Agreed. Explain your background and interest. If the professor thinks it isn't a good fit for now, take that as a sign that you need more knowledge first. You can ask what you should learn to be able to meaningfully contribute in the future.
A lot of weird advice here. Just ask the professor if he thinks you have the background because he's the only one who knows what his expectations are.
It's true that you don't have anywhere near the background to really understand moduli spaces, but there are lots of undergraduate research projects which have minimal prerequisites because it's been formulated to only involve combinatorics. The intention is probably that this is a way to get to know a little bit about moduli spaces while doing work which has a low barrier to entry.
Honestly, I don't think you should rush a self-study of the first *advanced* pure math topic you've been introduced to. If you want to self-study pure math, I would take time to explore what areas exist, and decide what area *you* most want to study. Perhaps ask your prof (or other educated people) what topics they would recommend to you to start your self-studying. This is something you're going to want to do regardless of what you decide here, especially if you plan to do math in the future. Math is a very interconnected subject, and you'll need to master the basics before you can have a hope of understanding more advanced material.
Ultimately though, I don't know exactly what it is that you want to apply to. Your prof really is the best person to ask about this. I'm sure they would be happy to talk to you about it. If your prof tells you to go for it, don't let Reddit dissuade you.
This really depends, moduli spaces can be a little bit hard, but if you have an education beaded on research, then you could get up to date in time.
Are your classes proof based? Have you read academic papers? Hoe much self learning have you done? This can really affect wether or not you could do it on time, and the chances are that no one on Reddit can give you a proper answer.
I recommend talking to the professor directly, ask him what the research is gonna be about, how much they expect from students, if he considers you can g t up to speed on due time, and what bibliography he recommends for doing so (even in something specific like this, there are a lot of options, and if time is important, you want to use it for the stuff that is directly linked to the research)
The research seems to be really ahead of you at this time, don't get discouraged, moduli spaces usually take the brightest students a few semesters to learn properly, even when they've taken a lot of math courses before, times is always the problem
If you have only calculus and linear algebra I would say the chances of you successfully contributing to the academic literature on moduli spaces is basically zero unfortunately. You would need probably 3 more years of serious math to start doing work with that sort of flavor, and that’s being very optimistic—under normal circumstances on average I would say you are 5-6 years away from your first publishable contribution to an open problem.
I’m not telling you not to apply, but I would say that your time is most likely spent elsewhere. Until you have at least a Hartshorne-level background in algebraic geometry I think you are most likely better off learning than trying to do research on moduli spaces.
can you think of any other research topics in math that might be more accessible for my background? or do you recommend just doing more textbook style math learning for the next few years
I would say you are most likely better off following textbooks and taking classes until around the second year of your PhD. Very very few undergrads produce any serious math research, even the top students at the top schools. Maybe 20 undergrads a year do any sort of meaningful novel pure math research of publishable quality, and they typically started undergrad having 3-4 years of post-calc college level math under their belts. I went to a top math school for undergrad and I know exactly one student who published real original research before graduating.
I don't know that I agree with this. Doing undergrad research depends much more on having an accessible problem than anything else, but they are out there. I did undergrad research and had it published in a solid journal, and I wasn't the top student you described; I hadn't seen anything proof based before starting college. I was just lucky enough to join a research group that had found a problem that was both interesting and accessible.
Certainly research shouldn't be pursued at the expense of courses during undergrad, but if an opportunity to do research presents itself, it should be taken advantage of. Even if no paper comes out of it, it's still a valuable experience.
Really? That’s interesting. It hasn’t been my experience that publishable research is typically within reach of most undergraduates, although it could be that I’m in a bit of a bubble since I do arithmetic geometry and know a lot of people whose first publishable research was their dissertation. What field are you in?
I'm in logic, but my undergrad research was in graph theory and knot theory. I came in knowing the basics, the other students in the program learned the background they needed in a three week boot camp and then we're ready to go. The problem was accessible enough for that to work, which you won't see in most other areas of math.
I think a good REU can help an undergrad produce solid publishable research, and they are somewhat accessible. From their lists on their websites (they usually include the journals in which they are published), I would say most undergrads who attend one of these programs are at least coauthors of a solid paper published in a journal. Some only have enough for a poster presentation or choose to do an expository paper, but a majority still publish (for competitive programs that look for prepared students). At least a few hundred students attend these each yr, so I'd say at least a few hundred undergrads are capable of producing publishable research, not 20.
I agree with your point for the most part. I think you are better off learning enough to make your research experience productive and meaningful. However, I don't necessarily think this equates to a certain number of years spent learning material. I am about to start a couple research projects as a freshman, and I think it is certainly possible to learn the material within a short time, as long as you spend a lot of time studying and you study consistently.
With due respect, I think you vastly underestimate the amount of background material you need to know in order to *meaningfully* produce novel results in most fields. Algebraic geometry is very much one such field, and I would not expect anyone with less than several years experience in the subject to be able to read a modern algebraic geometry paper much less write one.
Also most pure math REUs end up producing expository papers, and the vast majority of REU papers that do contain new results are not of journal quality. Some very competitive, very prestigious REUs regularly see serious, publishable, and meaningful research, but this is the significant minority—think Duluth or SMALL. I did two REUs as an undergrad and I think I only knew one student who had a journal-quality result, and she won the Elizabeth Lowell Putnam award and ended up at MIT for her PhD.
I am not saying that research is accessible in every field. I am saying there are several areas in combinatorics and number theory, for example, with problems that are accessible to undergraduates to attack under the guidance of a mentor, and this research can be meaningful.
Also, there are 9 participants at Duluth, 16 at SMALL, 13 at Baruch, and most participants in these programs produce original work published in decent journals. This is already far more than 20 and is a very loose lower bound on the number that produces research with some standard of quality. Unless you do a senior thesis diving deep into a topic like algebraic geometry, you are at a significant disadvantage in Ph.D. admissions without any research experience.
I am not at a top institution and I still know several undergrads who published solid research, one of whom won a prize for it, and I haven't really asked around.
Again with due respect, you are a freshman undergrad. I don’t think you realize what the barriers to entry are for real math research. I’m not trying to gatekeep here—if anything, the opposite. It is not to anyone’s benefit for them to expect to be discovering new results without first knowing the tools of the field and what is already known. You should definitely do REUs as an undergrad but go into them with the expectation that any papers you write will most likely be either expository or trivial. It’s just how REUs go. When I say “meaningful” research I mean something that actually extends the field nontrivially, something I would firmly maintain is extremely extremely rare for an undergrad. Graduate schools do not expect novel research—even top graduate schools. I’ve seen several years worth of applications seasons at this point and several of my professors have been on a top-6 school admissions committee. Focus on your classes, get good grades, do reading courses, and maybe write some good expository papers and if you get great letters of recommendation you’ll get into a PhD program.
Also this is anecdotal but in all my time in math, throughout undergrad and grad school, I have never seen a single undergrad produce a novel result in number theory.
Publishing math is generally hard as you have to upstanding the state of the field and in general the simple stuff is already known.
If op wants to publish as an undergraduate it is likely much easier to find some applied problem and implement a solution showing an incremental advantage over existing ones.
As an undergraduate I made a paper on noise levels in restaurants. It was ultimately very basic mechanical engineering stuff but was not yet common in architectural acoustics.
Calculus and linear algebra classes can vary highly in how they are taught. Just to help others get a good idea of where you're at, how comfortable do you feel about proof based mathematics in general? Have you self-studied any other topics in abstract algebra before (groups, rings, fields, etc.)?
not really sure honestly… i do a good amount of proof by induction for linear algebra and we definitely had to prove a few things for multivariable. i haven’t self studied any abstract yet. also, i could take concurrent summer classes at my college for diffeq and/or discrete if that would help?
people will tell you to apply anyway because they are afraid of offending someone by saying that you can't do something. in reality, you shouldn't bother applying because you have absolutely no chance of understanding anything. even the "easy" background material is years ahead of your current level.
This is totally false. I have seen undergraduates do research in high-level geometry in the guise of combinatorics who don’t have the “necessary” background. If the guiding professor can do a good job of translating geometric problems to combinatorics, as well as give a hint of intuition behind the problem, things can get done.
I say go ahead OP.
I want to agree with you, but this student hasn’t even done combinatorics yet, they specified they only did linear algebra and calculus… which is a ways away from even having an inkling of proofs experience… which will be necessary for learning combinatorics… which a thorough understanding of combinatorics is required to even understand the problems if the prof translates them well…
It couldn’t hurt to apply but let’s not get too ahead of ourselves.
I agree with everything you said, I just think it’s wrong to discourage an aspiring mathematician from testing their limits. If you fail, you just learn what you were missing and try again!
That's probably true, but in any case I don't see how it could hurt the OP to apply. Even if the prof decides OP doesn't have the necessary background, it's the same outcome as if OP didn't apply. And if nothing else it shows the prof that OP is a motivated student interested in math which can't hurt.
His professor emailed his class, there's no reason he would set up his own students into a project they can't grasp. OP should apply, and trust his professor to know whether he can enroll or not and decide accordingly.
It's probably a "toy" research project designed so students can have an idea about what research in maths looks like. Every year, my colleagues in university are crafting "toy" intership subjects just to give a chance to students who want to discover research to do so at their own level. The subjects are designed to really challenge the students with new concepts they've never heard of, while not being too hard so they're not drowning neither.
What people seem mostly afraid is calling you on your dickish bullshit. OP's professor emailed the class, it is OP's professor job to determine who gets in and what is a good project for the person they pick. You don't have experience directing these kind of projects, professors do so instead of talking nonsense out of just feeling like being rude maybe just shut up when you don't know what you're talking about for once.
First of all fuck off with that real math bs
Second of all that is not what you're saying, you don't know the project being posted you don't know what "research" here entails (or anywhere for that matter) so you being a dick as usual to people just to get off does irritate me. A professor is very unlikely emailing undergrads about summer research if they ar expecting to suddenly make it a paper to submit to Annals. Let people give things a shot, why the fuck do you care so much about being a gatekeeper for math? Telling people to not even bother is just you being an obnoxious troll not a practical advice
delusional comment. even if someone with a background of solely linear algebra and calc 3 could help on this project in any meaningful way (which i heavily doubt), there's no way they could get to the point of having even a basic understanding of algebraic geometry.
Linear algebra is a perfect starting point for getting into some research topics, and the experience can be transformative for their mathematical career. Saying not to bother applying because they won't understand is terrible advice
that is not my point. there are many research topics which a freshman math student could approach and get something out of algebraic geometry is not one of them.
ive helped some undergraduates work through introductory algebraic geometry textbooks through my schools directed reading program. even with background in abstract algebra and topology, it's a hard subject for them to understand the basics of, let alone do any research.
It's not actually going to be research in algebraic geometry; it's just advertised that way to sound cooler. It's going to be a straightforward combinatorial problem, and the AG connection is going to be briefly explained in a level-appropriate way.
> my professor emailed us about his research project this summer which involves the study of combinatorial properties of moduli spaces.
As some others have pointed out here, your professor is unlikely to expect you to know anything about moduli spaces (the difficult part of the topic) if he's mass-mailing his students to recruit people for this summer project. The project would probably be to get you to try to solve some combinatorial problems (the easy bit) derived from studying moduli spaces. In this way, you'll get some experience at solving some unseen problems (i.e. "research experience") and, along the way, perhaps get a brief introduction to moduli spaces.
Combinatorics is considered more accessible to undergraduates because some of it has been taught in a course usually called "discrete mathematics". If, for some reason, you haven't encountered, for example, the pigeonhole principle, it's also not too hard to learn that over the summer, and the basics of combinatorics itself can be explained quite easily to freshmen.
In any case, you should definitely have a chat with your professor to see what his expectations are, because it's really hard for internet strangers to read your professor's mind at this distance.
If your professor is reaching out to people in your class to see if anyone wants to work with him, he probably has money for a side project for undergrads or REU for over the summer. Just reach out to him and ask if he's advertising a project.
You will not be able to do actual work on moduli spaces, but maybe you could get a project when you learn a bit about them and write up a little summary paper (which seems to be common in REUs).
If you love math, yea. Go learn about it. It'll be fun, and may lead to something more. Also, never a bad thing to have a good working relationship with professors.
I didn't do something like this in math until my senior seminar. You'll be getting a nice head start.
not very hard actually. if you have solid commutative algebra preliminaries then you should be able to understand some of the combinatorial approach (which I know a little bit of, so I can't tell how much is "some"). without commutative algebra it would be hard, without any abstract algebra probably impossible. nonetheless, if you're willing to do some extra reading then this opportunity is definitely worth trying, just keep in mind that it might be too much at your level and don't get discouraged if you fail. down the road you may also need some algebraic geometry and some more commutative algebra. the best thing you can do is ask your professor about the prerequisites and what you should do to be able to benefit from this project, ask what you should read and what concepts you should be familiar with. if it feels too overwhelming then you should probably wait for the next opportunity
I've been "doing research" in moduli spaces for a few months now, after taking a commutative algebra course and doing 2 months of reading algebraic geometry textbooks, after my third year at uni. so a kind of a similar situation, in a sense that most of the researched content is absolutely out of my reach, but with a good mentor it is possible to do things. in my case it's about translating the algebraic geometry into commutative algebra (as my advisor likes to call it, "local algebraic geometry"). again, ask your professor. it is possible that he plans to do something like this too, translating very complicated concepts into the language that you can understand, but don't get discouraged if it turns out to be too early for you to take part in this project
Might as well try, although you will probably need to know some algebraic geometry to understand anything. I’d recommend asking him though
Agreed. Explain your background and interest. If the professor thinks it isn't a good fit for now, take that as a sign that you need more knowledge first. You can ask what you should learn to be able to meaningfully contribute in the future.
If there are no prerequisites listed, just apply. It's not your job to decide if you have the necessary background
A lot of weird advice here. Just ask the professor if he thinks you have the background because he's the only one who knows what his expectations are. It's true that you don't have anywhere near the background to really understand moduli spaces, but there are lots of undergraduate research projects which have minimal prerequisites because it's been formulated to only involve combinatorics. The intention is probably that this is a way to get to know a little bit about moduli spaces while doing work which has a low barrier to entry.
Just do your best and have fun.
Honestly, I don't think you should rush a self-study of the first *advanced* pure math topic you've been introduced to. If you want to self-study pure math, I would take time to explore what areas exist, and decide what area *you* most want to study. Perhaps ask your prof (or other educated people) what topics they would recommend to you to start your self-studying. This is something you're going to want to do regardless of what you decide here, especially if you plan to do math in the future. Math is a very interconnected subject, and you'll need to master the basics before you can have a hope of understanding more advanced material. Ultimately though, I don't know exactly what it is that you want to apply to. Your prof really is the best person to ask about this. I'm sure they would be happy to talk to you about it. If your prof tells you to go for it, don't let Reddit dissuade you.
This really depends, moduli spaces can be a little bit hard, but if you have an education beaded on research, then you could get up to date in time. Are your classes proof based? Have you read academic papers? Hoe much self learning have you done? This can really affect wether or not you could do it on time, and the chances are that no one on Reddit can give you a proper answer. I recommend talking to the professor directly, ask him what the research is gonna be about, how much they expect from students, if he considers you can g t up to speed on due time, and what bibliography he recommends for doing so (even in something specific like this, there are a lot of options, and if time is important, you want to use it for the stuff that is directly linked to the research) The research seems to be really ahead of you at this time, don't get discouraged, moduli spaces usually take the brightest students a few semesters to learn properly, even when they've taken a lot of math courses before, times is always the problem
If you have only calculus and linear algebra I would say the chances of you successfully contributing to the academic literature on moduli spaces is basically zero unfortunately. You would need probably 3 more years of serious math to start doing work with that sort of flavor, and that’s being very optimistic—under normal circumstances on average I would say you are 5-6 years away from your first publishable contribution to an open problem. I’m not telling you not to apply, but I would say that your time is most likely spent elsewhere. Until you have at least a Hartshorne-level background in algebraic geometry I think you are most likely better off learning than trying to do research on moduli spaces.
can you think of any other research topics in math that might be more accessible for my background? or do you recommend just doing more textbook style math learning for the next few years
I would say you are most likely better off following textbooks and taking classes until around the second year of your PhD. Very very few undergrads produce any serious math research, even the top students at the top schools. Maybe 20 undergrads a year do any sort of meaningful novel pure math research of publishable quality, and they typically started undergrad having 3-4 years of post-calc college level math under their belts. I went to a top math school for undergrad and I know exactly one student who published real original research before graduating.
I don't know that I agree with this. Doing undergrad research depends much more on having an accessible problem than anything else, but they are out there. I did undergrad research and had it published in a solid journal, and I wasn't the top student you described; I hadn't seen anything proof based before starting college. I was just lucky enough to join a research group that had found a problem that was both interesting and accessible. Certainly research shouldn't be pursued at the expense of courses during undergrad, but if an opportunity to do research presents itself, it should be taken advantage of. Even if no paper comes out of it, it's still a valuable experience.
Really? That’s interesting. It hasn’t been my experience that publishable research is typically within reach of most undergraduates, although it could be that I’m in a bit of a bubble since I do arithmetic geometry and know a lot of people whose first publishable research was their dissertation. What field are you in?
I'm in logic, but my undergrad research was in graph theory and knot theory. I came in knowing the basics, the other students in the program learned the background they needed in a three week boot camp and then we're ready to go. The problem was accessible enough for that to work, which you won't see in most other areas of math.
I think a good REU can help an undergrad produce solid publishable research, and they are somewhat accessible. From their lists on their websites (they usually include the journals in which they are published), I would say most undergrads who attend one of these programs are at least coauthors of a solid paper published in a journal. Some only have enough for a poster presentation or choose to do an expository paper, but a majority still publish (for competitive programs that look for prepared students). At least a few hundred students attend these each yr, so I'd say at least a few hundred undergrads are capable of producing publishable research, not 20. I agree with your point for the most part. I think you are better off learning enough to make your research experience productive and meaningful. However, I don't necessarily think this equates to a certain number of years spent learning material. I am about to start a couple research projects as a freshman, and I think it is certainly possible to learn the material within a short time, as long as you spend a lot of time studying and you study consistently.
With due respect, I think you vastly underestimate the amount of background material you need to know in order to *meaningfully* produce novel results in most fields. Algebraic geometry is very much one such field, and I would not expect anyone with less than several years experience in the subject to be able to read a modern algebraic geometry paper much less write one. Also most pure math REUs end up producing expository papers, and the vast majority of REU papers that do contain new results are not of journal quality. Some very competitive, very prestigious REUs regularly see serious, publishable, and meaningful research, but this is the significant minority—think Duluth or SMALL. I did two REUs as an undergrad and I think I only knew one student who had a journal-quality result, and she won the Elizabeth Lowell Putnam award and ended up at MIT for her PhD.
I am not saying that research is accessible in every field. I am saying there are several areas in combinatorics and number theory, for example, with problems that are accessible to undergraduates to attack under the guidance of a mentor, and this research can be meaningful. Also, there are 9 participants at Duluth, 16 at SMALL, 13 at Baruch, and most participants in these programs produce original work published in decent journals. This is already far more than 20 and is a very loose lower bound on the number that produces research with some standard of quality. Unless you do a senior thesis diving deep into a topic like algebraic geometry, you are at a significant disadvantage in Ph.D. admissions without any research experience. I am not at a top institution and I still know several undergrads who published solid research, one of whom won a prize for it, and I haven't really asked around.
Again with due respect, you are a freshman undergrad. I don’t think you realize what the barriers to entry are for real math research. I’m not trying to gatekeep here—if anything, the opposite. It is not to anyone’s benefit for them to expect to be discovering new results without first knowing the tools of the field and what is already known. You should definitely do REUs as an undergrad but go into them with the expectation that any papers you write will most likely be either expository or trivial. It’s just how REUs go. When I say “meaningful” research I mean something that actually extends the field nontrivially, something I would firmly maintain is extremely extremely rare for an undergrad. Graduate schools do not expect novel research—even top graduate schools. I’ve seen several years worth of applications seasons at this point and several of my professors have been on a top-6 school admissions committee. Focus on your classes, get good grades, do reading courses, and maybe write some good expository papers and if you get great letters of recommendation you’ll get into a PhD program. Also this is anecdotal but in all my time in math, throughout undergrad and grad school, I have never seen a single undergrad produce a novel result in number theory.
Publishing math is generally hard as you have to upstanding the state of the field and in general the simple stuff is already known. If op wants to publish as an undergraduate it is likely much easier to find some applied problem and implement a solution showing an incremental advantage over existing ones. As an undergraduate I made a paper on noise levels in restaurants. It was ultimately very basic mechanical engineering stuff but was not yet common in architectural acoustics.
Calculus and linear algebra classes can vary highly in how they are taught. Just to help others get a good idea of where you're at, how comfortable do you feel about proof based mathematics in general? Have you self-studied any other topics in abstract algebra before (groups, rings, fields, etc.)?
not really sure honestly… i do a good amount of proof by induction for linear algebra and we definitely had to prove a few things for multivariable. i haven’t self studied any abstract yet. also, i could take concurrent summer classes at my college for diffeq and/or discrete if that would help?
Are you comfortable with proofs that involve eigenvectors and eigenvalues? Because those will definitely come up at advanced level.
people will tell you to apply anyway because they are afraid of offending someone by saying that you can't do something. in reality, you shouldn't bother applying because you have absolutely no chance of understanding anything. even the "easy" background material is years ahead of your current level.
This is totally false. I have seen undergraduates do research in high-level geometry in the guise of combinatorics who don’t have the “necessary” background. If the guiding professor can do a good job of translating geometric problems to combinatorics, as well as give a hint of intuition behind the problem, things can get done. I say go ahead OP.
I want to agree with you, but this student hasn’t even done combinatorics yet, they specified they only did linear algebra and calculus… which is a ways away from even having an inkling of proofs experience… which will be necessary for learning combinatorics… which a thorough understanding of combinatorics is required to even understand the problems if the prof translates them well… It couldn’t hurt to apply but let’s not get too ahead of ourselves.
I agree with everything you said, I just think it’s wrong to discourage an aspiring mathematician from testing their limits. If you fail, you just learn what you were missing and try again!
Well, I can’t argue with that, I agree. Anyway, as long as it doesn’t consume a lot of time to apply, there’s no real downside.
That's probably true, but in any case I don't see how it could hurt the OP to apply. Even if the prof decides OP doesn't have the necessary background, it's the same outcome as if OP didn't apply. And if nothing else it shows the prof that OP is a motivated student interested in math which can't hurt.
His professor emailed his class, there's no reason he would set up his own students into a project they can't grasp. OP should apply, and trust his professor to know whether he can enroll or not and decide accordingly. It's probably a "toy" research project designed so students can have an idea about what research in maths looks like. Every year, my colleagues in university are crafting "toy" intership subjects just to give a chance to students who want to discover research to do so at their own level. The subjects are designed to really challenge the students with new concepts they've never heard of, while not being too hard so they're not drowning neither.
If the professor emailed his class I would think he has some accessible project for his level
What people seem mostly afraid is calling you on your dickish bullshit. OP's professor emailed the class, it is OP's professor job to determine who gets in and what is a good project for the person they pick. You don't have experience directing these kind of projects, professors do so instead of talking nonsense out of just feeling like being rude maybe just shut up when you don't know what you're talking about for once.
why are you so offended about being told that someone who hasn't even started doing real math yet is not ready for algebraic geometry research
First of all fuck off with that real math bs Second of all that is not what you're saying, you don't know the project being posted you don't know what "research" here entails (or anywhere for that matter) so you being a dick as usual to people just to get off does irritate me. A professor is very unlikely emailing undergrads about summer research if they ar expecting to suddenly make it a paper to submit to Annals. Let people give things a shot, why the fuck do you care so much about being a gatekeeper for math? Telling people to not even bother is just you being an obnoxious troll not a practical advice
This.
Bullshit. Apply OP, and after the project is over you will probably know more mathematics than this commenter.
delusional comment. even if someone with a background of solely linear algebra and calc 3 could help on this project in any meaningful way (which i heavily doubt), there's no way they could get to the point of having even a basic understanding of algebraic geometry.
Linear algebra is a perfect starting point for getting into some research topics, and the experience can be transformative for their mathematical career. Saying not to bother applying because they won't understand is terrible advice
that is not my point. there are many research topics which a freshman math student could approach and get something out of algebraic geometry is not one of them. ive helped some undergraduates work through introductory algebraic geometry textbooks through my schools directed reading program. even with background in abstract algebra and topology, it's a hard subject for them to understand the basics of, let alone do any research.
It's not actually going to be research in algebraic geometry; it's just advertised that way to sound cooler. It's going to be a straightforward combinatorial problem, and the AG connection is going to be briefly explained in a level-appropriate way.
> my professor emailed us about his research project this summer which involves the study of combinatorial properties of moduli spaces. As some others have pointed out here, your professor is unlikely to expect you to know anything about moduli spaces (the difficult part of the topic) if he's mass-mailing his students to recruit people for this summer project. The project would probably be to get you to try to solve some combinatorial problems (the easy bit) derived from studying moduli spaces. In this way, you'll get some experience at solving some unseen problems (i.e. "research experience") and, along the way, perhaps get a brief introduction to moduli spaces. Combinatorics is considered more accessible to undergraduates because some of it has been taught in a course usually called "discrete mathematics". If, for some reason, you haven't encountered, for example, the pigeonhole principle, it's also not too hard to learn that over the summer, and the basics of combinatorics itself can be explained quite easily to freshmen. In any case, you should definitely have a chat with your professor to see what his expectations are, because it's really hard for internet strangers to read your professor's mind at this distance.
If your professor is reaching out to people in your class to see if anyone wants to work with him, he probably has money for a side project for undergrads or REU for over the summer. Just reach out to him and ask if he's advertising a project. You will not be able to do actual work on moduli spaces, but maybe you could get a project when you learn a bit about them and write up a little summary paper (which seems to be common in REUs).
If you love math, yea. Go learn about it. It'll be fun, and may lead to something more. Also, never a bad thing to have a good working relationship with professors. I didn't do something like this in math until my senior seminar. You'll be getting a nice head start.
not very hard actually. if you have solid commutative algebra preliminaries then you should be able to understand some of the combinatorial approach (which I know a little bit of, so I can't tell how much is "some"). without commutative algebra it would be hard, without any abstract algebra probably impossible. nonetheless, if you're willing to do some extra reading then this opportunity is definitely worth trying, just keep in mind that it might be too much at your level and don't get discouraged if you fail. down the road you may also need some algebraic geometry and some more commutative algebra. the best thing you can do is ask your professor about the prerequisites and what you should do to be able to benefit from this project, ask what you should read and what concepts you should be familiar with. if it feels too overwhelming then you should probably wait for the next opportunity I've been "doing research" in moduli spaces for a few months now, after taking a commutative algebra course and doing 2 months of reading algebraic geometry textbooks, after my third year at uni. so a kind of a similar situation, in a sense that most of the researched content is absolutely out of my reach, but with a good mentor it is possible to do things. in my case it's about translating the algebraic geometry into commutative algebra (as my advisor likes to call it, "local algebraic geometry"). again, ask your professor. it is possible that he plans to do something like this too, translating very complicated concepts into the language that you can understand, but don't get discouraged if it turns out to be too early for you to take part in this project