Should I Become a Mathematician?

[mathwonk]

Torelli's theorem is really interesting and simple. You have a geometric shape defined by polynomials, and therefore it is entirely determined by those polynomials, and hence by their coefficients, which is a finite set of numbers.

But is there some more natural way to get the numbers that determine the shape? Riemann assigned numbers as follows to a complex curve, or riemann surface. he chose a basis of loops one around each hole in the surface, and then proved there is one independent holomorphic differential form for each hole. so if the genus of the curve is g, by integrating each form around each loop, one gets a g by 2g matrix of numbers.

the torelli question is whether these numbers determine the curve back. the first step is taken by riemann. i.e. to recover some geometry from that matrix, mod out the space C^g by the lattice generated by the 2g columns of the matriux.

That gives a g dimensional complex torus. Then to go further, and this is the key step, riemnann wrote down a theta function using the matrix as the quadratic coefficient in a quadratic form, then using z as the linear coefficient, then summing iover integer arguments. this gives a function of z, whose zero locus is periodic for that lattice, and defines a "theta divisor" inside the complex torus.

this pair is called the jacobian variety of the curve. Moreover riemann showed this divisor was highly singular. then to recover some geometry further, Andreotti and Mayer showed that if one intersects the tangent cones to all the singular points, after translation to the origin, one gets the curve back in general! but for just which curves it worked was unknown.

That this recovery procedure holds for all complex curves except those on a specific short list was proved by Mark Green in 1981. Then Robert Varley and I generalized the proof to hold over fields of any characteristic but 2.

The next step is to see how this proof procdedure generaliuzes to the relative case of prym varieties (kernel of map between jacobian varieties) for a double cover C'-->C. It is known agaion that the procedure works in general but not precisely which covers it works for. And in this business, not knowing exactly which curves it works means not being abkle to identify any for which it works. they just form some shadowy unknown open set.

Varley and I have shown that one can recover the double cover from the prym variety, i.e. a certain complex torus plus theta divisor, if one also knows a certain natural P^1 bundle over the theta divisor, the so called "abel map".

but how to recover this map is a probem. And in fact we have not found the key to understanding just when the procedure works, because our proof also works in cases where the tangent quadric construction is known not to work.

so there is more data in the abel map than in the theta divisor itself. so which prym theta divisors allow one to recover the abel map uniquely? or even a finite set of possibilities?

the approach we took involved delicate questions of deformation theory, a useful technique analogous to differential calculus, but for functors of geometric objects rather than functions of numbers. Then, ironically considering a thread elsewhere on interrupted research time, teaching and administrative duties intervened, and we have not had time to pursue it to a conclusion.

 

[Edgardo]

Mathematics - Career Advice by Terence Tao

I thought the following might fit in this thread:

Mathematics - Career Advice by Terence Tao
http://www.math.ucla.edu/~tao/advice.html

 

[mathwonk]

that is great advice!

 

[Mororvia]

And, valid for more than just mathematics too, for most of it. Very nice

 

[mathmuncher]

Terry Tao is not just mathematically creative, but charmingly articulate.

Thanks for sharing!

 

[mathwonk]

searching for other good sources of advice from professionals, i found this book that looks pretty good, by steven krantz.

A Mathematician's Survival Guide: Graduate School and Early Career Development
Steven G. Krantz

 

[mathwonk]

here is a review of krantz's book that let's us know that krantz has a good section on preparing for quals.

A Mathematician's Survival Guide:
Graduate School and Early Career Development
by Stephen G. Krantz

Reviewed by Ioana MihailaA Mathematician's Survival Guide is a new addition to Stephen's Krantz "How To", series, the best known of which is How To Teach Mathematics. The present book, written in the same strong and opinionated tone, is meant to guide prospective mathematicians through the process of getting a graduate degree in mathematics, getting a job, and obtaining tenure.

The book is based on the author's extensive experience as faculty at several well-known institutions, and on his role as advisor to many graduate and undergraduate students. The first eight chapters are devoted to providing the reader with step-by-step descriptions of the various stages of the process of becoming a mathematician: how to prepare for graduate school, how to choose a graduate school, how to pass the qualifying exams, how to choose an advisor and a thesis topic, how to write your dissertation, how to get a job, and how to get tenure.

The ninth chapter has a different flavor. The author gives lists of typical subject area contents for the qualifying exams, exemplified by the current requirement at Washington University — Stephen Krantz's home institution. In addition, he writes a very brief synthesis of each subject listed, highlighting the main ideas. Even though the subject areas of the exams and their contents vary somewhat from institution to institution, I found this last chapter of the book very valuable for the prospective student who wants to know what will be expected of him. For those interested in actual examples of qualifier-type problems, the authors cites Berkeley Problems in Mathematics by Silva and De Souza, but this overview of topics definitely has a place in a graduate school self-help book.

As with any guidance book, the essential question to be answered is "Is it helpful?" To summarize it in one sentence, this book tells the reader to "work hard", "communicate with others", and "plan ahead". All of these are most definitely good advice, and I find it refreshing that Stephen Krantz is actually putting it in writing that you should work hard in order to succeed (although most people either do work hard without being told, or pay no heed to this kind of advice). But the book's main value is in the "plan ahead" part, because it gives the reader the necessary information to do so. It takes tremendous effort and organization to make it to the final step of getting a job and obtaining tenure in today's world, and being uninformed can be very costly. I would definitely recommend to every mathematics department to keep a copy of this book for their undergraduate and graduate students

 

[mathwonk]

here is an anonymous readers review from amazon.com

Useful and comprehensive, November 11, 2003
Reviewer: A reader
It tells you most of the basic things of becoming a mathematician, more specifically, an professor of mathematics. I was quite passionate for a career in the academics. But after reading the book, I decided to try other choices because I don't want to spend 4 years as an undergraduate, 5 years OR MORE as a graduate student, and then 2 years or more as postdoc and another 5 or more years as an assistant professor waiting for the tenure while being paid like $50,000-55,000 a year! This is crazy and unfair! My friend with an MBA has $70,000 as beginning salary! Suppose I can be a full professor at an above average research university, I probably would get paid just $70k a year! Why should I waste so much time (at least 20 years!) earning so little and engaging myself into such a fierce competitive academic world! I do thank the author for telling us about the path of being a professor of math. He also lists many other governmental and private companies' positions that are more attracting than the professorship, and that's what I am going for!
In summary:
First of all, you got to pass the qualifying exams, which are harder than the William Lowell Putnam Mathemtics Competition. Then, you have to spend 4-6 years writing a good thesis! If your thesis is not significant enough, you may have a very tough time finding a plum job. Even if you have produced an important thesis, you have show yourself constantly and actively working on your field subjects. You then may have an assistant professorship on the tenure track--waiting for the vote of the faculty members, and approval of the dean, the provost and then the Chancellor. The process is kinda of harsh.
But as the author says "One of the best things about a degree in mathematics--at any level--is that it opens many doors and closes few of them. It gives you a world of opportunities from which to choose." Good luck to us who love math! Wish us well.

 

[mathwonk]

here is charming interview with two very smart and upbeat mathematicians who work with computers, algorithms, ...

http://www.pbs.org/wgbh/nova/sciencenow/3210/04-chudnovsky.htmland this link from stony brook looks good as a guide to their department and to research in general.

http://www.cfkeep.org/html/snapshot.php?id=97990319782859

 

[mathwonk]

i have refrained from citing posts where the author, often a famous old mathematician, gives advice like: publish a lot of papers on fashionable stuff, so you will get cited a lot and win a lot of grants. or be sure to plug the work of some member of the audience during your talk so you will kiss up to him/her and he will enjoy your talk.

this sort of cynicism is i suspect meant partly in jest, but young people will take it too literally and it is unhealthy to be so shallow at a young (or old) age.

terrence tao's advice you will note is more honest and straight forward, full of positive enthusiasm. my suggestion is to take his advice, or raoul bott's or the chudnovsky's, instead of that by john baez or gian carlo rota.

of course even those semi cynics and comedians give some good advice too, but you have to know which is which.

 

[mathwonk]

Anyone in the Athens Georgia area this week will want to consider attending the lecture by Michael Spivak Wednesday night 7pm on how Newton himself solved the problem of the geometry of planetary motion.

there is also a colloquium thursday.

March 1, 2007
3:30pm, Room 328
Speaker: Michael Spivak
Title of talk: Physicists’ Rigid Bodies With Mathematician’s (Being Lesson 1 of Physics Without Tears)
Abstract: Newton's laws apply to "particles" or "point masses," which can also be considered to apply to the objects of astronomical problems, but you can't do most other physics problems without considering larger (rigid) bodies.

Newton never discussed rigid bodies (smart man). Euler's pioneering treatment, the basis for the elementary undergraduate hocus-pocus, regards solid bodies as continuous expanses of matter, a rather disconcerting view in the atomic age, whereas the advanced graduate hocus-pocus considers a collection of particles bound by "constraints" in a manner sufficiently abstract to hide all the difficulties in a haze of generalities.

This lecture attempts to give a coherent exposition of the subject, essentially explaining and giving meaning to some of the strange things that physics textbooks contain.

 

[mathwonk]

This is not advice, just a reaction to tonight's talk by Spivak. I loved it. Continuing his history of making the writings of riemann and Gauss understandable to the rest of us, in his differential geometry book, he spoke about Newton's Principia tonight.

He clarified how Newton proved some significant and interesting results on planetary motion using simple Eucldean geometry. he also explained how foolish some recent writers were who claimed Newton made a mistake in claIming certain results. I.e. he explained how Newton was right.

Now you might think few people would be so foolish as to claim Newton was wrong, but such people exist. And when you reflect how smart Newton was, you realize how hard it is to provide the details that Newton omittted in his arguments.

But Spivak did it and made it look easy.

He struck me profoundly as a real scholar, a man who seeks to understand significant ideas, and to clarify the works of the greatest minds, rather than to play a game of competitive publication of trivial facts.

Very inspiring. (He also gave me some of the best chocolate I have ever tasted.)

 

[mathwonk]

proofs of, big theorems in calculus

i have posted in the calulus section, proofs of the main theoretical results of calculus that are usually thought too difficult to include in standard course. I believe this is nonsense, and would like to know what you think. Please take a look and report back. (I started to post it here too, but realized that double posting wastes space.)

 

[ircdan]

I think it's great you got to hear Spivak speak. I've looked at the material you posted and to be honest I have mixed feelings about whether it should be included in a standard calculus course. I personally would have benefited from something like that, but not everyone is like me. I think mathematics is hard, and at first people may find proofs intimidating and it may turn them away. It takes a special kind of person to appreciate it. I really don't know what would be best.

 

[mathwonk]

Actually Spivak is an old friend of mine but I had not been in contact for a long time and had no chance to hear him for that long time. So it was doubly great for me to hear and see him again.

As to the big theorems in intro calc, I agree with you. In fact I have not included them in the intro course I am teaching now. I did sketch the proof of IVT there though but only briefly.

Nonetheless it was while teaching this course and rereading the material as it is presented in the book, that led me to write down these arguments.

my point is that if they are going to put this stuff in the book at all, for someone to read, they might as well do it better than they do.

e.g. the MVT is a silly theorem that students try to memorize, focusing on whether it is f(b)-f(a)/[b-a] or f(a)-f(b) etc... when this formula is not the important part.

the important part are the corollaries. so those should be stated as the theorems, and this silly formula should be relegated to the proof as I have done.

i.e. there are several key parts to these results:
1) the IVT, that if a function is continuous on an interval then its values also form an interval.

2) next if a function is continuous on a closed bounded interval. then the values also form a closed bounded interval, in particular there is a max and a min.

those are the foundational results, easily proved by just constructing an infinite decimal that works, step by step. still the proof of the second one is a little tedious.

these results are already sufficient to handle almost all max min problems, using the endpoint limit test to deal with open intervals. no MVT or first or second derivative tests are needed. just test critical points and endpoints, or on an open interval, if the limit is infinite at any missing endpoints just test critical points.

using those however, the next results, the MVT type ones, are easier, and should be stated as follows, not as they usually are.

3) a diff'ble function that is not monotone on an open interval, has a critical point on that interval, (at any local extremum). i.e. a function is strictly monotone on any interval where there are no critical points.

this is the way the rolle thm should be stated as it is stronger than that usual statement. this then gives the basic principle of graphing, that a graph is monotone between two successive critical points.

it also gives the first derivative test in a form stronger than the usual corollary of the MVT, namely with this version you only need to test the sign of the derivative at one point of an interval in which there are no critical points, or not at all if you know the value of the function at two successive critical points.

It also follows that if the derivative of a function does not change sign on an interval the function is strictly monotone there, without the MVT.the MVT is just a variation on the rolle thm of course, but there is no point in limiting it to the case usually stated, since that obscures the idea, and makes the statement an unelightening formula instead of a simple statement.
i.e. instead of stating then MVT as it usually is, one should just apply the previous "Rolles" theorem to the difference of two functions.

i.e. the following is a corollary of rolle:
3b) two diff'ble functions that agree at two points, have the same derivative at some intermediate point.

now the whole point of the MVT is the following result which should thus be stated as a theorem:
4) two diff'ble functions with the same derivative on an interval, differ there by a constant, in particular only a constant function can have derivative identically zero.

this is proved as usual by observing that any function agrees at two points with a linear function, hence 3b) implies that a function which is not constant has somewhere the same derivative as a non constant linear one, i.e. has non zero derivative somewhere.

there is no real need for the explicit formula [f(b)-f(a)]/[b-a] to enter the statement at all.my point is this: first the technicalities in the statements of MVT have been allowed to overwhelm the essential results we want people to remember. so for the average students the usual statements are pedagocially flawed.

the statements should be linked to their applications. thus above statement 1 is enough to "solve" equations. statement 2 is enough to do max min problems. 3) is enough for graphing. 4) is enough to do antiderivative problems, like the FTC.

second, for those students who might appreciate the proofs, those proofs have been allowed to become too abstract to present in an elementary course. this too is unnecessary. the arguments using infinite decimals for statements 1, 2 are completely rigorous, and more intuitive than the more abstract least upper bound arguments usually reserved for advanced analysis courses. i.e. one easily constructs least upper bounds as decimals, rather than postulating them for abstractly defined sets.

these are my suggestions.

 

[hrc969]

I had failed to notice that you posted about Krantz's book. I have read that book and it is really good.

I just have one question:
I don't know if you have read his book. But one of the things he mentions is along the lines of: undergraduate research is not representative of research done by mathematicians and therefore not so good for undergrads to do "undergrad research".

[I let one of my classmates borrow my copy of that book, so I can't give an exact quote, sorry.]

What I wanted to ask is whether you agree with that idea.
He suggest that what we should do is learn as much math as we can before we get into grad school. That's what I have been doing right now. But I want to start doing some research, maybe starting this summer or starting next fall.

I am a third year and what I want to study is complex manifolds(complex differential geometry) and Several complex variables. I have taken several grad courses (actually I'll be done with the geometry courses my school offers other that the "topics courses" offered). I was wondering what defines undergraduate research. I think that the research that I would do might not be considered undergraduate research, but I don't know.

Thank in advance.

 

[mathwonk]

well i used to agree with him, but some of my colleagues seem really good at giving undergrads the feel of doing research.

as long as everyone is enjoying it, there seems no harm in encouraging people to try to think up and solve problems as early as possible.

so while i myself am not that great at directing undergrad research, that is probably because i am also not that great at directing grad research either.

so i say go for it, and maybe i could learn something about directing theses if i tried to guide some undergrad research myself.

i think krantz wants to remind you that if you want to do really impressive research, you will indeed need to know as much as possible. so keep working at learning too.

 

[mathwonk]

The leading engineer thread seems to have begun in 2004 and this one in 2006. Any bets on when the number of hits in this one surpasses that one? June 2007? Come on, get in the pool! A bottle of wine to the winner over 21.

 

[d_leet]

The leading engineer thread seems to have begun in 2004 and this one in 2006. Any bets on when the number of hits in this one surpasses that one? June 2007? Come on, get in the pool! A bottle of wine to the winner over 21.

What if the winner isn't over 21?

 

[pivoxa15]

MathWonk, how hard is it usually for a recently graduated pure maths Phd student (from Austrlalia) to find work as a postdoc? It seems there aren't many pure maths post doc positions on offer in Australia so traveling overseas seems essential. What happens if you get accepted into a university in a country which you don't speak the lanugage? One of my lecturers said he once had a permanent job in a French university but I don't think he could speak fluent french as he was grown up in Australia.

Which major area in pure maths is employment usually easiest? Which area usually the least? And why?

 

[mathwonk]

phooey my response got erased. I mentioned you should ask Adam Parusinski, who is a prof in Australia, for some advice. let me know what happens. Also the US is good source of postdoc jobs, as we are fond of exploiting talented young people without giving them permanent status.

 

[mathwonk]

oops, Adam moved to Angers, France. but you might search on opportunities for postdocs in math.

there is a central site for members of the AMS, and many scattered individual sites offering jobs. sorry i can't help more precisely.

 

[pivoxa15]

Have you seen or heard of newly graduated Phds in pure maths who cannot obtain a postdoc position? If so what do they do in the mean time? How often does this occur? Are they forced to go into a nonacademic field with more demand? What is usually the reason for them not able to get an academic position?

Which major area in pure maths is usaully most in demand? Which is least? Or do they not exist because whenever an area has a lot of demand, the students in that field increase and so you are back at an equilibrium?

 

[mathwonk]

I have not heard of PhDs unable to find a postdoc, but anything is possible.

there are many reasons for a job glut, lack of congressional support for science, influx of foreign candidates, cutback by state colleges in science faculty, excess of graduates because of erroneous job projections,...but these things all balance out in time. I myself came out in a time of few jobs but because I loved the area I persisted and found a job. I started as an instructor at a small college. Some of my friends were teachers at private prep schools.

if you are primarily interested in earning a good living i recommend going into business or medicine. Math is for the few who will not be denied the right to do it.

numerical analysis, and statistics, computer science, and other applied areas are usually better paying than pure maths, but in the past few years there have also been people in Comp Sci losing jobs.

evenso, eventually we tend to come out ok. sorry if this is not useful.

I am not so much a survival guru as a math guru, but of course I have learned something about survival. basically to survive, never give up, and remember your "boss" is just a man like you. He needs you or he would not have hired you. Never forget you have as much to offer the company as they have to offer you. And you can always take it elsewhere.

some companies always want to make you feel inferior, as if they are doing you a favor by hiring you at low wages. hang in there. you are a good man (or woman?) and you will be fine.

 

[james5094]

Mathwonk,

How would you rate mathematics at UGA? for someone who likes physics? I'm on the fence between pure math and physics.

 

[pivoxa15]

I have not heard of PhDs unable to find a postdoc, but anything is possible.

there are many reasons for a job glut, lack of congressional support for science, influx of foreign candidates, cutback by state colleges in science faculty, excess of graduates because of erroneous job projections,...


but these things all balance out in time. I myself came out in a time of few jobs but because I loved the area I persisted and found a job. I started as an instructor at a small college. Some of my friends were teachers at private prep schools.

if you are primarily interested in earning a good living i recommend going into business or medicine. Math is for the few who will not be denied the right to do it.

numerical analysis, and statistics, computer science, and other applied areas are usually BETTER PAYING THAN pURE MATHS, but in the past few years there have also been people in Comp Sci losing jobs.

evenso, eventually we tend to come out ok. sorry if this is not useful.

That is very useful. I think there are no end to money. I have heard of a senior academic physicst who was claiming that the pay isn't fulfililing so maybe people have to go into the private sector if they are looking for money.

 

[mathwonk]

Well I think the math faculty at UGA is very strong. You can certainly get a good PhD there in several areas, including number theory, algebraic geometry, arithmetic geometry, representation theory, topology, differential geometry, integral geometry, analysis, and applied math.

The undergrad program is also very good and very hands on. It is small but this is a plus for the students as there enough other students to talk to, the comraderie is good, and the faculty know the students and care about them. There is an active Putnam team, a good selection of advanced courses and a math club.

We are one of the places that still offers a Spivak type calculus course for strong math major types, (like Chicago, but unlike Harvard or Stanford).

And of course there are seminars such as the one featuring Spivak a couple weeks back, and this one coming up in 2 weeks:

http://www.math.uga.edu/%7Evalery/conf07/conf07.html

 

[mathwonk]

some of the UGA math faculty are actively involved in mathematical physics such as Malcolm Adams, Robert Varley, Cal Burgoyne, Dave Edwards, and there is a quantum physics seminar going on.

 

[J77]

That is very useful. I think there are no end to money. I have heard of a senior academic physicst who was claiming that the pay isn't fulfililing so maybe people have to go into the private sector if they are looking for money.

Being an academic doesn't mean you can't work with industry; eg. gaining research money or even doing consultancy.

The UK research funding agencies are very big on industrial collaboration.

 

[pivoxa15]

Being an academic doesn't mean you can't work with industry; eg. gaining research money or even doing consultancy.

The UK research funding agencies are very big on industrial collaboration.

True but if you research in pure areas of physics or maths than industry won't be as interested and the academic in turn wouldn't be too interested in working with them either.

 

[J77]

True but if you research in pure areas of physics or maths than industry won't be as interested and the academic in turn wouldn't be too interested in working with them either.

What do you mean by pure areas of physics?

And, there is work in industry even for pure mathematicians eg. in defence/cryptogrpahy or finance/actuary.

 

[pivoxa15]

What do you mean by pure areas of physics?

And, there is work in industry even for pure mathematicians eg. in defence/cryptogrpahy or finance/actuary.

pure physics being answering questions in physics the solution of which does not have any remote use in society yet. i.e where mass came from. Or unification theory or doing experiments to verify or disprove a theory.

True about pure mathematicians being able to find work elsewhere but that involves retraining and not doing pure maths while doing the work which isn't ideal for some pure mathematicians.

 

[J77]

True about pure mathematicians being able to find work elsewhere but that involves retraining and not doing pure maths while doing the work which isn't ideal for some pure mathematicians.

That's a very broad statement!

And I would immediately contradict it by giving the example of number theorists working in cryptography -- I'm sure they would think their work ideal for their branch of "pure" mathematics.

And I would say by definition that all physics is applied -- I'll ask some of the particle guys whether they consider themselves as pure because their work shows no "remote use in society yet"

 

[pivoxa15]

That's a very broad statement!

And I would immediately contradict it by giving the example of number theorists working in cryptography -- I'm sure they would think their work ideal for their branch of "pure" mathematics.

And I would say by definition that all physics is applied -- I'll ask some of the particle guys whether they consider themselves as pure because their work shows no "remote use in society yet"

What do you mean by "...their work ideal for their branch of "pure" mathematics"?
Crytography is an application so if any person is working on it, they would be considered doing applied work even if they were donig a proof in the theory of crytography.

Since the discussion was orginally about job prospects, I used the definition of applied as "Use to society as part of motivation of research". Pure as "Use to society is not a motivation of research ". However, pure maths can become applied when someone concerned with applications realize the use of this piece of pure maths.

society is a broad word but it can describe people in other academic disciplines.

In the context maths only, I think this has some correctness to it. Many areas of applied maths has use to society or other disciplines. i.e PDE theory, statistics, Operations research, contiuum mechanics, even mathematical physics because it helps the physicists or has physical theories as a motivation.

If we are talking about physics, one might argue that it is all applied maths but if we follow the definition I gave above than you could separate pure and applied physics. Such distinction exists with the existence of the International Union of Pure and Applied Physics.

wiki has
"Applied physics is a general term for physics which is intended for a particular technological or practical use."

http://en.wikipedia.org/wiki/Applied_physics

If we say all physics is either pure or applied than Pure physics would be physics not intended for a particular technological or practical use. These definitions is in line with the general one I gave earlier.

 

[mathwonk]

Introduction to algebraic geometry 1

For people who might wish to go into algebraic geometry, my area, I am going to repost my introduction to that subject, since it seems to belong here rather than in the specific thread on that question. (I tried to delete the earlier post but cannot now do so.)

Naive introduction to algebraic geometry: the geometry of rings

I used to say algebraic geometry is the study of the geometry of polynomials. Now I sometimes call it the "geometry of rings". I also feel that algebraic geometry is defined more by the objects it studies than the tools it uses. The naivete in the title is my own.

I. BASIC TOOL: RATIONAL PARAMETRIZATION
Algebraic geometry is a generalization of analytic geometry - the familiar study of lines, planes, circles, parabolas, ellipses, hyperbolas, and their 3 dimensional versions: spheres, cones, hyperboloids, ellipsoids, and hyperbolic surfaces. The essential common property these all have is that they are defined by polynomials. This is the defining characteristic of classical algebraic sets, or varieties - they are loci of polynomial equations.
A further inessential condition in the examples above is that the defining polynomials have degree at most 2 and involve at most 3 variables. This limitation arose historically for psychological and technical reasons. Before the advent of coordinates, higher dimensions could not be envisioned or manipulated, and even afterwards it was commonly felt that space of more than 3 dimensions did not "exist" hence was irrelevant.
The dimension barrier was lifted by Riemann and Italian geometers in the 19th century such as C. Segre, who realized that higher dimensions could be useful for the study of curves and surfaces. Riemann's use of complex coordinates for plane curves simplified their study, and Segre understood that some surfaces in 3 space were projections of simpler ones embedded in 4 space.
One reason for restricting attention to equations in (X,Y) of degree at most 2 is a limitation of the basic method of "parametrization", expressing a locus by an auxiliary parameter. E.g. the curve X^2 + Y^2 = 1 can be parametrized by the variable t by setting X(t) = 2t/[1+t^2], Y = [1-t^2]/[1+t^2]. This substitutiion, along with dX = 2[1-t^2]dt/[1+t^2]^2, allows one to simplify the integral of dX/sqrt(1-X^2), to that of 2dt/[1+t^2] = 2d[arctan(t)].
The cubic Y^2=X^3 can also be parametrized, say by X = t^2, Y = t^3. But to simplify in this way the integral of dX/sqrt(1-X^3), requires us to parametrize the cubic Y^2 = 1-X^3, a problem which is actually impossible. These questions were considered first by the Bernoullis, and resolved by new ideas of Abel, Galois, and especially Riemann as follows. (Interestingly, in three variables the difficulty arises in degree 4, and 19th century geometers knew how to parametrize most cubic surfaces.)