/sci/ - Science & Math

I don't know about the most enjoyable (I like it, though), but Linear Algebra and Tensor Calculus is very useful if you are into Physics. I learn't that in school, and it was massively helpful for extra study.

>>4141888
Trig is the best field of math

>>4141907
BORING
try topology. shits cool. algebraic topology in particular
ooooh or set theory

>>4141907
What's tensor calculus? I've already taken cal 1 & 2, multivariable/vector cal, and DE, and am signed up for linear algebra and applied stats, but I've got 2-4 more math classes in my future, dunno what to take. They all sound pretty exciting.

>>4141915
How about knot theory? That's related to topology right? My favorite math professor is all about some topology and knot theory.

>>4141913
Trig? Trig is easy to master, and boring.

>>4141917
Tensor calculus can be described as transforming between field.

For instance, if a function is in the space X, with n coordinates, it can be transformed into space Y, Z or a spherical co-ordinate system, etc.

This is really useful for general relativity.

>>4141929
Oh I think we just touched on that in vector cal slightly. Sounds like the sort of stuff I like.

>>4141922
yeah knot theory seems fun. I never got deep into it. but it seems kind of like graph theory, fun and not too hard

>>4141932
Take a look at the transformation rules; if you've done multi-variable calculus and Linear Algebra, and Curvilinear co-ordinates (very important), it's definitely worth studying.

Anyone know about differential geometry? Is that any good?
Maybe we should have a poll, determine once and for all /sci/'s mathematics tier list lol.

I really enjoyed Partial Differentials, if that is available.

>>4141948 Anyone know about differential geometry? Is that any good?
That's where my first real "wtf math" moment came from. And the second.

>>4141947
I've done all of that except linear, which will be remedied in the spring. I'll definitely look into that, hopefully my school has a course on it. It would go pretty well with a modern physics class I reckon? Speaking of that, I wish I could CLEP all my dumbass gen eds and take more interesting classes. But no, I have to take art appreciation and social geography and other such bullshit.

Had to learn differential geometry for general relativity class - i mean, GR is pretty cool and all, but the differential geometry part of it is... bleh

>>4141954
What was your 'wtf math' moment about?

>>4141976
I doubt you've been doing real differential geometry in a GR lecture, because it's simply impossible to cover both the mathematical side and then applying it to GR - and the "working knowledge" of differential geometry is ugly as hell.

>>4141977
The first one was Stokes, the second one doesn't have a specific formula, it's more like Lie groups, principal bundles and all that combined. (Although I'm far from understanding those.)

>>4141977
inb4 Stokes

>>4141981
Yea, it was really half-assed differential geometry XD
Actually, that was probably the problem with it - our prof skimmed through it so fast b.c he wanted to get to the physics that 95% of the class was completely confused (or at least, until everybody read the course notes to do the first problem set...)

>>4141984
Oh, Lie, damn you and your maths...

>>4141985
Getting to "whatis a Christoffel symbol" is usually taught in the second half of math classes on differential geometry at the very earliest, at least from what I've learned. (Makes sense if you ask me.)

>>4141986
YOU DICK

>>4141985
I would probably agree with you lol... A lot of us thought that it would've made sense to cover the material over two semesters rather than one... But the prof wanted to get onto really advanced GR topics the second sem, so yea.

>>4142002
I always tend to think that the interesting things about GR are already contained in a room. Here "really advanced topics" probably refer to some black hole metrics, cosmology, redshift, etc., but I'm far more interested in the differential geometry stuff and the axioms/principles.

I'm not really sure if you guys are counting topics from differential topology into differential geometry. If so, you're under a grave misapprehension for thinking that working diffgeo is ugly. I not however, I agree in terms of basic stuff. At the beginning it's usually a lot of manual checks in local coordinates which do suck. But later on work solely intrinsically you tend to work in a more "algebraic" manner, not that you're really using algebra though.

My vote for most interesting undergrad area goes to topology. Just because I work in symplectic topology and because of its many connections to areas like algebra, category theory, mathematical logic, probability theory and number theory.
The only thing that has similar properties in terms of connections might be measure theory, though it does stay in the rather analytic areas.

>>4142031
So... What is topology all about?

>>4142037
open sets.

>>4142031
what is the connection between symplectic topology and propability theory???

>>4142037
Essentially geometry without having to mess with explicit "numbers", if you like. For me it's simply the most interesting aspects of geometric problems. A bit more formally it's a kind of global geometry.

As an undergrad faggot who's barely taken babby calc, I find combinatorics and group theory to be very fascinating, but I'm probably chump change compared to Josef and the other math high-rollers in here

>>4142043
>>what is the connection between symplectic topology and propability theory???
I was referring to the connections of topology and probability theory. I personally am not aware of any notable connections between symplectic topology and probability theory.

>>4142031
measure theory is gritty. it's like topology but no fun

>>4142048
I bet you'll love the statement and proof of the Nielsen-Schreier theorem. Mathematical beauty at its best.

>>4142042
That sounds awesome, I hate dealing with messy numbers. Symbolic and abstract is the way to go.

>>4142061
Ha, I agree wholeheartedly!

Imo, combinatorics and set theory.

>>4142052
tell a bit more about symplectic topology. sounds quite specific to me, what are you working on and btw. what is the starting point in symplectic topology - I can't really think away the differential geometric structure.
Or let's formulate it this way: In which way does "symplectic" specify the topology, compared to an general abstract topology theory. What do you have to work, which isn't usually there.

>>4142048 Josef and the other math high-rollers in here
I'm not a math high roller, all I have is disconnected half-knowledge.

>>4142072
Who are /sci/'s resident mathematicians then?
I think I've seen more sober and insightful math posts from you than anyone else, except one guy who's always remained anonymous and another guy who hasn't been around in a long time

>>4142064
Yeah, basically, you only keep concepts like "this is between this and that", "this is close to that", "the border of this object is of this or that kind", etc, and most of the time, if you can draw the objects you're working on, roughly straight lines and whatever looks like a potato will do simply because the exact shape doesn't really matter.

>>4142072
>I have is disconnected half-knowledge.

I have connected half-knowledge

but it's not path-connected.

>>4142078
I don't know any mathematician tripfags, but I've gotten some pretty good answers from anons here.

>>4142072
You were involved in about half the interesting math talks I've had on /sci/, though.

>>4142081
Dude, I didn't make the connection joke after >>4142043, why did you do that!!

>>4142088
I'm a bad person I'm sorry.

>>4142100
As such, what do you fell is most important for an undergrad to learn?

>>4142088
Come on, let's be more open-minded with these puns. Or countable union of opens, at least.

I'm an Applied Math grad student, but I'm not a huge enough faggot to need a tripcode.

>>4142073
Manifolds equipped with a closed non-degenerate 2-form. It's more differential geometry than you might think. In the beginning one first realizes some basic implications of the existence of such a 2-form e.g. that a symplectic manifold always has even dimension. I currently work in an intersection of symplectic topology, algebraic topology and partial differential relations. More precisely I am considering the use of pseudo-holomorphic curves in cases where Gromov's homotopy principle fails (which is a very powerful geometric method for solving PDR's). These pseudo-holomorphic curves turned out to be extremely important when interacting with symplectic forms leading quickly to things like the Gromov-Witten invariants which should be a known buzzword among the string theory buzzword-people. Then there's also Floer homology which can be constructed using moduli spaces of pseudo-holomorphic curves; Floer homology itself has been used to study fixpoints of Hamiltonian flows and symplectic topology has its roots in the considerations of Hamiltonian systems. I hope this rather brief and disconnected overview will shed some light onto all the connections.

>>4142079
mhm, well what I don't really understand is how you reformulate the whole 2-form business? I mean if you don't use charts etc. what does it mean that it's a set with a symplectic strucure? Maybe I'm missing something, but I just assumed you're throwing something away what symplectic manifolds in diff geo usually have. or don't you? Maybe it just means that you just specify on the topological structures, even if the other things are still there - is it like that?

>>4142103
we topologize for our bad sense of humour

>>4142116
Yeah, we know we are being dense, sorry about that.

>>4142106
It depends on what your goals are. Typically someone with a 4 year degree in math should have had exposure to multivariable calculus, modern algebra, real and complex analysis, topology, ODEs, proof based probability theory, graph theory, and a few programming courses (any decent mathematician should be a good programmer). In addition they should have at least a couple graduate level courses, if only to see what is beyond basic undergrad stuff.

>>4142114
are you a post-doc?

Well I'm not a topologian, but as a statistician, I find your jokes to be very mean.

>>4142128
fuck programming. I regret learning programming

>>4142115
Read >>4142114. Additionally I need to say that the reason it's called symplectic topology is because just as with the dimension of symplectic manifolds it turns out that there's not really much interesting local geometry to be done on those. Rather all the results are of a more global nature and thus it's called symplectic toplogy and rarely only symplectic geometry. Or at least that's how it is where I'm from.

>>4142114
okay, yeah that gives me some sort of picture. When I browsed wiki for symplectic topology I discovered the semi-holomorphic curves thing and spottet "Witten" too. But I mean I'm never surprised anymore to see that name in anything related to the last 30 years, especially classifications of semi-manifoldian things. In any case, I have to say that I can't quite imagine what these curves are. at least without reading about them. Strangely enough, I usually avoid complex function theory.

I'm asking these things because I recentry worked with symplectic structures myself, more on the physical side though. I think Josef is interested in these kind of things as well.

>>4142128
I never learnt about graph theory. ;_;

>>4142137
But that's the markov all the great private jokes.

>>4142131
Not yet, I'm currently working on my Ph.D.

>>4142139
kk, so it's pretty much called topology to emphasis non-local effects (because of the Darboux theorem, I'd guess)

>>4142128
I'm not a glorious math major, merely a lowly math minor geologist peasant lol. So what constitutes modern algebra, besides basic highschool algebra and linear algebra?

(I mean emphasise)

>>4142142
It's also somehow less complex than one might think, or at least that's how I felt about it. I too used to shun anything analytic or holomorphic but somehow what I'm currently doing feels way more algebraic than it probably should. Also these infamous pseudo-holomorphic curves aren't really difficult to define or understand (at the very basic level at least) once you know about Riemann surfaces and some basic function theory. What's really difficult about them are the many seemingly unrelated connections to so many other things, doing any research with them is somehow really intimidating, but maybe that's just my supervising professor.

>>studying symplectic topology
>>studying fish-bone topology

>>4142165
lol

>>4142158
Precisely.

>>4142168
post arXiv plox

>>4142171
>>4142165

I don't get it....

>>4142174
I haven't published anything yet.

>>4142159
I use the term "modern algebra" when it is more commonly called abstract algebra. It is the study of algebraic structures, such as groups, rings, fields, etc. There are some really neat applications for it too, such as cryptography. Here is a free book!

http://abstract.ups.edu/download.html

>>4142179
Ah, knew there had to be more to it.
And on an unrelated note... I wonder how difficult it would be to make a LaTeX -> Mobi/AZW compiler....

>>4142178
Na, you mentioned the prof, that's what I ment.