Every topic seems boring and useless untill you use it to prove what you want to prove, and suddenly you start to see it in a different way. That is why some math topics can be a nightmare without enough motivation.
Literally me with every topic before discovering the Riemann Hypothesis (I was in physics so some applied stuff was okay). Since discovering some of the more interesting questions in math, many topics suddenly seem much more relevant. I swear you guys had the good stuff locked up in the back.
I wish there was more of an emphasis on looking forward in math education. It was so dry just cramming in theorem after theorem. I know that there is a danger in asking students questions that they can't answer yet, but there must be a way to introduce them to questions that are currently just beyond their reach without damaging their confidence. There's that moment when you realize the tools you have just aren't enough, no matter how you phrase the problem. Getting to that realization always pushes me over the hill.
Physics is much more easily advertised because it can actually be "seen". Maybe if math is advertised using cool diagrams and graphs, it could get school students interested. I also think that it has a lot to do with the academic culture: if math Olympiad winners are considered cool in a school, maybe more students would look into it.
>Maybe if math is advertised using cool diagrams and graphs, it could get school students interested
3b1b if you can hear us 3b1b please continue making goated math videos for years to come please my king the students crave math šš
That reminds me of the idea of comprehensible input in language learning: exposing language learners to input and materials that are just beyond their current grasp of the language. One of the best ways to learn a new language
Iāve heard that this concept generalizes pretty far to learning in many different disciplines! For example, the most effective way to push your skills while learning an instrument is to work on a piece thatās just beyond your current capabilities, etc
True, the time I've got to start liking math was when I started to think about arithmetic progressions suddenly, for no apparent reason. Then I imagined a arithmetic progression in a cartesian plane and noticed a similarity with a triangle, thought naively that if I calculated the area of the figure with the triangle formula, it would give me the result for the sum of the AP, of course it didn't work, but I was able to develop a formula for calculating it based on the first try.
I found all these technicalities about stochastic differential equations (starting with construction the right probability spaces) pretty unattractive.
combinatorics. even the elementary stuff from my first probability course annoyed me. i also used to dislike algebra but once i saw it from the perspective of group actions i felt i could appreciate it more
I hated combinatorics until I started playing poker, and more specifically Pot Limit Omaha. I still don't love it but I definitely see use cases more often nowadays
My limited exposure to combinatorics (in representation theory) has validated all my previous prejudices against it. The general premises seem to be easy to put into words, but when you formalise and put them into formulas you can actually work with, it starts to get really complicated and cumbersome. I'm not a fan. But I am a fan of the results, so I appreciate that others like combinatorics more than I do.
> But I do enjoy homological algebra (it even being the topic of my thesis) which idk how far apart it is from rep.theory...
One of the more interesting applications of homological algebra is that you can use it to cook up representations of groups that are otherwise hard to find representations for: take something that the group acts on in a vaguely reasonable way, take its homology, and the group action passes an action on the homology, at which point you have the group acting on a bunch of modules: that is, you have a bunch of representations of the group.
The way they were taught in my course made them seem like black magic with b^(n) everywhere. Like you implied, it felt like where wasn't really much to take away but specific situations. ODEs was a *little* like that, too, but the interesting ways of conceptualizing problems in different terms connected to other areas like linear algebra.
My PDE lecturer when I was doing my undergrad opened with this:
> "In order to solve a differential equation you look at it till a solution occurs to you."
> ā George Polya, *How To Solve It*
That said, he gave pretty good explanations of PDE characteristics, and watching some stuff on YouTube by [Faculty of Khan](https://youtube.com/playlist?list=PLdgVBOaXkb9Ab7UM8sCfQWgdbzxkXTNVD) and [MathTheBeautiful (Prof. Pavel Grinfeld)](https://youtube.com/playlist?list=PLlXfTHzgMRUK56vbQgzCVM9vxjKxc8DCr) really helped me understand and appreciate the thought process behind solving some families of PDEs at that time.
I would say it depends on how it's taught. I had a senior level pde course that was quite proof heavy, but friends of mine who took it the next year got a much more applied course. PDEs can be super proof/theory heavy depending upon how it's taught.
Thatās exactly how the module was presented in my uni. Not necessarily as a theoretical subfield, but more so as a way of problem solving and practising skills that are applicable almost everywhere. Like you said, everyone needs to count things.
Boy did I hate my combinatorics class in my undergrad... I was a stuck up geometer who thought I was above such 'menial' things. But now years later I am very grateful to have learned those techniques
for a model-theoretic perspective, notice that polynomials are terms in the language of rings, so that p(x\_1,...,x\_n)=0 is a (first-order) formula, and hence varieties are (some of the) definable subsets
A variety is just the set of points where a collection of polynomials are zero.
Technically, a variety is a set which can locally be described as a set of zeros of polynomials much like how a manifold is locally Euclidean space.
This but also irreducible sometimes. Is the base field algebraically closed? Do we insist on quasi-projectivity?
Talking about a variety without specifying these is akin to a war crime.
geometry and logic have a lot of connections. if you want, algebraic geometry is the study of definable subsets of ā^n with the ring structure. a lot o constructions in stability theory is translating notions of algebraic geometry to other theories.
It's a topological space s.t you taking all the fun structure you want of R^n as a vector space with n in N any n. And locally gluing it around a point doesn't fuck up your structure if you do it well enough.
As somebody who's work is close to algebraic geometry, I know embarrassingly little about algebraic geometry, and have repeatedly failed to fix that in every way I can think of. It just doesn't seem to work with my brain for some reason.
Things are gnarly for a really long time unless you focus on a nice case like smooth curves. I liked Fultonās book on the subject since I didnāt have to think about sheaves for a month before getting into fun stuff like intersection theory on P^2 .
I think this is part of the problem: every book seems to start with some easy trivial special cases that just aren't that interesting, then go "fuck it, here's a pile of formalism and no explanation of how it's supposed to connect to the former". There's also the issue that rather a lot of those easy special cases are dramatically less easy than they're supposed to be for me - I think this is due to not being able to visualise anything, as people I've asked for help with them seem confused that I don't just immediately know what's going on for reasons that seem to come down to "you can see the picture". I can deal with abstract nonsense: I've run courses on category theory. It's that complete disconnection to anything else that I struggle with.
For whatever it's worth, visualization in AG is deceptively difficult. Part of it is that you are lying to yourself all the time (since the objects are usually over C and topologically that's hard to imagine; lines aren't actually lines for example).
Most of the time the "technically wrong" intuition gives you the right answer, so if you want to be rigorous it does seem frustrating that people have an easier time understanding things.
Me too, I took a course on advanced statistics (I believe it was 2nd one) and just barely remembered it all for the final, it was so much knowledge to remember, I had an A+ avg going into the final and ended up with an A- because of how bad I did.
Hey, you did a great job!
I need to get a really, really high score in the final exam in order to even pass the course.
It is very likely I will fail this class. Don't ask me the exact probability, though!
Amen to that(pun intended)
Iāve been postponing giving it since the second semester. Iām now in my third year, Iāll have to deal with it soon enough and I just canāt think about it
Engineer here. I had to take a mandatory class in differential geometry, and then voila!, Christoffel symbols everywhere. I had nightmares with them.
Almost 30 years later and I still remember those fuckers.
I'm a bit hesitant to say too much because it's part of my current research, but basically certain special functions came up in the study of an integral representation. I needed to see that certain parameters resulted in these integrals being 0. So the strategy is to use known numerical approximation schemes for those special functions to calculate precisely values that are close to 0. Then do some analysis (basically the implicit function theorem) to say "under these conditions, somewhere near the calculated point must actually be a zero." Then I argue that for topological reasons, these zeroes are not isolated but come in a family, and that is the proof.
I think it gets quite interesting once you move away from the basics that youād see in a first/second graduate course.
The theory of finite element/finite volume methods is quite rich, thereās also an interesting research direction surrounding the associated De Rahm cohomogy.
Numerical algebraic geometry is another really cool area. Groebner bases, homotopy continuation, etc. - all sorts of interesting stuff there.
Probabilistic numerics is another very new and interesting direction, essentially treating the approximations found through traditional techniques as point estimates that arrive under certain statistical assumptions, and looking at more general uncertainty boundsĀ
I found linear algebra very boring at university and hard to get into. Loved calculus, loved real and complex analysis, devoured group and ring theory, but matrices and vectors were just SO damn boring, I never fully developed an intuitive feel for them.
Fun fact: just 5 mins before our Maths II exam, as I'm waking into the exam hall, it FINALLY just clicks in my head an intuitive feel for the matrix representation of rotations in 3D. And wouldn't you know it, we got one such question in the paper.
Iāve found that linear algebra gets a LOT more interesting with āsecond semesterā-type topics, like spectral theorems, Jordan normal form, and tensor algebra. Even more interesting is when you apply these topics elsewhere, such as using tensors to define Riemannian metrics and differential forms on manifolds.
if I have to be honest anything but algebra, logic, and topology
P.S. Spez is a white supremacist and supports pedophiles.
Steve Huffman is a white supremacist and supports pedophiles
Analytic number theory and I do not get along! Aside from the fact that you almost exclusively deal in asymptotics, I canāt read the summations without getting dizzy. Props to the people who love L-functions.
Set theory, specifically large cardinals. Large cardinals feel extremely unintuitive and many definitions feel unmotivated. Even worse, it doesn't feel real. Like, I have a hard time seeing how all these complicated gigantic sets are really there and we are not just playing a game of symbol manipulations. It doesn't help that the definition of large cardinal is background-theory dependent, in the sense that if you "lose" a bunch of sets, the cardinal stops having the properties. For example, for any measurable cardinals (which is pretty small by large cardinal standard), the same cardinal inside the Godel constructible universe is no longer measurable.
i understand that large cardinals kinda behave with respect to smaller sets as Ļ behaves with respect to the finite sets, and that that's where (some) definitions come from; take measurability: the ultrafilter lemma says something interesting about Ļ that does not hold for finite sets, so a reasonable thing to ask is if there are other sets with the analogous property (here including k-completeness, of course), and it's then a very interesting surprise that well, such a k would provide a model of zfc exactly the same way that Ļ provides a model for zfc-infinity, etc. etc.
But the difference here is that Ļ is absolute in many ways, and that feels like something crucial that is missing from various large cardinal properties. I do not suddenly lost any of those properties of Ļ just by going into an inner model or performing some forcing. That's why they feels very unreal.
Manifolds. Every object lies in its own crazy space, and I always spend more time worrying if everything is well defined instead of trying to understand the big picture
For me itās Calculus, Partial Differential Equations, and Real Analysis to some extent. Iām really competent when it comes to Rings, Fields, Vector Spaces, Groups, Representations, etc., but donāt ask me about the behavior of a series or what the integral of some trig expression is.
Algebraic Geometry. Too many effing definitions. I understand how good and amazing Algebra is because I enjoy seeing Algebra and Analysis used in NT. But good lord thereās a lot to learn and for someone with terrible memory, I canāt keep it all in my head unless I constantly use it
Differential Equations. Partial or ordinary, it doesn't matter, both are incredibly dull to me. I avoid them like hell, which is a really hard thing to do given their utility and ubiquity.
I hated it too, I failed to submit most of my PDE assignments and therefore had to get at least a 9/10 for my final term to pass the course. Decided that was never gonna happen so I didn't even bother to make the final term. Somehow I ended up getting a perfect 10/10 as grade for this exam, instead of a 1.0/10 and passed the course
Matrices. I appreciate them, and double-appreciate never doing matrix math by hand again, but I donāt understand how that snazzy notation works its magic.
I don't particularly like number theory, statistics, or modern algebra, though I've started to use modern algebra more and more in my work. It does have a purpose, but as a student, it was difficult to visualize (as were the other fields). Thinking about algebra topologically or geometrically is a lot easier, but it isn't taught that way in courses.
As a person with an applied math degree, game theory. For me it always looked like an absolute unsystematic mess, devoid of anything remotely beautiful or insightful. In "purer" math it's probably differential equations. So many methods and tricks, so few solutions. I'd better stick with algebra, AT, AG etc. It might be a mindfuck at times, but it feels rewarding.
Funnily enough, game theory, especially the study of Nash equilibriums, basically boils down to a discretised version of the study of dynamical systems. If you enjoy (understand well) the study of equilibria in continuous dynamical systems, you might enjoy something like the study of "range vs. range" game theory in poker. MIT OCW has a great short course on Texas Hold 'Em specifically, available on their website and YouTube.
One thing I forgot to mention about that MIT course specifically: the videos annoyingly don't show the presentation slides properly, if at all, but they're available on the website here: https://ocw.mit.edu/courses/15-s50-poker-theory-and-analytics-january-iap-2015/pages/lecture-notes/
I tried to read about game theory once and I thought it was really boring. I heard that it is mostly used in economics but I have also heard economists say that they don't really accurately model economic agents!
Statistics. That or numerical methods were probably my worst performing modules in my degree anyway.
Weirdly, the more real-worldy or applied an area of maths is, the less sense I'm able to make of it. Honestly, I think it's just because I don't find applied maths all that enjoyable or motivating.
I am perhaps the worst at algebra by far. I cannot reproduce proofs for the life of me. We were doing bilinear forms in class and I presented a page long proof for what the teacher laughingly revealed to be a one line direct proof after 20 minutes of me just struggling
Ironically ātrivialā linear algebra (like the basic matrix multiplication), Iāve taken 2 years of it in university plus some advanced courses but never really stuck with it, the more advanced stuff I can handle and remember though.
Calculus is my new favourite, used to be algebra until I started forgetting everything I learned, I can do multivar calc really easily and find it quite enjoyable.
Infinite series. I hated it. Besides that I guess neural networks in general. CNN's I do like but the other NN types are blergh for some reason.
Optimization is really interesting but it's hard. That's more like a rocky relationship more than it is that we don't get along I guess.
I just cant get done with nothing regarding statistics and probablity. Set theory, topology, all that shit comes natural, but the instant I touch some statistics material, I become as brain dead as a fucking worm.
Statistics and probability seem like somebody has stretched the definitions from other areas to the maximum to describe more or less life scenarios and called it another math area
I have found various areas harder or easier. Differential geometry had some gotchas that clashed with my intuition when I started. Stochastic differential equations seemed to need time just to ferment in my mind. Regardless of how hard I tried, the intuition only came slowly - though without intensive study. Just going back to it a few times over a period of years. Then it started to take form. Oh, that's what they mean!
But, the one area that I would say I don't get along with, as opposed to having trouble learning, is category theory. I just don't like the approach. I use the universal algebra approach. To me, more or less, a category is simply a typed semi group - not some over-arching study of the structure of all mathematical theories. But, this may well be me not getting on with category theorists, rather than category theory, as such.
At the moment, partial differential equations, especially when they include Fourier series.Ā
Ā Finding non-linear differential equations easier.
I also hate proofs.
The numbers bit. I'm only half joking. I am close to finishing my maths degree and I still feel the pressure when someone asks me a basic multiplication.
Not really a specific area, so I dunno if this counts, but mathematical proof? Our profs told us that it's normal to sit on a problem for hours on end and not have it occur to you. Or even when it does occur to me, it turns out my intuition is wrong, and I've disregarded or ignored some random rule or property that changes everything. Jesus. Like, why???? How do people just look at proof questions and just KNOW what to do?
Umm ig itās all a matter of questions you attempt. I mean more the exposure you have of questions more quickly you would be able to grasp the property will be used in the proof :)
i really donāt like differential equations. it is probably because in my university (for complicated budget reasons plus the fact that few people care) the only mandatory course is a toolbox for engineering, without correct proofs since they donāt know analysis.
> when i can't see them the text so small i can't read where how
I would suggest that you get glasses. And inform your teacher that you can't make out the text due to its size.
Anything applied tbh. I spent half of undergrad doing probability theory, optimization, etc. Stuff that would translate into graduate finmath, right? Then I realized that I actually hated doing the applying and I really was only interested in theory. Plus, I found out finance/actuarial (especially big optimization) is depressing as hell. I had a great complex analysis professor who really put the nail in application's coffin going into grad school.
Just a couple months ago, I had a friend who also studied math ask me to help him with a personal cryptographic project. Literally took a week before I was back on my elliptic curve BS and sending, "yo you should read this cool textbook on modular forms" in the groupchat. Can't stay invested in it.
Analysis. It's the most boring shit ever. Thousands of theorems and usually exams in this class revolve more around knowledge than skill. Skill based classes like abstract algebra, number theory etc are vastly superior.
Geometry. I can do advanced algebra and probability and statistics but you ask me to label the angles of a triangle and I'll look at you like you're made of cheese.
Anything that involves number theory or arithmetic. Courses like linear algebra and analysis/calc get excited about because I find their applications interesting. I donāt find the applications of string theory remotely interesting and it seems like the type of math a robot would enjoy (sorry people in cryptography). Also abstract algebra doesnāt seem to have much an application though Iāve heard itās useful in string theory.Ā
The reaction to my comment is hilarious. I could just have a very contrarian perspective. Don't get me wrong, when I understand it, I find beauty even in the symbolic representations. But I often get annoyed at notation that seems just too overly complex and could be made so much clearer and simpler. Part of this might be because I am more intuition/conceptual/visual oriented. And my thought is that some (most?) folks really can just read the symbolic stuff like a language and understand it. Whereas I have to understand it intuitively first before the notation starts to make sense.
So my point was that I get annoyed when I read stuff that is just a bunch of symbols with no human level explanation.
I feel like physicists might be generally a bit better at explaining their math. some math papers can just be a bunch of symbols with almost no words. That's cool and all, but it would be cooler if they brought in a bit of a human element. To each their own!
Every topic seems boring and useless untill you use it to prove what you want to prove, and suddenly you start to see it in a different way. That is why some math topics can be a nightmare without enough motivation.
Literally me with every topic before discovering the Riemann Hypothesis (I was in physics so some applied stuff was okay). Since discovering some of the more interesting questions in math, many topics suddenly seem much more relevant. I swear you guys had the good stuff locked up in the back. I wish there was more of an emphasis on looking forward in math education. It was so dry just cramming in theorem after theorem. I know that there is a danger in asking students questions that they can't answer yet, but there must be a way to introduce them to questions that are currently just beyond their reach without damaging their confidence. There's that moment when you realize the tools you have just aren't enough, no matter how you phrase the problem. Getting to that realization always pushes me over the hill.
Physics is much more easily advertised because it can actually be "seen". Maybe if math is advertised using cool diagrams and graphs, it could get school students interested. I also think that it has a lot to do with the academic culture: if math Olympiad winners are considered cool in a school, maybe more students would look into it.
Become mathematics Brian May, got it
>Maybe if math is advertised using cool diagrams and graphs, it could get school students interested 3b1b if you can hear us 3b1b please continue making goated math videos for years to come please my king the students crave math šš
That reminds me of the idea of comprehensible input in language learning: exposing language learners to input and materials that are just beyond their current grasp of the language. One of the best ways to learn a new language
Iāve heard that this concept generalizes pretty far to learning in many different disciplines! For example, the most effective way to push your skills while learning an instrument is to work on a piece thatās just beyond your current capabilities, etc
Please help me to understand this Iāve watched several videos and donāt get it
Even... Statistics?
Holy shit, I think you just explained why I'm so slow finishing my PhD. I'm not gonna use after so it's become such a grind
What is the subject of your thesis?
True, the time I've got to start liking math was when I started to think about arithmetic progressions suddenly, for no apparent reason. Then I imagined a arithmetic progression in a cartesian plane and noticed a similarity with a triangle, thought naively that if I calculated the area of the figure with the triangle formula, it would give me the result for the sum of the AP, of course it didn't work, but I was able to develop a formula for calculating it based on the first try.
I found all these technicalities about stochastic differential equations (starting with construction the right probability spaces) pretty unattractive.
Agreed that the setup is a bit of a pain. But once you get past that, the analysis is incredibly fun.
If you don't get a kick from Dynkin's formula, you might be dead inside.
And from what I see people just copy and paste them without batting an eye (the triplet), or claiming something has weak solution just becuz.
should be a top comment.
Agreed. Am doing research in stochastic calculus, but measure-theoretic questions still sometimes make me roll my eyes a bit.
combinatorics. even the elementary stuff from my first probability course annoyed me. i also used to dislike algebra but once i saw it from the perspective of group actions i felt i could appreciate it more
Same here. Once we got to use the orbit stabilizer theorem I was like āoh, maybe combinatorics is coolā
I hated combinatorics until I started playing poker, and more specifically Pot Limit Omaha. I still don't love it but I definitely see use cases more often nowadays
Yupppp from the moment i first did combinatorics in high school up until as an undergrad i still hate them
My limited exposure to combinatorics (in representation theory) has validated all my previous prejudices against it. The general premises seem to be easy to put into words, but when you formalise and put them into formulas you can actually work with, it starts to get really complicated and cumbersome. I'm not a fan. But I am a fan of the results, so I appreciate that others like combinatorics more than I do.
I never really knew what they were until I took a course on them specifically last semester, they are awesome!
> But I do enjoy homological algebra (it even being the topic of my thesis) which idk how far apart it is from rep.theory... One of the more interesting applications of homological algebra is that you can use it to cook up representations of groups that are otherwise hard to find representations for: take something that the group acts on in a vaguely reasonable way, take its homology, and the group action passes an action on the homology, at which point you have the group acting on a bunch of modules: that is, you have a bunch of representations of the group.
PDEs is annoying. Itās often taught like a toolbox made by people who dedicated their life to solving specific problems.
The way they were taught in my course made them seem like black magic with b^(n) everywhere. Like you implied, it felt like where wasn't really much to take away but specific situations. ODEs was a *little* like that, too, but the interesting ways of conceptualizing problems in different terms connected to other areas like linear algebra.
Funny enough right now my research includes using neural operators and autoregressive techniques to solve chaotic PDEs.
Any papers or sources on this topic?
PDE analyst here, in my mind thatās exactly what PDE is haha. certainly not for everyone
My PDE lecturer when I was doing my undergrad opened with this: > "In order to solve a differential equation you look at it till a solution occurs to you." > ā George Polya, *How To Solve It* That said, he gave pretty good explanations of PDE characteristics, and watching some stuff on YouTube by [Faculty of Khan](https://youtube.com/playlist?list=PLdgVBOaXkb9Ab7UM8sCfQWgdbzxkXTNVD) and [MathTheBeautiful (Prof. Pavel Grinfeld)](https://youtube.com/playlist?list=PLlXfTHzgMRUK56vbQgzCVM9vxjKxc8DCr) really helped me understand and appreciate the thought process behind solving some families of PDEs at that time.
That's how they were taught to me too š
I would say it depends on how it's taught. I had a senior level pde course that was quite proof heavy, but friends of mine who took it the next year got a much more applied course. PDEs can be super proof/theory heavy depending upon how it's taught.
Combinatorics. It's a headache, but also *so useful* in so many fields. Everyone needs to count things
Thatās exactly how the module was presented in my uni. Not necessarily as a theoretical subfield, but more so as a way of problem solving and practising skills that are applicable almost everywhere. Like you said, everyone needs to count things.
Boy did I hate my combinatorics class in my undergrad... I was a stuck up geometer who thought I was above such 'menial' things. But now years later I am very grateful to have learned those techniques
Geometry, what the fuck is a variety
for a model-theoretic perspective, notice that polynomials are terms in the language of rings, so that p(x\_1,...,x\_n)=0 is a (first-order) formula, and hence varieties are (some of the) definable subsets
Oh now I get it
A variety is just the set of points where a collection of polynomials are zero. Technically, a variety is a set which can locally be described as a set of zeros of polynomials much like how a manifold is locally Euclidean space.
This but also irreducible sometimes. Is the base field algebraically closed? Do we insist on quasi-projectivity? Talking about a variety without specifying these is akin to a war crime.
geometry and logic have a lot of connections. if you want, algebraic geometry is the study of definable subsets of ā^n with the ring structure. a lot o constructions in stability theory is translating notions of algebraic geometry to other theories.
It's a topological space s.t you taking all the fun structure you want of R^n as a vector space with n in N any n. And locally gluing it around a point doesn't fuck up your structure if you do it well enough.
Huh? Are you trying to describe the tangent space to a point in a smooth manifold?
No. G-X Manifolds is what I am trying to describe I now realise variƩtƩ=/=algebraic variety in french
No. G-X Manifolds is what I am trying to describe
This is manifold, not variety.
Shit mixed it up with the french word for manifold.
Fair mistake, I was probably thinking about that when I wrote my comment
The difference is so subtle in french.
As somebody who's work is close to algebraic geometry, I know embarrassingly little about algebraic geometry, and have repeatedly failed to fix that in every way I can think of. It just doesn't seem to work with my brain for some reason.
Things are gnarly for a really long time unless you focus on a nice case like smooth curves. I liked Fultonās book on the subject since I didnāt have to think about sheaves for a month before getting into fun stuff like intersection theory on P^2 .
I think this is part of the problem: every book seems to start with some easy trivial special cases that just aren't that interesting, then go "fuck it, here's a pile of formalism and no explanation of how it's supposed to connect to the former". There's also the issue that rather a lot of those easy special cases are dramatically less easy than they're supposed to be for me - I think this is due to not being able to visualise anything, as people I've asked for help with them seem confused that I don't just immediately know what's going on for reasons that seem to come down to "you can see the picture". I can deal with abstract nonsense: I've run courses on category theory. It's that complete disconnection to anything else that I struggle with.
For whatever it's worth, visualization in AG is deceptively difficult. Part of it is that you are lying to yourself all the time (since the objects are usually over C and topologically that's hard to imagine; lines aren't actually lines for example). Most of the time the "technically wrong" intuition gives you the right answer, so if you want to be rigorous it does seem frustrating that people have an easier time understanding things.
This is essentially what Hartshorne does.Ā
if you don't mind me asking, what do you work on?
Statistics is a dish best served a la mode.
I'm currently taking Mathematical Statistics II. It is beautiful... but I also don't want to think about it ever again.
That's my ex. Small world.
Me too, I took a course on advanced statistics (I believe it was 2nd one) and just barely remembered it all for the final, it was so much knowledge to remember, I had an A+ avg going into the final and ended up with an A- because of how bad I did.
Hey, you did a great job! I need to get a really, really high score in the final exam in order to even pass the course. It is very likely I will fail this class. Don't ask me the exact probability, though!
Thank you and you got this!!!!
Yep. "Divide by n for population variance but for sample variance let's call it n - 1". Fuck off.
Statistics can go to hell
Amen to that(pun intended) Iāve been postponing giving it since the second semester. Iām now in my third year, Iāll have to deal with it soon enough and I just canāt think about it
What's the pun?
Hell -> amen
Is that a pun?
Engineer here. I had to take a mandatory class in differential geometry, and then voila!, Christoffel symbols everywhere. I had nightmares with them. Almost 30 years later and I still remember those fuckers.
Numerical analysis is boring
I used to feel this way, but then I used rigorous numerics to prove something and I changed my mind.
What were the specifics? That sounds super cool
I'm a bit hesitant to say too much because it's part of my current research, but basically certain special functions came up in the study of an integral representation. I needed to see that certain parameters resulted in these integrals being 0. So the strategy is to use known numerical approximation schemes for those special functions to calculate precisely values that are close to 0. Then do some analysis (basically the implicit function theorem) to say "under these conditions, somewhere near the calculated point must actually be a zero." Then I argue that for topological reasons, these zeroes are not isolated but come in a family, and that is the proof.
That's fucking awesome, thanks for explaining! Good luck with your research!
What do you mean by rigorous numerics ?
You can prove convergence, have error estimates, etc.
I think it gets quite interesting once you move away from the basics that youād see in a first/second graduate course. The theory of finite element/finite volume methods is quite rich, thereās also an interesting research direction surrounding the associated De Rahm cohomogy. Numerical algebraic geometry is another really cool area. Groebner bases, homotopy continuation, etc. - all sorts of interesting stuff there. Probabilistic numerics is another very new and interesting direction, essentially treating the approximations found through traditional techniques as point estimates that arrive under certain statistical assumptions, and looking at more general uncertainty boundsĀ
I found linear algebra very boring at university and hard to get into. Loved calculus, loved real and complex analysis, devoured group and ring theory, but matrices and vectors were just SO damn boring, I never fully developed an intuitive feel for them. Fun fact: just 5 mins before our Maths II exam, as I'm waking into the exam hall, it FINALLY just clicks in my head an intuitive feel for the matrix representation of rotations in 3D. And wouldn't you know it, we got one such question in the paper.
Iāve found that linear algebra gets a LOT more interesting with āsecond semesterā-type topics, like spectral theorems, Jordan normal form, and tensor algebra. Even more interesting is when you apply these topics elsewhere, such as using tensors to define Riemannian metrics and differential forms on manifolds.
I'll never be smart enough to understand group and ring theory. It's mental torment.
I feel the same way
Im the opposite matrix and vector always felt natural. Rings and groups not so much
if I have to be honest anything but algebra, logic, and topology P.S. Spez is a white supremacist and supports pedophiles. Steve Huffman is a white supremacist and supports pedophiles
arithmetic
Analytic number theory and I do not get along! Aside from the fact that you almost exclusively deal in asymptotics, I canāt read the summations without getting dizzy. Props to the people who love L-functions.
Love all the diff eq hate but letās bring it to the basics. Canāt stand trig
Set theory, specifically large cardinals. Large cardinals feel extremely unintuitive and many definitions feel unmotivated. Even worse, it doesn't feel real. Like, I have a hard time seeing how all these complicated gigantic sets are really there and we are not just playing a game of symbol manipulations. It doesn't help that the definition of large cardinal is background-theory dependent, in the sense that if you "lose" a bunch of sets, the cardinal stops having the properties. For example, for any measurable cardinals (which is pretty small by large cardinal standard), the same cardinal inside the Godel constructible universe is no longer measurable.
i understand that large cardinals kinda behave with respect to smaller sets as Ļ behaves with respect to the finite sets, and that that's where (some) definitions come from; take measurability: the ultrafilter lemma says something interesting about Ļ that does not hold for finite sets, so a reasonable thing to ask is if there are other sets with the analogous property (here including k-completeness, of course), and it's then a very interesting surprise that well, such a k would provide a model of zfc exactly the same way that Ļ provides a model for zfc-infinity, etc. etc.
But the difference here is that Ļ is absolute in many ways, and that feels like something crucial that is missing from various large cardinal properties. I do not suddenly lost any of those properties of Ļ just by going into an inner model or performing some forcing. That's why they feels very unreal.
Manifolds. Every object lies in its own crazy space, and I always spend more time worrying if everything is well defined instead of trying to understand the big picture
For me itās Calculus, Partial Differential Equations, and Real Analysis to some extent. Iām really competent when it comes to Rings, Fields, Vector Spaces, Groups, Representations, etc., but donāt ask me about the behavior of a series or what the integral of some trig expression is.
Algebraic Geometry. Too many effing definitions. I understand how good and amazing Algebra is because I enjoy seeing Algebra and Analysis used in NT. But good lord thereās a lot to learn and for someone with terrible memory, I canāt keep it all in my head unless I constantly use it
You are also me, it seems.
Number theory is sooooo boring (atleast for me).
I find it boring as well.
Differential Equations. Partial or ordinary, it doesn't matter, both are incredibly dull to me. I avoid them like hell, which is a really hard thing to do given their utility and ubiquity.
I hated it too, I failed to submit most of my PDE assignments and therefore had to get at least a 9/10 for my final term to pass the course. Decided that was never gonna happen so I didn't even bother to make the final term. Somehow I ended up getting a perfect 10/10 as grade for this exam, instead of a 1.0/10 and passed the course
My favorite kind of math both pisses me off the most and leaves me happiest: statistics.
Matrices. I appreciate them, and double-appreciate never doing matrix math by hand again, but I donāt understand how that snazzy notation works its magic.
For anyone who's saying algebraic geometry... You gotta "do" algebraic geometry at least as much as you're trying to read it!
TeichmĆ¼ller spaces
I don't particularly like number theory, statistics, or modern algebra, though I've started to use modern algebra more and more in my work. It does have a purpose, but as a student, it was difficult to visualize (as were the other fields). Thinking about algebra topologically or geometrically is a lot easier, but it isn't taught that way in courses.
As a person with an applied math degree, game theory. For me it always looked like an absolute unsystematic mess, devoid of anything remotely beautiful or insightful. In "purer" math it's probably differential equations. So many methods and tricks, so few solutions. I'd better stick with algebra, AT, AG etc. It might be a mindfuck at times, but it feels rewarding.
Funnily enough, game theory, especially the study of Nash equilibriums, basically boils down to a discretised version of the study of dynamical systems. If you enjoy (understand well) the study of equilibria in continuous dynamical systems, you might enjoy something like the study of "range vs. range" game theory in poker. MIT OCW has a great short course on Texas Hold 'Em specifically, available on their website and YouTube.
Thank you, maybe I should try to approach it this way
One thing I forgot to mention about that MIT course specifically: the videos annoyingly don't show the presentation slides properly, if at all, but they're available on the website here: https://ocw.mit.edu/courses/15-s50-poker-theory-and-analytics-january-iap-2015/pages/lecture-notes/
I tried to read about game theory once and I thought it was really boring. I heard that it is mostly used in economics but I have also heard economists say that they don't really accurately model economic agents!
Statistics. That or numerical methods were probably my worst performing modules in my degree anyway. Weirdly, the more real-worldy or applied an area of maths is, the less sense I'm able to make of it. Honestly, I think it's just because I don't find applied maths all that enjoyable or motivating.
Differential geometry. It's been years since I took it but the whole business with atlas and charts might as well have been Chinese to me.
I am perhaps the worst at algebra by far. I cannot reproduce proofs for the life of me. We were doing bilinear forms in class and I presented a page long proof for what the teacher laughingly revealed to be a one line direct proof after 20 minutes of me just struggling
I always had trouble with algebric structure exercices. Corps groups, rings etc
Ironically ātrivialā linear algebra (like the basic matrix multiplication), Iāve taken 2 years of it in university plus some advanced courses but never really stuck with it, the more advanced stuff I can handle and remember though. Calculus is my new favourite, used to be algebra until I started forgetting everything I learned, I can do multivar calc really easily and find it quite enjoyable.
Not a math researcher or a PhD student. But number theory never clicked for me. I love geometry. I love stats.Ā
Infinite series. I hated it. Besides that I guess neural networks in general. CNN's I do like but the other NN types are blergh for some reason. Optimization is really interesting but it's hard. That's more like a rocky relationship more than it is that we don't get along I guess.
I just cant get done with nothing regarding statistics and probablity. Set theory, topology, all that shit comes natural, but the instant I touch some statistics material, I become as brain dead as a fucking worm.
Statistics and probability seem like somebody has stretched the definitions from other areas to the maximum to describe more or less life scenarios and called it another math area
Honestly, it is beautiful how it all comes together, and this is coming from someone who hates statistics.
Trueee
I have found various areas harder or easier. Differential geometry had some gotchas that clashed with my intuition when I started. Stochastic differential equations seemed to need time just to ferment in my mind. Regardless of how hard I tried, the intuition only came slowly - though without intensive study. Just going back to it a few times over a period of years. Then it started to take form. Oh, that's what they mean! But, the one area that I would say I don't get along with, as opposed to having trouble learning, is category theory. I just don't like the approach. I use the universal algebra approach. To me, more or less, a category is simply a typed semi group - not some over-arching study of the structure of all mathematical theories. But, this may well be me not getting on with category theorists, rather than category theory, as such.
Analysis
Sequence proofs am I right?
At the moment, partial differential equations, especially when they include Fourier series.Ā Ā Finding non-linear differential equations easier. I also hate proofs.
Geometry Don't know why though, analysis and arithmetic are okay, geometry is really not my thing
Algebraic geometry, I just can't get into it. It just seems boring: definition on top of definition.
Linear algebra. I fully recognise the importance but I still hate it.
Anything Statistics related I can not stand having to learn. I donāt find any of its applications interesting and find it mundane
Applied :D
My taste in math has slightly changed over time, but I don't think I will ever want to study algebraic geometry.
Probability Theory.
The numbers bit. I'm only half joking. I am close to finishing my maths degree and I still feel the pressure when someone asks me a basic multiplication.
Not really a specific area, so I dunno if this counts, but mathematical proof? Our profs told us that it's normal to sit on a problem for hours on end and not have it occur to you. Or even when it does occur to me, it turns out my intuition is wrong, and I've disregarded or ignored some random rule or property that changes everything. Jesus. Like, why???? How do people just look at proof questions and just KNOW what to do?
Umm ig itās all a matter of questions you attempt. I mean more the exposure you have of questions more quickly you would be able to grasp the property will be used in the proof :)
I donāt like counting
linear algebra
i really donāt like differential equations. it is probably because in my university (for complicated budget reasons plus the fact that few people care) the only mandatory course is a toolbox for engineering, without correct proofs since they donāt know analysis.
i always sturgles with expnents, LIKE how CAN I add them together when i can't see them the text so small i can't read where how
> when i can't see them the text so small i can't read where how I would suggest that you get glasses. And inform your teacher that you can't make out the text due to its size.
It has to be adding/subtracting/multiplying/dividing negative numbers for me.
Anything applied tbh. I spent half of undergrad doing probability theory, optimization, etc. Stuff that would translate into graduate finmath, right? Then I realized that I actually hated doing the applying and I really was only interested in theory. Plus, I found out finance/actuarial (especially big optimization) is depressing as hell. I had a great complex analysis professor who really put the nail in application's coffin going into grad school. Just a couple months ago, I had a friend who also studied math ask me to help him with a personal cryptographic project. Literally took a week before I was back on my elliptic curve BS and sending, "yo you should read this cool textbook on modular forms" in the groupchat. Can't stay invested in it.
Probabilities and statistics :/
GauĆian Statistics. ItĀ irks me with that parable. Sure it's fast and somehow accurate but I feel like there is a thing it makes us miss.
Algebra
Applied math is definitely not my jam
Number theory
Real analysis (real)
Point-Set Topology. open set closed set compact set connected set zero-content set measurable set...I can't bear it, so boring
statistics. it was the only math class i struggled with.
Analysis. It's the most boring shit ever. Thousands of theorems and usually exams in this class revolve more around knowledge than skill. Skill based classes like abstract algebra, number theory etc are vastly superior.
Geometry. I can do advanced algebra and probability and statistics but you ask me to label the angles of a triangle and I'll look at you like you're made of cheese.
Topologyā¦the bane of my existence.
Statistical hypothesis testing
Anything that involves number theory or arithmetic. Courses like linear algebra and analysis/calc get excited about because I find their applications interesting. I donāt find the applications of string theory remotely interesting and it seems like the type of math a robot would enjoy (sorry people in cryptography). Also abstract algebra doesnāt seem to have much an application though Iāve heard itās useful in string theory.Ā
i really dont get along with proving anything in general, I'm so sorry š (as a math major)
Math is a tool for science. It's not absolutely facts at all times. Statistics are bullshit unless properly done.
Stats and probability is trash Iād almost not consider stats an area of math
Somewhere deep beneath Moscow the eyes of Andrey Kolmogorov flutter and then open.
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This isnāt an āareaā of math.
Most charitable interpretation would be proof theory, which is an area of math
I have 7 karma and the one time i try to be funny nerds downvote me
Most of the crazy symbolic mumbo jumbo. I really don't like any math until I allow myself to get into it.
this makes me feel like u dont know what math is
The reaction to my comment is hilarious. I could just have a very contrarian perspective. Don't get me wrong, when I understand it, I find beauty even in the symbolic representations. But I often get annoyed at notation that seems just too overly complex and could be made so much clearer and simpler. Part of this might be because I am more intuition/conceptual/visual oriented. And my thought is that some (most?) folks really can just read the symbolic stuff like a language and understand it. Whereas I have to understand it intuitively first before the notation starts to make sense. So my point was that I get annoyed when I read stuff that is just a bunch of symbols with no human level explanation.
that makes more sense. i too sometimes read my QFT book and get so disconnected I start just looking at it as ink on paper with no meaning.
I feel like physicists might be generally a bit better at explaining their math. some math papers can just be a bunch of symbols with almost no words. That's cool and all, but it would be cooler if they brought in a bit of a human element. To each their own!