Math General

It's that time of the month again:
>what are you researching?
>what are you studying?
>any good problems?
>book recommendations?
>cool theorems?

To get the ball rolling I've been doing some categorical logic, namely reading about the Curry-Howard-Lambek correspondence, focusing on Lambek's side of work. Essentially we have an isomorphism of a certain sort of category (cartesian-closed categories), a limited version of intuitionist logic, and typed lambda calculus. Essentially you get a "proofs as types" relation from the Curry-Howard, but adding Lambek's work you get categories also!

Also been reviewing Milnor's topology from a diff. viewpoint before I head back into differential topology.

Other urls found in this thread:

en.wikipedia.org/wiki/Zeta_function_universality
ncbi.nlm.nih.gov/pmc/articles/PMC330698/pdf/nar00192-0217.pdf
en.wikipedia.org/wiki/Q-construction),
twitter.com/NSFWRedditVideo

This correspondence fascinates the shit out of me. Luckily I'm specializing in software verification, which has to do with types, logic etc.

>any good problems?
the collatz conjecture is so fun to think about, yet not an inch of real progress can be made on any question related to it. the dynamical behaviour of maps of such simple maps on the natural numbers just turn out to be so strangely impenetrable

>book recommendations?
Kazimierz Kuratowski - Half Century of Polish Mathematics: Remembrances and Reflections

>cool theorems?
en.wikipedia.org/wiki/Zeta_function_universality

>>what are you researching?

The same as always. I am researching whether or not I am actually a genius and not just "above average" so that I can know if I should stay in mathematics for a career and do great things, or if I should settle down and then get a masters in mathematical finance like the low IQ cuck I am.

Why is this so hard? Why is there no definitive genius test?

I do well in calculus, but then that could just be that calculus is my intelligence limit

I do well in analysis, but then it could just be that analysis is my intelligence limit

Fuck it all. But this semester I am taking number theory. I've heard that this field is brainlet-proof. To be good at number theory you HAVE to be a genius, right?

Right?

This time this will finally tell me if I am worth anything, right?

Does anyone know someone who was good at number theory but then later showed their brainletness?

sounds like you're an undergrad who hasn't even attempted research yet

have you never applied for a summer research position with a professor?

I have not done research because I am
RE
TAR
DED

Seriously. What are you supposed to research when you are just a sophomore? All I can research right now is if I am smart enough to do research in the future.

i attempted a research project doing some time series analysis in the summer after my second year

it went terribly but it was still a worthwhile experience

does your school not have any list of professors with projects that they'll fund undergrads to spend a summer on?

a) you are going about the decision the wrong way.
b) most schools offer undergraduate research programs, usually over the summer or sometimes during the semester.

I would suggest pursuing an undergraduate degree in mathematics, using your electives to do some finance stuff. If you do not choose to pursue pure mathematics at the end of your degree, you can always just apply for a masters in financial mathematics and get your secondary dream.

Just went over the projective model structure on chain complexes with my advisor yesterday, he gave me a lot of good insight on the small object argument. finishing off this intro homotopy theory paper then moving onto more simplicial things in the summer.

Who else /homotopy theory/ here?

>it went terribly but it was still a worthwhile experience

You could have spent that time reading books so maybe it wasn't that worthwhile.

>does your school not have any list of professors with projects that they'll fund undergrads to spend a summer on?

I don't know a list, but I do know that professors do research but they mostly act as advisors for senior students and then master students.

I've never heard of them working with someone who was below senior level and for good reason. Last year I got to talk with a senior student who was going to defend his senior thesis that day and the difference is huge. How am I supposed to even compete? He knows galois theory. I barely know what the definition of a field is.

I am already studying specifically mathematics. I don't care about studying finance stuff in undergrad. I want to use undergrad to find out if my IQ is high enough to be a mathematician. If I fail by the end I have masters degrees in this very university that will serve me as backup.

well goodluck with that, m8

Done a little HoTT and also some homotopy theory in algebraic but not too much beyond the usual texts for algebraic topology which focus much more on homology theory.

Got any good homotopy theory books?

Hatchers book does basic higher homotopy groups and other early results pretty well. It's a bit harder to find good sources for the categorical approach to homotopy theory though, I'm reading "homotopy theories" by Dwyer and spalinksi, and so far it's been really helpful, but it will assume a bit of knowledge about categories and homological algebra (ext, tor, projective modules, etc...)

>You could have spent that time reading books so maybe it wasn't that worthwhile.
there's a huge difference between reading a book and attempting to genuinely solve a research problem

i did another project the next summer that went considerably better

>I've never heard of them working with someone who was below senior level and for good reason. Last year I got to talk with a senior student who was going to defend his senior thesis that day and the difference is huge. How am I supposed to even compete? He knows galois theory. I barely know what the definition of a field is.
if you're american they have some sort of system for getting undergraduates into research (REUs)

it's hard to compete but you'll never succeed if you don't even try, i felt like a fool emailing every single professor in the department but the one who replied and gave me a chance made it worth it

Out of curiosity, what was the research like?

the first project was analyzing trends of the proton density in solar wind (learned some Matlab along the way, did a ton of organizing data)

the second project was studying the arithmetic properties of the zeros of certain functions related to elliptic curves (learned some Sage along the way)

not sure what specifics you're interested in but i was pretty out of my depths for both projects, it's more about getting the experience at that point rather than expecting publications

Not the guy you asked, but I have a pretty great project where I got to do original research.

It was not trivial research, but it wasn't exactly at the forefront. The professor already established a result which guaranteed another one, and I and another student formalized this secondary result. It got published in an undergraduate journal and was my first research experience.

Another research project I did was less original, but it was proving a well-known theorem with new machinery. This one was also enjoyable for the paper writing aspect. Did not publish this one.

Protip: you don't have to be a genius to be a mathematician.

But, you have to do mathematics in order to develop genius.

>what are you researching?

Following my nose haphazardly. Reading list in pic attached. Not a student or smart enough to work in business so this is all just a hobby.

I'm also typesetting a small book on sets and proofs to learn LaTeX. Which... I really gotta say this.... Having just started using it, LaTeX is just absolutely fucking awesome. I'm sure this is just a honeymoon period, but after years of drawing equations by hand, and getting awful cramps and having to take breaks in between long passages of tedious writing, being able to type a couple lines of code (and copy and paste already written code) and then have any equation I want appear in perfect crisp detail at almost any size has kept me in an almost permanent state of joy and relief for the past couple days. It's just... awesome and I wish I had started using it sooner. ...That's all I have to say.

>analyzing trends of the proton density in solar wind
>studying the arithmetic properties of the zeros of certain functions related to elliptic curves

Do you have any background in physics? Would that have helped at all with what you were asked to do? How much direction were you given? Did you like who you were working with? Did you know what you were going to be doing before you started? Which one was more challenging or straightforward in what you needed to do, or is it difficult to compare because they presented different challenges? I know that's probably a lot of questions so no pressure to answer any, I'm just very curious about your experience.

>Do you have any background in physics?
at that point i had taken a first year physics class and a second year E&M class

>Would that have helped at all with what you were asked to do?
not really, but i think the ideal person for the project would be someone with lots of knowledge about the sun (solar cycles...) and data analysis techniques, some knowledge of statistical methods

>How much direction were you given?
1st project not nearly enough, or I just wasn't asking the right questions (more likely)
2nd project it was always clear what needed to be done, it was just a matter of knowing enough mathematical theory to get through it

>Did you like who you were working with?
in both cases yes (both were genuine experts, one old, one young) so the limitations on both projects certainly came from my own lack of knowledge/experience

>Did you know what you were going to be doing before you started?
1st project was just listed as 'project in data analysis' and I got to pick between a couple different topics (I think one of the other one was ECG data or something like that)

I sort of accidentally got involved with the 2nd project after hearing someone give a lecture about it in a seminar and asking for references after, later he came and just asked me if I wanted to work on it with him over the summer

>Which one was more challenging or straightforward in what you needed to do, or is it difficult to compare because they presented different challenges?
The data analysis was far less straightforward (the nature of applied mathematics I guess, wide variety of tools) but the sort of 'ceiling' for the elliptic curves project was higher (nature of number theory...), the material you need to progress further ramps up in terms of prerequisites and depth quickly. There was a much more concrete goal for the elliptic curves than just 'play around with big data set'

>what are you researching?
I've been developing a method to shrink categories. It allows one to use the properties of small categories "locally" in categories that need not be smal themselves, such as the Mitchell embedding theorem. No idea if this has already been done by someone, but I'm going to see a lecturer of mine next tuesday and see what she has to say about this.

>what are you studying?
Some algebraic topology and a bit of differential equations, and pointless topology on my own.

>any good problems?
Using the category of Hausdorff spaces and comtinuous maps, show that an arrow can be both an epimorphism and a monomorphism without being an isomorphism. (Hint: the inclusion [math]\mathbb{Q} \to \mathbb{R}[/math].

>I've been developing a method to shrink categories. It allows one to use the properties of small categories "locally" in categories that need not be smal themselves
could you elaborate on this a bit? sounds interesting

Shit, I left out the most important thing. Until I've heard a pro opinion on this thing, and whether I should publish it or not, I'm gonna be a bit vague, but the basic idea is that I have a category C, and a certain type of subcategory C' of C, and these are ALWAYS equivalent. The "lesser" categories, as I have called them, are such that this C' is small. Clearly, small categories are lesser, but I have constructed several large lesser categories, too. I'm not completely sure if one can have a Grothendieck topology for any category, but this would allow one to take a (suitable) large category, get an equivalent small category, and topologize that one.

Sorry for being handwavy. This might be new, and I've seen the pic in which someone has posted on Veeky Forums how some quaternionic thing resembles the Lorenz attractor, and then this one guy has replied something like "Thanks for the free paper to publish".

Possible brainlet question:

How does someone "do research"? How much math is needed to "do research"?

I am taking Algebra and Real Analysis next academic year, with a proofs class this summer. Did well in my lower division coursework which I finished up.

How do you even find research problems to work on? I'm just a pleb undergrad right now

Why is every single fucker here obsessed with category theory

there are no analysts, no geometers, no topologists, there's never anything but category spergs on this board

wit ct you do geometry and topology, not that 20 yo undergrad can know this

>I'm not completely sure if one can have a Grothendieck topology for any category
Canonical topology.

Yes, that is a topology that would be easily used anywhere, but one source says categories must be small, and the other that the can be whatever. For example, Johnstone defines (pre)sheaves for small categories in his book on topos theory.

I'm more upset by the fact that all the category theorists are animufags.

I am OP and I am a geometer.

There is nothing wrong with appreciating all parts of mathematics.

>>what are you researching?
Nothing at the moment but I'm desperate to get into symbolic dynamics, specially after reading this paper:
ncbi.nlm.nih.gov/pmc/articles/PMC330698/pdf/nar00192-0217.pdf

The thing is that it seems like there is not much research going in right now ):

I was good in number theory in undergrad.
Now I'm the brainlet of my algebraic geometry class in grad school.

>mfw reading about quantum link invariants
>mfw the Verlinde formula for conformal field theories could be obtained as an operator link invariant on a ribbon category
>mfw there is a deep connection between conformal field theory and links on a 3-manifold despte them having seemingly no relation to each other
>mfw category theory does it again
I haven't been able to stop cumming since two days ago when I read about this.

tfw you cant know whether you are a brainlet until it's too late and you have invested years and years

I'm not even a mathematician rofl

"brainlet" is the biggest meme ever and isn't actually a thing.

>Protip: you don't have to be a genius to be a mathematician.
But I don't want to be a mediocre mathematician. I want to get a fields medal.

I wish I could know if I was fields medal material before it was too late.

Fugg. But maybe that has to do with the change of subject.

Are you still good at number theory? As in, grad level number theory? Algebraic Number Theory? Analytic Number Theory? Are you good at those?

I don't particularly care about geometry and even though I got an A in my last "geometry" class, I know that I am bad at it compared to algebra and analysis, so I don't care about it.

I just care about number theory my maaan.

Not sure if this is the right place to post
I'm an electrical engineering student and I am "bad at math".
I want to learn calculus, differential equations and Laplace z and Fourier transforms.
Is there a diagnostic test available online to highlight my gaps in knowledge?

>I wish I could know if I was fields medal material before it was too late.
>Fugg.
Yeah I can tell you right now, you are not Fields medal material. Enjoy your degree in software engineering.

>Enjoy your degree in software engineering.
topkek I am already in pure mathematics motherfucker.

Also, it is just a meme holy shit this is Veeky Forums.

any canadians in? Advice for getting an NSERC?

Be aboriginal, female, disabled, and top grades in your class.

If you are white male, just get A+ in everything and that should be decent enough, assuming you are good at writing the essays.

which NSERC brainlet? i got one for both USRA and CGSM if you need tips

how do you guys find managing studying math and having a social life? i never see my flatmates and they always go out without me, it's kinda depressing but at least they're around so im not too lonely.

It's the journey that counts.

Make friends who do math.

>I would suggest pursuing an undergraduate degree in mathematics, using your electives to do some finance stuff. If you do not choose to pursue pure mathematics at the end of your degree, you can always just apply for a masters in financial mathematics and get your secondary dream.

I would really counsel against this advice.

Reading OP's statements I think the best place for him right now is outside of the mathematics ghetto. As a mathematician, he would be better served joining a research group in a field working on a real problem, biology, molecular bio, biochem, physics- nearly every project needs some mathematics.

One, it will broaden your horizons, two, you will exercise some respect for other branches of the sciences. Three, you will realize there is not nearly as much opportunity in mathematics as there is in real research fields, and four, you will be able to exercise and improve your socialization skills.

The messiness of real world data integration should be more than worth it form an experience standpoint alone.

I can't emphasize enough how important an undergrad research experience could be, especially if you are able to participate in publishing.

So take a shower, get a haircut, wear clean clothes, look people in the eyes, knock on doors and talk to profs and express some fucking human interest in life.

Why would you counsel him to get outside math when math is what he wants to confirm talent in?

Nice: any space can be made sober by taking the spectrum of its locale.

>what are you reading ?
Topology book and some analysis. Undergrad level.
>what are you studying ?
Materials engineering.
>any good problems ?
Not that I'm aware off.
>book recommendations ?
Enclyclopedia of Mathematics seems pretty dope, even though I can't understand most of it.
>cool theorems ?
Bolzano-Weirstrass, either in the form that any bounded sequence (in R) has at least one convergent subsequence, or in the form that in any closed bounded interval [a, b] in R, any infinite subset X of that interval has at least one accumulation point in [a, b].

I don't know why, but I feel the two mean the exact same thing. Might try to do some work on it later that night.

Oh, and I'm doing a small presentation of the Riemann zeta function to my comrades engineers Monday. Don't know how far into the details I should go, they usually don't like maths :/

Of course the namefag says something retarded.
There are geniuses who are not good at math.

>I want to get a fields medal.
I want that to, but you gotta be realistic.

My plan right now is to finish my engineering degree and my master in applied physics, find a nice work in a laboratory of a university, get to know the professors, work there and study pure maths.

what's the highest level math you'll take during eng?

The highest I have taken. I don't do math anymore at this level, except on my own.
I've done a fair amount of calculus (culminating to Stokes theorem and other Green-Riemann stuff about surface integrals and line integrals), linear algebra (including a little bit with differential equations), and an introduction to Analysis (series, sums, complex transforms, a little bit of functional analysis, a little bit of Fourier analysis.)

I lack training in formal topology, formal algebra (except the very basic stuff like bijections and maps), geometry (only ever touched that with physics classes), and, of course, "proving" stuff. Only the class in Analysis required us to prove stuff (namely, we had to prove Dirichlet theorem and the existence/convergeance of Fourier series in a project), the rest was just learning tools, combine them, apply them.

you're going to wait until you already have an applied physics job 10 years from now to start? you can't be serious. if you want to learn math there's no excuse not to start right now

I've started, reading topology books and other stuff on the side. I also did 2 courses in "applied" maths (Optimisation and Probabilities) where I got pretty good grades.

>you already have an applied physics job 10 years from now
Hopefully it'll be shorter. I'm graduating in 2-3 years, and the average period before finding a job is 2 months after graduation.

oh in that case that's ok. books alone won't take you very far if you don't talk to someone to get a "big picture" kind of insight, so try to talk to a professor soonish

I'm lucky on that front, dad is a math teacher with pretty good insights.

Holy shit please be joking

Why being smug ? He have a diploma more or less equivalent to a master in pure maths. He has the qualifications to teach in university if he wanted to.

And he can answer questions.

@8734353
Confirmed bait. Nobody can be this stupid lmao.
No (You) for you.

Illuminate me. I'm not referring to him as the ultimate reference, of course, and yes, I'll get in touch with other professors as well, but it's always nice to have somebody to help you figure out some stuff.

Care to elaborate?

Im the typical comp sci major in a calc class. Its going ok.

Revising high school math because had shitty high school teacher that explained everything way to complicated.
23 years old now and it's easier to follow khan & basic math book.

Take any topological space [math]X[/math], sober or not. You then have have the lattice [math]\Omega (X)[/math] of open subsets, giving you the frame of the space. Then, the locale [math]L(X)[/math] is the corresponding object in the category of locales, or the dual category of frames, and you now have a functor [math]L \colon \textbf{Top} \to \textbf{Loc}[/math]. For any locale, one can define a spectrum by considering its set of all completely prime filters, and equiping this with a suitable topology. Now, this gives rise to a functor [math]S \colon \textbf{Loc} \to \textbf{Top}[/math], and all the spectra themselves are sober spaces, so now the composite of these functors defines a sobrification [math]X \mapsto SL(X)[/math].

Ok. A lot of these functors aren't clearly defined and I don't see how the sobrification relates to the original space. Anyways what this looks like is a space structure if the functor[math]L:\mathscr{T}\rightarrow \mathscr{S}[/math] is covariant, and it seems to be compatible with disjoint unions and associative/commutative on the nose as well. If we can endow this [math]L[/math]-space structure with a cobordism theory [math](M,L)[/math] then we can define a topological quantum field theory [math](\mathfrak{T}_K,\tau_K)[/math] on it where [math]K[/math] is a ring. Depending on what [math]L[/math] preserves this TQFT can probably tell us something about how locales affect the structure of the modules.

Starting from the frame functor [math]\Omega[/math], we have, for any continuous map, [math]\Omega (f)(U) = f^{-1} U[/math] for all [math]U \in \Omega (X)[/math]. Since [math]\textbf{Loc} = \textbf{Frm}^{op}[/math], we set [math]L(X) = \Omega (X)[/math] and [math]L(f)=\Omega (f)_*[/math], where [math]\Omega (f)_*[/math] is the right Galois adjoint of the frame homomorphism [math]\Omega (f)[/math], namely [math]f \colon X \to Y \Rightarrow L(f)(U)=Y \setminus \overline{f[X \setminus U]}[/math] for all open sets [math]U[/math].

Then, the other functor. For any locale, say [math]M[/math] now that I was stupid enough to use L already LOL, define [math]S(M)=\{\text{all completely prime filters in } M\}, \{ \Sigma_a\ |\ a \in M\})[/math], where [math]\Sigma_a = \{ F\text{ a completely prime filter in } M\ |\ a \in F\}[/math]. That's the space. For the morphisms, set [math]S(f)(F)=(f^*)^{-1}F[/math], where [math]f^*[/math] is the left adjoint of [math]f[/math].

Does this clarify stuff? Both functors are covariant. No idea about those field theories, though.

I fucking hate set theory

>Does this clarify stuff?
No lmfao god forbid I actually find an application of Galois theory to physics.

How do you guys come up with research questions? Just familiarity with your field?

Friendly reminder that this is a math thread, not a fiction thread. We have a separate board for fiction discussions.

Feel free to continue your discussion there.

t. engineer

So do I sometimes. I don't like the fact that my glorious plans can be ruined by having a proper class where I'd like to see a set.

I was just doing my master's thesis and got a bit derailed, and then eventually I had accidentally redone what Quillen did (en.wikipedia.org/wiki/Q-construction), and then went the other way around and will hopefully get to publish what I have or atleast put it on Arxiv.

Fug

Bully. Stop or be stopped.

>I don't like the fact that my glorious plans can be ruined by having a proper class where I'd like to see a set.
What does it even mean ? You prove something on set(s) but if I give the set(s) a certain property it all falls apart ?

Proving stuff, and using old results. Some require the "smallness" of sets, and then I need sets and still encounter classes, and this forces me to set some restrictions. It sucks when you lose generality like that. Nevertheless, I have been able to (give a method to) generalize several results that would be size queens.

No need to use a gun on me, you psycho.
And I'm here, taking an hour to see that
(x, y) => (x + y, xy) is surjective but not injective.

(and the answer was so easy I cried in shame after)

Before I go ahead and take a nap I'd just like to say that making category theory my bitch is an incredible turn on.

I'm just beating the shit out of those classes. And don't worry. Sometimes the most concrete of problems are the hardest.

Physicists are always so physical.

>A lot of these functors aren't clearly defined
yes they are
in classical math, a space is sober iff the space is T__0 + the irreducible closed subsets are the closures of a singleton

a topological space is sober iff the counit is an isomorphism ie the set of the space is the spectrum

the complete prime filters are not good when you deal with locales
if you care about locales, the spectrum of X is LOC(1, X) with topology
{ {f^* such that f^*(u) = 1} for any open u}

also, on the logical side of the propositional theory, a topological space is sober when there is an isomorphism between the models of the theory and the points, instead of only having that points are models.

I only started reading about these things today, so I'm just a beginner. The sobrification thing was pretty cool, though, and I decided to share it. What do you mean by LOC(1, X)?

>LOC(1, X)?
the arrows of the category Loc of locales and locale arrows, from the initial locale to X

it is natural to build this, rather than the classical way of Y∖closure[f[X∖U]]

so you go
Top->Loc
X->ΩX
f->f^*

Loc->TOp
L -> [Loc ( 1 ,X), { {f^* such that f^*(u) = 1} for any open u} ]
g-> composition

>f->f^*
must be f-> (f^*) formally reversed

...

Right. That makes sense, as it makes points generalized elements. Thanks for sharing this. I'll let it digest for a while so that it becomes more intuitive.

I am woke and ready to go.

Welcome back.

how many layers of abstraction are you on right now

If you want to do compelling math use a computer with a massive gay men GPU and generate some pretty fractal

So many I don't even know what I'm talking about my dude

I just circlejerk with an anime avatar

The fuck you on about? I know what a sober space is you fuckwit.
Put down the gun boy.

like,, maybe 5, or 6 right now. my dude

You first.

>god forbid I actually find an application of Galois theory to physics.

It already has applications via Langlands

How many of you can/can't program?

What would you program if you were really good at programming?

I am using category theory to study knot/link theory to study gauge theory to study AdS/CFT to study strongly correlated fermions. So I guess 4.
Are you resisting arrest?

I can't, but I would probably program some program to make hyperbolic tesselations.

A.C.A.B.

I know Mathematica, Maple, C++, Python, FORTRAN, and I'm currently learning Golang. I've made SCFT, Lanczos-type calculations, DMRG, QMC simulations, odeint, and dozens of other root finding algorithms.
You're being very naughty right now.

I'm reasonably competent with a few languages. Probably could not hack it in a professional programming position but I can function on small-scale projects.

Unless you go to a place that lets you jack off in 100% theory all day (most don't) and you are only interested in doing that (you shouldn't be) you'll end up learning at least some R or Matlab or something and some C out of necessity.

I'd like to learn how trading algorithms work but my knowledge of finance is weak and I'm too lazy to improve it right now.