusting the font size made the paper exceed the page limit by three pages. I reluctantly decided on turning most of the display-style math equations into inline equations, which made the paper look quite ugly, but at least the paper was exactly twenty pages again. (do you understAND my frustrATION?) I didn’t realize that we weren’t supposed to include any external links in the paper until the day of the deadline, so there’s that as well. In short, if you are applying, you should actually read the requirements document.
All I have left to do is to have the paper posted on the PRIMES website (edit: done!) and submitted to the Arxiv. The official end date of the program is January 15th, 2021. I find it slightly unfortunate that the directed graph argument was never realized, but my mentor said that if I ever wanted to continue the project outside of the PRIMES program, we certainly could!
Concluding Remarks
First of all, I want to say that I was quite lucky with the mentor assignment. My mentor was quite fun to chat with, and he was genuinely interested in helping me understand the material to the best of my ability. Especially since I want to go into research mathematics in the future, the experience was certainly an enlightening one which helped me better understand what a possible career in academia might look like.
Secondly, I know that my research was nowhere near spectacular, but it was definitely a worthwhile experience due to the mathematics I learned, the practice I had with writing and editing a research paper, and my participation in a mathematics conference. I was not chosen
Lastly, I know a fair number of you reading were in PRIMES last year as well; you should share your thoughts in the comments as well Thank you for reading!
PRIMES-USA is a year-long, mentored mathematics research program sponsored primarily by the Massachusetts Institute of Technology’s mathematics department. Students in the program work closely with mentors to produce a research paper. You can find more detailed information about the program here.
What was my research about?
My research concerned the connection between constraint satisfaction problems and universal algebra. Informally, constraint satisfaction problems are decision problems that take the form of combinatorial puzzles; the goal is to determine whether or not there is an assignment of the variables such that each constraint is satisfied. For example, Sudoku is a constraint satisfaction problem; in Sudoku, there are 81 variables, and each constraint asserts that the nine variables the constraint applies to are distinct. Though solving a constraint satisfaction problem is generally NP-hard, certain constraint satisfaction problems can be solved in polynomial time by their linear programming relaxations (read: systems of linear inequalities). A fairly recent paper has characterized such constraint satisfaction problems as those for which the clone
Some definitions:
of operations preserving the relations of the constraint satisfaction problem contains symmetric operations of every arity. However, no algorithm has been discovered that determines whether or not a clone contains symmetric operations of every arity.- An operation over a domain is a function ; the number of inputs, , is known as ’s parameter count, or arity. For example, addition is an arity-2 (e.g. binary) operation over the real numbers, and the logical NOT operation is an arity-1 (e.g. unary) operation over the Boolean domain .
- An operation is symmetric if its output does not depend on the order of its inputs. For example, addition is symmetric by the commutative property, but subtraction is not.
- A clone over a domain is a set of operations over closed under composition. The precise definition requires defining what are known as projection operations, which I won’t cover here.
My mentor conjectured that every clone containing symmetric operations of arities , , , , where is the domain of the clone, contains symmetric operations of every arity, which would imply the existence of a deterministic algorithm that can determine whether or not a constraint satisfaction problem is solved by its linear programming relaxation. The goal for my project was to prove this conjecture. While I ultimately was unable to prove the conjecture in full generality, I proved the conjecture for clones over a domain of size at most four, and I obtained strong evidence supporting the truth of the conjecture for a domain of size five.
Applying to the program
The PRIMES-USA application opens in mid-September of each year and is open to high school juniors in the United States. The application consists of three main parts: an entrance exam (which consists of around six “general” math problems and six “advanced” math problems), a personal statement, and two recommendation letters.
To an experienced olympiad contestant, the “general” math problems are mostly approachable. When I applied, the last “general” problem gave me a bit of trouble since I didn’t know the Ford-Fulkerson algorithm, but I eventually discovered it through numerous Google searches.
The “advanced” math problems require knowledge of basic abstract algebra, linear algebra, and calculus to solve. To prepare for the application, I studied sections I, II, and IV of Evan Chen’s Napkin. To my surprise, the material I learned was sufficient to solve every problem except for the second-to-last one
you caN’T just put a in the problem without USING IT
. I tried my best to write my solutions clearly; however, my solution to the last problem was heavily rushed, as I only realized my solution was incorrect on the day of the deadline.For the questionnaire and personal statement, one essentially explains why one wants to participate, what makes one a good candidate, and so on. I tried to finish it as early as possible so it didn’t have to be on my mind while working on the problem set. I suggest giving the question “what is your preferred area/topic of research?” some serious thought, because this is what you will be working on for an entire year.
I asked my school math team coach and Professor Loh, the director of a summer camp I attended, for recommendation letters. I originally asked Evan but I was bad and didn’t read his website, which said I couldn’t ask him for a letter since he was involved with admissions
Here are some tips to prospective applicants (these are just my opinions, so take these with a grain of salt):
- Edit your solutions several times, as you would do with any research paper. After all, PRIMES-USA is a research program, so they want their students to be able to present their work clearly.
- Even if you didn’t solve a problem, write up your progress. In my case, I included my own remarks as an addendum to each of my solutions. Maybe this helped my application? I’m not sure.
- Don’t be afraid to cite the sources (e.g. textbooks, websites) you used! After all, this is typical in research.
- Do NOT apply for the sole purpose of making your college applications look good. If you were truly focused on building a good college application (which you probably should not spend your high school life doing, anyways), your time is better spent somewhere else. For example, as a result of PRIMES, I barely spent any time on olympiad math during the latter half of my junior year.
Experience
As listed on the PRIMES website, the program is broadly split into three periods; the reading phase, the research phase, and the writeup phase. Though most students have weekly calls or meetings with their mentor, my mentor and I decided to have text conversations instead. The website says a dedication of around 10 hours per week is a minimum, and in my experience this is quite accurate. During the first month of the pandemic, I probably spent over double this amount, much of which was spent writing code.
I received an email from my mentor in early January, which detailed the background I needed to learn and the goals for the project. I initially found the email quite intimidating, since it contained numerous links to pretty much incomprehensible papers. However, after two months of painstakingly going through the material and asking my mentor for help when I couldn’t understand things (I’m not exactly the fastest at learning new material), I finally finished the reading phase. I mostly learned from my mentor’s constraint satisfaction problem notes, which were dense and daunting, but thankfully my mentor was patient and explained some of the concepts more thoroughly than he did in his notes document.
During the research period, I realized I didn’t need most of the background to be able to finish my mentor’s first goal: to classify the clones over a domain of size four containing a unary, binary, ternary, and quaternary symmetric operation, and then to prove that each one contained symmetric operations of every arity. I wrote a computer program to assist me with the classification, and proved, by hand, that each clone in the classification indeed contained symmetric operations of every arity. After I completed the domain of size four case, I rewrote the program twice more, making optimizations and adding functionality each time, and eventually my program was able to classify all clones over a domain of size 5 that satisfied the conjecture’s hypothesis (i.e. symmetric operations of arities 1, 2, 3, 4, and 5). However, there were too many clones to verify by hand, so I wrote additional code that allowed me to check that each one contained symmetric operations of arities up to twenty. Perhaps a brute-force proof that proves the existence of symmetric operations of all arities is possible? It’s likely doable, but it would take a long time and I don’t think it would be that instructive.
I completed the classification shortly before the interim report was due in May, so I hastily put some notes together and used it as my interim report. In retrospect, I probably should have put more effort into it so I could have received more valuable feedback. In a typical year, the interim report is used as a way to make sure that students have material to present at the annual PRIMES conference in May, but this year’s conference was moved to October. I am unsure if it will stay this way.
After completing the classification, my mentor and I used the classification to attack the general conjecture by looking at special subalgebras of the algebras corresponding to each clone. At one point I thought I had proof that any minimal counterexample to the conjecture required , but my mentor found a flaw a few weeks later and I was pretty disappointed. The method I used concerned a directed graph of subalgebras of an algebra; a similar method was able to show that every Taylor algebra (an algebra containing operations which satisfy an identity that cannot be satisfied with projection operations) contains cyclic operations of all arities greater than the size of its domain. However, summer break started before I could make any tangible progress. Since I was participating in four other summer programs, I put my PRIMES project on a hiatus, which in retrospect was a bad idea.
When summer break ended, I realized I had forgotten much of the work I had done previously, and I felt pretty unmotivated to put additional work into my project since I barely remembered how the directed graph argument worked and I didn’t write my progress
(note to self: Tanya was right! Always, always write progress down!)
down in full detail. I also convinced myself that the classification I completed wasn’t significant, especially given that the subalgebra progress was quite promising. At this point, I felt hopeless since I wanted to leave the program with a strong research paper, but I felt like I didn’t make any nontrivial progress on the problem. Eventually, my mentor was able to convince me to write my final paper about the classification instead of trying to make the directed graph argument work, since time was running out.I started preparing my presentation a month before the October PRIMES conference. I spent a lot of the time writing and editing my presentation, which, save for the last slide, was entirely background material. My mentor made it clear to me that the goal for the presentation was to leave the audience with something memorable and not to present the material at a pace that half of the audience probably wouldn’t keep up with.
I gave my talk during a Saturday morning (at least in Central time) in a Zoom meeting. For the presentation, I wrote a script outline because I was afraid of forgetting some important detail that I didn’t list on the slides. In my opinion, the presentation went well. At the end, a professor asked some questions (if I recall correctly, one asked if considering an infinite domain was an interesting problem). After the PRIMES conference, I set immediately to work on writing up the paper so I could submit it to the Regeneron Science Talent Search, the application of which was due mid-November.
I made the mistake of not writing up my progress for the domain of size four case in full detail earlier. If I did, I could have applied to the Yau High School Science Award and could hae taken my time on the Regeneron Science Talent Search application. After all, I had all the material I included in the final paper in April, so there wasn’t any good reason to wait six months.
I ended up spending approximately three weeks writing the paper, and I went through approximately six revisions. I wrote the essays in three days (which is something you should NOT do). Two days before the deadline, I realized that my paper had to be in 1.5 spacing, but changing the spacing made the paper exceed the page limit of twenty pages! Out of frustration, I took out all the tables, which made the paper exactly twenty pages long. A day before the deadline, I realized that my paper was in 11pt font instead of the 12pt minimum font size, but adjusting the font size made the paper exceed the page limit by three pages. I reluctantly decided on turning most of the display-style math equations into inline equations, which made the paper look quite ugly, but at least the paper was exactly twenty pages again. (do you understAND my frustrATION?) I didn’t realize that we weren’t supposed to include any external links in the paper until the day of the deadline, so there’s that as well. In short, if you are applying, you should actually read the requirements document.
All I have left to do is to have the paper posted on the PRIMES website (edit: done!) and submitted to the Arxiv. The official end date of the program is January 15th, 2021. I find it slightly unfortunate that the directed graph argument was never realized, but my mentor said that if I ever wanted to continue the project outside of the PRIMES program, we certainly could!
Concluding Remarks
First of all, I want to say that I was quite lucky with the mentor assignment. My mentor was quite fun to chat with, and he was genuinely interested in helping me understand the material to the best of my ability. Especially since I want to go into research mathematics in the future, the experience was certainly an enlightening one which helped me better understand what a possible career in academia might look like.
Secondly, I know that my research was nowhere near spectacular, but it was definitely a worthwhile experience due to the mathematics I learned, the practice I had with writing and editing a research paper, and my participation in a mathematics conference. I was not chosen
probably because I didn’t explain why my research was important well enough, according to my mentor
as a Regeneron Science Talent Search semifinalist, but ultimately, there are bigger fish than STS.Lastly, I know a fair number of you reading were in PRIMES last year as well; you should share your thoughts in the comments as well Thank you for reading!
This post has been edited 1 time. Last edited by The_Turtle, Jan 22, 2021, 2:45 AM
by anser, Jan 21, 2021, 5:10 AM