@Daminark @user2860452 but do you think it's beneficial to pick the most influential research category in mathematics or to choose something that speaks to you personally? I believe that if I was born around newtons time I would not for instance focus on analysis, however "sexy" that area of research was at the time.
Like, fields will look really nifty at first glance or even at the beginning of the first class you take on it
But as you go deeper things change
I was gung ho about calculus and analysis when I first got into it, and I now almost dislike calculus and mostly like to think about analysis through topology
So give it some time, at this point there's likely a natural progression of classes that you'll have to do anyway, like after getting through calculus you'll do multivariable calculus/analysis and also abstract algebra (plus linear algebra in the meantime)
(If you aren't currently doing proof-based math, you'll probably have an "intro to proofs" class, absolutely take that)
And then at that stage you'll have a nice overview of a few fields and can start to zoom in a little bit to that which interests you. But try not to gain too many prejudices yet, things can change for better or worse
@Dodsy I think algebra and topology are my preferences right now
Like, the only classes I've taken so far outright are first year calculus, then analysis (including some measure theory and functional), and differential topology
And I've picked up some group theory on my own. This fall I'm doing group theory, bio (gotta get that requirement done eventually), a reading course on algebraic topology of finite spaces, and likely model theory
@user2860452 so you're doing still single variable calculus
Quality book, but it tries too hard to avoid determinants so that it makes the exposition on the characteristic polynomial awkward. Also, it's afraid of algebra, which I think is kind of a bad thing
I'm not particularly fond of it myself, but it's good for getting a feel about how to do things using linear maps more and matrices less
And in general I'm quite fond of coordinate-free methods when applicable, so I'll give it points on that note
Now they're using the book that they started supplementing second quarter of algebra with after Hoffman and Kunze went out of print: Charles Curtis' Linear Algebra: An Introductory Approach
130s are if you're shaky on precalc, 150s are if you're solid in precalc, and 160s is honors calc (you prob wanna have some calc background going in) and is proof-based
For the general audience that needs to satisfy core but doesn't want calculus, we have two "studies in math" classes, one on number theory and one on geometry/symmetry
20300-20400-20500, the standard analysis class. Does preliminaries on metric spaces, convergence, and linear algebra in 203, followed by differentiation and optimization in 204, and integration and vector calc in 205
But yeah, if you don't do honors analysis you have to do linear algebra before 20400/20410. Then there's the algebra sequence, regular and honors, and after that a bunch of electives
@Daminark your school sounds awesome + is considered one of the best schools for math in the entire world. So you're probably getting a first class education :D
so farewell. Probably might come back, but I probably won't. It's better that way for everyone. Also, good luck on that spam flagger issue if it's still ongoing. Ugh.
So this past summer I was in this bootcamp thingy, and complex analysis was one of the topics, but it was set up by a prof who's very old-fashioned and wanted us to get familiar with that part of math, including computing series directly and whatnot
Which didn't click with my style too much, and I don't think I put as much in as I should've, I got distracted by other stuff like algebraic topology and the like
So next quarter I'm gonna sit in on complex analysis by a prof who apparently goes a bit quickly and gets through good ground by the end of the quarter, including basics of Riemann surfaces
Tentatively, we're working out of a book that our prof wrote called "Finite Spaces and Larger Contexts"
Basically, you have a subject called algebraic topology
Which, at the beginning of your studies, is essentially that you can give certain algebraic objects to topological spaces which are invariant under the kinds of maps that you use to classify spaces
For example, similar to how isomorphism is what you use to determine if 2 groups are "the same", there are 3 ways that I know of in topology that you might wanna do this
One is homeomorphism, which is that you have an invertible map such that it and its inverse are continuous
One is homotopy equivalence, so two maps $f:X\to Y$ and $g:Y\to X$ such that $f\circ g$ is homotopic to the identity map on $Y$ and $g\circ f$ is homotopic to the identity map on $X$
(Homotopy is a technical condition, in a nutshell think of 2 curves, you can "continuously slide" one until it becomes another. In $\mathbb{R}^n$ you can always do this, but in other spaces, you can't. Imagine the two circles of the donut, you can't do it)
Then there's an even more technical condition called weak homotopy equivalence
But basically, you can put algebraic objects that are tied to these spaces
There are these things called homology and cohomology groups
And if two spaces are either homeomorphic, homotopy, or even (I think) weak homotopy equivalent, then they must have the same homotopy, homology, and cohomology groups
That's why these are often useful tools
Now the subject is not precisely that, you tend to focus on a certain class of spaces called CW complexes
Where CW essentially means "not utter shit"
And it eventually balloons a lot, gets into category theory, and whatnot (also btw simplicial complexes are more general than CW, I think, and are still studied)
The nifty thing is that finite topological spaces (and more generally, these things called Alexandroff $T_0$ spaces) actually capture a lot of information
In particular, once you go deep enough into algebraic topology, your criterion for equivalence really starts to zoom in around weak homotopy equivalence
Any finite simplicial complex (meaning, finitely many simplices, which are n-dimensional triangles) is weak homotopy equivalent to some finite topological space, and any simplicial complex is weak homotopy equivalent to some Alexandroff $T_0$ space
@Semiclassical so I tried looking for seminars and can't find anything :/ it's really ridiculous how hard it is to find stuff that is going on. I missed a talk today on some mathematical idea about plants or something. Then I went to the Putnam page and it's from last year!
even if they aren't involved themselves at this point, they can probably hand you off to someone who knows better
I think I'm spoiled as far as event calendars, though. the physics department web page here puts it front and center: physics.umn.edu/index.html
and there's a weekly email that we get which has all the events. (on that note, you should dig around and see if there's a similar mailing list for your department) @dodsy
I just logged in on research gate, and there was a researcher that had a name very close to mine so I tought someone used my name to publish something, I got really afraid
