Other Should I Become a Mathematician?

[mathwonk]

please, this is a family thread.

 

[pivoxa15]

Mathwonk, you said that you haven't heard of a Phd not being able to find post doc work but the woman in that ad said academic jobs in universities are hard to find. Does she have a Phd? If not than obviously it will be hard to find an academic job such as a research job. She has the option of being a teaching assistant at univeristy which is less hours than a school teacher. If she does have a Phd in maths than it would be 'low' teaching at school wouldn't you say? Do you know of any maths Phds teaching in a school?

 

[mathwonk]

no. but in the old days i heard that top private schools like andover, exeter, may have had science profs that were very well trained, possibly phd.

high, low, if you enjoy teaching good students, then top high school or prep school teaching might ring your bell.

i once taught high school students for free, for a year or so, 2 days a week, and a month in summer. although it was lower level maths than some of my uni teaching and my research work, i enjoyed it greatly because the students were more responsive.

two of my high school students later went to ivies and obtained phds in physics and maths. one of them is now a full prof at an ivy himself.

 

[verbasel]

I have a question about "math fatigue". I've been questioning whether or not I'm really cut out to be a Mathematician.

Back in my second year, for various unwise reasons, I binged on honours Math courses. I thought I was going to do a specialist degree, so I took 4 honours math & stats credits and overall I was taking 6 full credits, which is the maximum course load at my uni. My year was an academic disaster, resulting in problems with anxiety and depression and the only decent marks I got were in the non-math/related courses.

After burning myself that way & seeing counselors both academic and otherwise, I opted for a double major in Math and Economics instead (adding another year to my degree). I've basically taken only Economics courses since then, and have completed the courses required for my Economics major. While the so-called 'math' in Economics infuriates me and the pure math courses I took way back when interested me greatly, I'm really apprehensive about taking math courses again.

I've further downgraded Math to a minor and plan to take the easiest courses in order to finish my degree without any further mishaps, but I know I would eventually like to return to the more rigorous math that fascinated and confounded me back in the early days.

How do I regain my confidence? Was I ever a mathematician to begin with? What's a good way to ease back into it?

 

[symbolipoint]

Verbasel,
Maybe you are not cut out to be a mathematician. A minor concentration in Mathematics might still be reasonable. What do you study between semesters? How much time (hours per week, and how many months) are you willing to dedicate to Mathmeatics? Are you willing to restudy courses which you already studied and earned credit in?

 

[J77]

Mathwonk, you said that you haven't heard of a Phd not being able to find post doc work but the woman in that ad said academic jobs in universities are hard to find. Does she have a Phd? If not than obviously it will be hard to find an academic job such as a research job. She has the option of being a teaching assistant at univeristy which is less hours than a school teacher. If she does have a Phd in maths than it would be 'low' teaching at school wouldn't you say? Do you know of any maths Phds teaching in a school?

Bear in mind, it was an advert for GCHQ -- therefore, she would likely say that academic jobs in universities are hard to find. However, there never seems any shortage of jobs available in the UK.

I would feel like teaching high school at some point -- my gf is one -- however, I would only like to teach kids who would be into it; ie. not there for "crwod control" -- which seems to be the norm in a lot of schools.

 

[pivoxa15]

I have a question about "math fatigue". I've been questioning whether or not I'm really cut out to be a Mathematician.

Back in my second year, for various unwise reasons, I binged on honours Math courses. I thought I was going to do a specialist degree, so I took 4 honours math & stats credits and overall I was taking 6 full credits, which is the maximum course load at my uni. My year was an academic disaster, resulting in problems with anxiety and depression and the only decent marks I got were in the non-math/related courses.

After burning myself that way & seeing counselors both academic and otherwise, I opted for a double major in Math and Economics instead (adding another year to my degree). I've basically taken only Economics courses since then, and have completed the courses required for my Economics major. While the so-called 'math' in Economics infuriates me and the pure math courses I took way back when interested me greatly, I'm really apprehensive about taking math courses again.

I've further downgraded Math to a minor and plan to take the easiest courses in order to finish my degree without any further mishaps, but I know I would eventually like to return to the more rigorous math that fascinated and confounded me back in the early days.

How do I regain my confidence? Was I ever a mathematician to begin with? What's a good way to ease back into it?

Interesting case. I took very much an opposite route to you in many ways than one. I didn't have a solid science maths background going into uni and enrolled in a commerce degree at first. But took some maths subjects in the first year. I loved it very much although found it extremely difficult especially the purer ones. I switched to a BSc in second year although decided only to take one maths subject and some other science and philosphy subjects. Looking back this may not have been a wise choice as I could be a better maths student had I done more maths in that year but at the time offcourse I wanted to explore other subjects and took physics for the first time so didn't know what to expect. I also felt that I didn't have enough mathematical maturity to do 2nd year maths even though I got 70 and 80 for 1st year pure and applied maths repectively. Now in my 4th year at uni, I am taking an overload (one extra subject) of 3rd year maths and physics subjects and although I also found it extremely challenging, have found that I enjoy it more than ever and can't wait to do higher maths in the future. But first thing is first, hopefully I complete this year successfully.

 

[mathwonk]

well life choices are not so easy. i suggest gradual movements. stay at least partially with what is working, and go gradually in the direction of what you hope will work.

you are young and strong, and smart, so there are lots of openings.

but temporary fears and insecurities are common, at least in my experience.

the key is to persist with what you love.

if you are working at it, you are a mathematician, regardless of your success rate.

 

[mathwonk]

one thing that relieves math fatigue is contact with other mathematicians. i am now enjoying my birthday conference at uga and am extremely grateful to the visiting speakers and others who came to provide stimulus to those of us here. but guess what? at least one speaker said he himself was feeling the same lift from being here that we are feeling from having him here!

so try to get together with people who enjoy discussing together, and they will stimulate you and each other.

 

[pivoxa15]

Good point. I relieve maths fatigue by hanging around here.

 

[J77]

one thing that relieves math fatigue is contact with other mathemticians. i am now enjoying my birthday conference at uga and am extremely grateful to the visiting speakers and others who came to provide stimulus to those of us here. but guess what? at least one speaker said he himself was feeling the same lift from being here that we are feeling from having him here!

so try to get together with people who enjoy discussing together, and they will stimulate you and each other.

Yeah -- conferences certainly give you a lift. eg. I've just come back from a physics conference -- explaining your (mathematical) work to physicists really gives you new insight/avenues to explore.

 

[mathwonk]

what did you talk about?

 

[pivoxa15]

How common are mathematicians who scarifice their 'life' to do maths? i.e live alone without a partner or children and maintain minimum personal social interactions? How productive are they in the long term? I know Newton was one and the Russian who solved Poincare's problem but they have extroordinary abilities. How do people with lesser abilities do?

 

[mathwonk]

most mathematicians i know are pretty ordinary, and have families, friends, children, etc. those people you mentioned are very unusual, and not usually better mathematicians then the ordinary ones in my opinion.

 

[J77]

what did you talk about?

I pretty much fall under the category of nonlinear optics.

(ie. as opposed to GR/SR, HEP, Nano... etc.)

Response to above, also -- most mathematicians can usually be found occupying the local drinking spot at one time or the other. Communication and social skills are way up there if you want to succeed, imo.

 

[mathwonk]

feel like summarizing what you said about non linear optics? even if it way over my head, someone will enjoy it.

i will tell you for example what my friend asked me atmy conference.

he asked about generalizing riemanns proof of "jacobi inversion" for a single curve, to the analogous result for a pair of curves, one doubly covering the other.

in algebra its like generalizing a result about one field to the case of a quadratic extension of fields.

riemann showed that you could parametrize the jacobian of a genus g curve, which is a complex torus of dimension g, almost one to one, by a map from the product of g copies of the curve, via some "abelian integrals".

his argument can be given in two ways, one by using his theta function, (a fundamental solution for the heat equation), but there is another way, more geometric and almost tautological, using the dual torus called the picard variety of the curve.

of course this uses riemann's and abel's proof that the two tori are in fact isomorphic.

anyway my friend had done the analytic proof in the relative case and I then did the geometric proof the other night, between blogs here.

 

[mathwonk]

Now that my conference has ended and I am still exhilarated by the experience of meeting again so many mathematical friends and hearing so much interesting math that it has literally jump started my math research thinking again, I wanted to extend my earlier advice on becoming a mathematician to include strong advice to attend conferences.

Then I ran across edgardo's link to terence tao's advice, which contains everything i would have said and much more, but said more clearly and succinctly. Plus it has the stamp of approval of a Fields medalist. In fact I myself just reread Tao's advice for my own benefit.

Everyone here should read the advice of Terence Tao, and try to heed it. This is the best article i have seen on how to become a mathematician.

I am going to send it to our grad students at UGA for their benefit too.

For reference again, (and with thanks to edgardo):

http://www.math.ucla.edu/~tao/advice.html

 

[pivoxa15]

most mathematicians i know are pretty normal, and have families, friends, children, etc. those people you mentioned are very unusual, and not usually better mathematicians then the normal ones in my opinion.

But do you think had them or you not had a family etc would have enabled you to go further with your maths? In other words you would have had less distractions. Or do you think these things are necessary to make a good mathematician or at least keeping a balanced life is necessary to becoming a successful pure mathematian?

 

[pivoxa15]

Response to above, also -- most mathematicians can usually be found occupying the local drinking spot at one time or the other. Communication and social skills are way up there if you want to succeed, imo.

But what if it's pure maths?

 

[mathwonk]

well i find myself thinking sometimes that if I had no family obligations, then I could work more. There is a joke that a mathematician needs both a mistress and a wife because then when he is not with the mistress she thinks he is with the wife, and vice versa, so then he can skip out on both of them and go to the office and get some work done.

But in truth I never found it possible to complete my own grad studies and become a mathematician until i got married and had a normal family life. The birth of my children energized me also in my math.

Hironaka, the fields medalist once told me a joke about mathematicians who found they proved good theorems on getting married would sometimes get married several times to have this experience over again.

It is reminiscent of a remark made to me by an advisor at Harvard college on students who wanted get away from Cambridge and all its distractions to study more, but when they returned they found that the students who had stayed, somehow had accomplished more, even with all the distraction.

I personally cannot bear to stay longer than one week alone at a meeting or summer session. I love my work, but not exclusively, and I work better in a normal environment.

Life is not easy or simple. As my yoga teacher said, one has a spiritual self, a physical self, an intellectual self, an emotional self, etc...

The task is to keep them all functioning in harmony.

good wishes.

 

[Beeza]

Mathwonk,
At any point did you ever doubt your ability to succeed in advanced math courses? I just began self studying Apostol's Mathematical Analysis with a professor of mine (whom offered to continue working with me over the summer when the spring semester is over), and find even the beginning exercises very fun, but often time consuming and difficult. I sometimes worry I won't live up to my own expectations, or even my professors'.

 

[pivoxa15]

well i find myself thinking sometimes that if I had no family obligations, then I could work more. There is a joke that a mathematician needs both a mistress and a wife because then when he is not with the mistress she thinks he is with the wife, and vice versa, so then he can skip out on both of them and go to the office and get some work done.

But in truth I never found it possible to complete my own grad studies and become a mathematician until i got married and had a normal family life. The birth of my children energized me also in my math.

Hironaka, the fields medalist once told me a joke about mathematicians who found they proved good theorems on getting married would sometimes get married several times to have this experience over again.

It is reminiscent of a remark made to me by an advisor at Harvard college on students who wanted get away from Cambridge and all its distractions to study more, but when they returned they found that the students who had stayed, somehow had accomplished more, even with all the distraction.

I personally cannot bear to stay longer than one week alone at a meeting or summer session, because I enjoy being around my wife too much. I love my work, but not exclusively, and I work better in a normal environment.

Life is not easy or simple. As my yoga teacher said, one has a spiritual self, a physical self, an intellectual self, an emotional self, a sexual self, etc...

The task is to keep them all functioning in harmony.

good wishes.

What kind of distractions exist in Cambridge?

Your point about having balanced life is extremely important I think because we have evolved evolutionary and people who do a wide range of things are rewarded psychologically as a way of our body thinking us for what we have done to prolong its existence. Having children is one of those things I think. And when we don't do these things, our body punish us by making us feel depressed.

From your wide observations, what kind of wife is best suited to an academic mathematician? i.e another mathematician, school teacher, etc. OR is it too wide ranging to say?

 

[morphism]

pivoxa15, from all your posts I gather that you have some weird, disturbing idea about what a "pure mathematician" is. Mathematicians are humans, not machines that do mathematics...

 

[tehno]

Now that my conference has ended and I am still exhilarated by the experience of meeting again so many mathematical friends and hearing so much interesting math that it has literally jump started my math research thinking again, I wanted to extend my earlier advice on becoming a mathematician to include strong advice to attend conferences.

How many conferences,on average,you attend per year?Just being curious.

 

[symbolipoint]

Mathwonk, I just checked your initiating message on this topic and found what you said regarding foreign languages:

learn to struggle along in French and German, maybe Russian, if those are foreign to you, as not all papers are translated, but if English is your language you are lucky since many things are in English (Gauss), but oddly not Galois and only recently Riemann.

What more can you tell us about the usefulness of knowing Russian for the purposes of reading articles written in Russian about any Mathematics? How valuable? Do significant articles exist which have not yet been translated which Mathematical specialists might want to read and understand? In other words, is there still significant Mathematics work written in Russian which have not been translated? Would knowing Russian then be a special qualification for gaining admission to even an undergraduate Mathematics program (AS A STUDENT)? Were Russian Mathematicians known for any significant contributions to field of Mathematics (in other words, what were Russian Mathematicians famous for creating/discovering?)

symbolipoint

 

[JasonRox]

From your wide observations, what kind of wife is best suited to an academic mathematician? i.e another mathematician, school teacher, etc. OR is it too wide ranging to say?

Um... that's like asking the average guy the same question.

You want a wife that you'd love. If she's not suited for your career, don't marry her. A girl you love suits within your life in every way.

I'd want a nice good looking girl who loves playing in the bedroom. I need to clear my mind once in awhile.

 

[mathwonk]

well I admit the standard russian math journals are regularly translated into english so maybe it is not too crucial to know russian for math. but every now and then I find a russian preprint or paper that is not translated and it helps that i read russian. this does not happen too often though.

i do have several russian math friends though and i enjoy at least being able to say hello.

there are a lot of outstanding russian mathematicians and their contributions are legion: novikov, arnol'd, postnikov, shafarevich, tjurin, shokurov, alexeev,
nikulin, margoulis, dolgachev, moishezon, pontrjagin, tichonov, urysohn, sobolev, lobachevsky, malcev, kac, kazhdan, efimov, markov, givental, voronoy, delaunay, lefschetz, kurosh, gromov, iskovkikh,...

 

[mattmns]

Here is a silly question for you mathwonk How do you pronounce Spivak?

 

[pivoxa15]

I wonder why there are so many outstanding Russian mathematicians. It seems like the fields medalist list are dominated by them and Americans. However the Americans tend to come from many different ethinic backgrounds.

Is it because they are biologically more adapted to abstract things like maths and chess or is it because of their communist ruling for most of the recent past so there isn't many things to do or not many distractions.

 

[matt grime]

More strange opinions there, pivoxa. I would suggest you might consider that in Russia (in the past), eduaction was just more highly valued than elsewhere, and especially mathematics. Similar things have happened throughout the world in a variety of arenas.

The Aussies put a lot of emphasis on sports now, as they saw it an arena where they could compete with the rest of the world. Consequently in the 80s they spent a lot of cash on the infrastructure to create cricketing, rugby league and swimming teams that are te envy of the world. Another case, albeit an odd one, is scrabble. Some of the best scrabble players in the world (in English) are from Taiwan (or do I mean the Philippines) even though they can't speak English - it is taught in schools for some reason.

The Russians invested heavily in mathematics. Now they don't spend that much on it and consequently a lot of the best Russians are no longer in Russia.

 

[MathematicalPhysicist]

I'd want a nice good looking girl who loves playing in the bedroom. I need to clear my mind once in awhile.

exactly!
(-:

 

[mathwonk]

I pronounce Spivak as Spih - vak, i.e. not Spee - vak.And it is interesting that although the Soviets did invest heavily in math and science and valued it greatly, the communist government often tried to prevent their jewish citizens from benefiting from these math opportunites.

In spite of many obstacles in their path, nonetheless many Jewish soviets still became mathematicians and outstanding ones.

I do not know in general which of those I named are Jewish, but I know Moishezon was, since Boris was a friend of mine. Also Kazhdan, since I knew him slightly.

I also omitted to name perhaps the most famous recent Russian mathematician, Grigory Perelman.

 

[MathematicalPhysicist]

the better question is how spivak pronounce his last name?

 

[mathwonk]

well, i apologize for being vague, but since he is a friend of mine you may assume my pronunciation is one he has heard a few times without objecting, and that i have also heard many other people pronounce it as i do over the past 40 years.

i cannot recall hearing him pronounce his own name in a long time since he knows I know what it is.

 

[MathematicalPhysicist]

so it's spy-vak, i thought it was spee-vak.

 

[mathwonk]

spih not spy or spee, but i think it is allowed to say it other ways.

 

[Werg22]

Mathwonk, what do you suggest for self studiers? Learn one branch at a time or learn them all together?

 

[mathwonk]

if you are like me i can only learn one thing at a time, at best. and not one branch, one fact!

 

[Werg22]

Hummm, often I find myself amputated when it comes to some subjects in mathematics (for example, I know very little about linear algebra) because I've put all my energy into number theory and analysis. Wouldn't it be better, for example, to learn the foundations of several branches before pressing on the mastery of one?

 

[Data]

Hummm, often I find myself amputated when it comes to some subjects in mathematics (for example, I know very little about linear algebra) because I've put all my energy into number theory and analysis. Wouldn't it be better, for example, to learn the foundations of several branches before pressing on the mastery of one?

You're still in high school; study whatever catches your fancy! You will get a much more general background when you start a university program in math (or math-physics). For now, if you think you'd like to learn something about linear algebra, go ahead and pick up a textbook.

Personally, I'm taking advantage of the small gaps between my exams to start learning a few topics that I haven't had a chance to pick up yet. For example, I've just read all of the elementary material on measure theory that I can find online; Tomorrow, I have half a dozen books to pick up at the library!

 

[J77]

feel like summarizing what you said about non linear optics? even if it way over my head, someone will enjoy it.

If I started, I fear I would lose anonymity somewhat -- which I prefer on bbs

I would defintely reiterate your advice of conferences -- sharing your ideas with others (in an informal way rather than peer-review) and getting criticism really helps you to spur you on.

I've done four so far this "season" , with four more to go -- including two long-haulers.

 

[mathwonk]

i enjoyed anonymity for a while, then decided to forego it. thought it might make me more responsible, but it hasn't worked yet.

 

[mathwonk]

perhaps most of us should learn one thing at a time, but then also think about how it relates to other things.

 

[mattmns]

How important is a basic differential equations class (not theory, but a computational class that engineers and physics majors would take) for grad school admissions (PhD in Pure math)? I have been looking at different programs and it seems many schools want you to have taken basic differential equations.

Personally, I have never taken the class, and it looks to be a boring class that I really don't care to take. I am basically done with all the requirements for my degree in pure math, so I could take the class if I absolutely needed to, but I would prefer to take a class on topology or a second course in abstract algebra, or some other upper level theoretical math class.

Your thoughts? Thanks!

 

[J77]

How important is a basic differential equations class (not theory, but a computational class that engineers and physics majors would take) for grad school admissions (PhD in Pure math)? I have been looking at different programs and it seems many schools want you to have taken basic differential equations.

Personally, I have never taken the class, and it looks to be a boring class that I really don't care to take. I am basically done with all the requirements for my degree in pure math, so I could take the class if I absolutely needed to, but I would prefer to take a class on topology or a second course in abstract algebra, or some other upper level theoretical math class.

Your thoughts? Thanks!

DE is a hard one.

I think most student's views would be that it's only about learning methods and applying them by rote. However, I believe this to be a bit naive... or moreover, students don't understand that a great many mathematical fields are about applying techniques -- the complication of the technique just means the subject requires longer to master.

Basic DE classes form the backbone of many physical applications -- including, for many, the first obvious use of calculus.

Furthermore, they form the backbone of everything higher -- which some would label as pure math -- eg. in the pursuit of solutions of PDEs.

I think the "pure" guys on here may like to get rid of simple DE courses -- and start on, say, waves and their instabiities.

However, I like the basic DE courses because they give students a sense of application for, eg., calculus and linear algebra.

Even if they may be easy -- imo, they are worth it

 

[mathwonk]

well DE is important. some intro de courses are really boring, but some are not. the book by devaney blanchard et al, is kind of fun, altho i criticized it.

and arnol'd's book is wonderful, and interesting too. i also recommend martin braun's book for interesting applications as well as computations.

for a classic book that explains everything basic as well as advanced in a traditional way, try pollard and tenenbaum.

or take whatever you find fun.

 

[mathwonk]

sorry i did not notice the question on number of conferences i attend. in 2001 i suffered an injury and was unable to travel easily for several years.

up until then i had attended roughly 23 mostly international conferences as an invited speaker since 1977, about 10 more as a participant (not a speaker), and given about 40 more invited or contributed seminar or colloquium talks at various places here and abroad, as well as three invited courses of a total of 25-40 lectures abroad.

in the last 5 years or so i have received 2 international conference invitatiions and several invitations to give seminars which i have not been able to accept. this UGA birthday conference is thus the first one i have been to in a while.

it reminded me how wonderful and stimulating conferences are and now i am very tempted to go to a couple more in europe in june. the problem is that as a senior participant and not an invited speaker, airfare and hotels are quite high now, especially in euros.
our travel budget is essentially nil right now, which reminds me to suggest you investigate such things when choosing a university job.

since conferences are so useful, a travel budget is one of the most important ways for a govt or university to suppoort research.

so i guess for the first 25 years or so i averaged abut one major conference a year. the only time i did not feel the need to go to them was when i was at harvard. the atmosphere there was so stimualting, especially talking to David Mumford, that it was actually better than an international conference.

in fact when i did leave Harvard to go to an international conference, i found that the speakers were behind the curve of what was gong on right in the department at Harvard. In fact one of the talks concerned a result I myself had worked out and reported on to a Harvard colleague some 18 months earlier. So you could be more up to date by asking questions from people standing around the coffee room at Harvard than going to a big conference of experts.

at that time (1979-1981) Mumford, Griffiths, and Hironaka were all at Harvard, making it the center of the algebraic geometry universe. and everyone who did anything notable in the area would either send a copy to people at harvard for their review and approval, or would actually come up to speak about it there first.

as to conferences, there is a difference between being a participant and being a speaker. i find being a speaker even more stimulating usually because you are motivated to think very hard about your work, and you get to present it to a usually appreciative audience. it can be a real high.

As a speaker you also get the chance to advertise yourself and your work, and it helps people get to know you, which helps you get jobs, invitations, and grants.

being a participant, i.e. mostly listening, is more of a job, since it is hard to really grasp the talks in depth. the good side is it keeps you up to date in a way reading cannot do as quickly. it also acquaints you with the young people in the field, allows you to assess how strong and personable they are, and this is crucial in planning your own hiring.

if the talks are really good, you may learn something that inspires research of your own. I heard a talk by Mumford once that did just that, and the work that grew out of it with my colleague Robert Varley is one of the things I am most proud of.

As it happens I also said something in my talk that Mumford turned into a nice piece of work himself, extending some other work he had recently done. it was real thrill to have mumford call me over at lunch the next day, and show me his result. i still have the handwritten version of the proof he gave me.

by the way David Mumford is being honored on his 70th birthday at a 2 day conference June 1-2, at Brown, for his work in both algebraic geometry and artificial intelligence and perception. It should be a nice occasion, and if you are able to be in Providence then, it would be a wonderful way to begin your conference attending career.

 

[mathwonk]

back to DE. This is an enormously important subject that everyone needs to know about. all questions here about vector fields and diferential forms and de rham cohomology, are actually questions about differential equations.

i.e. a vector field IS a differential equation, and vice versa. this is the way it is taught in arnol'd, and from a more elementary viewpoint also in blanchard, devaney et al.

learn it that way and it will be both interesting and useful.

i was speaking about ode. partial diff eq is equally important but harder, less well understood as a theory hence concerns a study of more special equations.

but these special equations are among the most fundamental objects in mathematics: the laplace equation, the heat equation, and the wave equation, to mention only the most classical ones. So it may be that people just study one important pde at a time. I myself feel I know essentially nothing about pdes, but have long used the several variable complex heat equation, since it is satisfied, as perhaps Riemann first showed, by the theta function in the theory of abelian varieties. As far as I know, the heat equation was first used in the study of the famous Schottky problem in algebraic geometry in the now classic paper of A. Andreotti and A. Mayer, or possibly earlier in the case of genus 4 by Mayer.

 

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[mathwonk]

what in the world was that? and how did it wind up on here?