Redpill me on topology, Veeky Forums. Point-set was kind of a drag, but I like the idea of studying very general and abstract spaces, manifolds, etc. and I don't want one class to turn me off it. Why do you like topology and where do you use it?
In particular, what's some cool stuff at the intersection between topology and generalized harmonic analysis? I am interested in studying L-functions by means of things like integral transforms (think Tauberian theory), and I want to know if there's a topological viewpoint in that direction.
I think topology is coolest from a homotopical perspective. Homotopy theory distills topology from the (ugly, in my opinion) pointset stuff to the abstract concept of connectivity and interactions between paths of different dimension. Things start behaving very, very nicely after you mod everything out by coherent homotopy.
First off, the homotopy hypothesis: infinity groupoids are models for homotopy types (spaces up to homotopy equivalence). An infinity groupoid has 0-cells (points), 1-cells connecting them (paths), 2-cells between these (homotopies), and you can compose all of these cells to get new ones (for example, you can whisker a 2-cell and a 1-cell to get a new 2-cell, which is "degenerate" along the 1-cell). The fundamental operations in homotopy are suspensions, deloopings, and the smash product (after passage to pointed homotopy types). There is a process of stabilization, producing things called spectra. Spectra can be thought of as higher algebraic things, though: they are infinity modules over the sphere spectrum, which has the integers for all of its positive homotopy groups. The sphere spectrum is thus the Eilenberg-MacLane spectrum for the integers, and the statement that spectra are modules over this thing is essentially equivalent to Brown representability: spectra and generalized cohomology theories can be weakly identified (the caveat being that cohomology cannot detect some maps, called phantom maps).
This algebraic approach gets cooler, though: there is a notion of "good" cohomology theories, called complex orientable cohomology theories. There is a universal one, called MU, which is a spectrum representing complex cobordism cohomology theory. This is called the Thom spcetrum, and its cohomology ring calssifies formal group laws. This ties chromatic homotopy theory to algebraic geometry in some nifty ways.
Then you have a ton of methods for modelling homotopy theory, many of which have combinatorial flavours. (continuing)
Christopher White
() My personal favorite combinatorial model for homotopy theory is the theory of symmetric simplicial sets, which are like standard simplicial sets without orderings on cells. These objects relate homotopy theory to linear algebra, because a symmetric simplicial set is to the field on one element as a finite-dimensional manifold is to the real numbers. From here one can find more ties back to algebraic geometry, differential geometry, and even knot theory.
Important algebraic facts can be derived using homotopy theory, such as the link between Frobenius' theorem and the Hopf fibrations.
Differential geometry can be described in terms of homotopy theory, a la synthetic differential geometry. Things such as de Rham's theorem are quite trivial to prove in this setting, as they use more simple constructions in geometric/classical homotopy theory and show that they are facts in all homotopy theories. We use modalities to relate certain theorems and to show that they are true in any infinity topos.
Homotopy theory is used in quantum field theory. Cobordism cohomology theories can be interpreted as placing limits on particle interactions; Witten et alii have used the theory of genera on cobordism cohomology rings to derive physical laws. Physicists also use vector bundles everywhere, and homotopy theory gives the tools for classifying such bundles. This leads to higher Chern-Weil theory and its relatives.
Homotopy Type Theory, which is the abstract language of any flavour of homotopy theory, has applications in computer science and logic, as well as philosophy. Voevodsky's univalence axiom has Leibniz's identity of indiscernibles as a corollary, for example. The homotopy theory of topological spaces is universal amongst homotopy theories.
Homotopy theory is used in number theory, where we want to find nice homotopical descriptions of higher stacks, which vastly generalize Grothendieck's scheme theory.
Just peruse the nLab, friend.
Jordan King
i wish i knew as much as you do about topology. how do you think topology got started? like what mathematician visualized all that shit and did the math behind all that?
Andrew Reyes
Thanks bro. What's the way "into" this stuff? My point-set class used Munkres; I liked that one so I'll probably use the second part to meet the algebraic stuff.
Are you a grad student/postdoc? Topology seems very popular here at Northwestern
Camden Flores
>Thread on Veeky Forums about logic, topology or type theory >Homotopy autists show up Why are the only people who share interesting knowledge on Veeky Forums unsufferable category theorists?
Samuel Davis
He's a very special case. He's also one of our local category theory memesters, so take his word with a grain of salt.
Dominic Rogers
>how do you think topology got started? like what mathematician visualized all that shit Perhaps from the notion that "a hole is a hole".
Camden Bailey
Euler basically founded the subject when he studied the Seven Bridges of Konigsburg problem. Topology as we know it today was mostly due to Poincare, Frechet and Hausdorff. Basically we wanted to understand the essential properties of metric spaces like Euclidean space and continuous functions between them, and it seems that our current definition of a topology is the "right" one for doing this. Topology is a very modern subject, most of the work has been done in the past 60 years.
Dominic Martinez
It's the closest thing math has to a Grand Unified Theory (tm), and people who care about mathematical foundations tend to be more insufferable than the average user.
Jaxson Brooks
>higher stacks, which vastly generalize Grothendieck's scheme theory.
Algebraic stacks generalize schemes. Stacks in general are just what you get by bringing a notion of descent into category theory, which is a notion that has been around longer than that of schemes.
Juan Carter
>Redpill
Julian Hernandez
>>>/reddit/
Alexander Davis
yeah point-set topology is pretty boring on its own. algebraic topology is pretty cool, especially doing computations with CW complexes. After you get the hang of it, it's almost like playing with legos.
I don't know much about L-functions, but you might be interested in the field of Geometric Analysis, which is broadly concerned with the study of topology of manifolds via solutions of "geometrically interesting" PDEs defined on them (kind of like how the Fourier basis functions [math] e^{in\theta}[/math] are eigenfunctions of the Laplacian on the circle). This leads to a bunch of high-brow stuff like Seiberg-Witten theory, Floer theory, etc., but the main idea is always to construct a topological invariant as the space of solutions to a PDE whose definition depends on some underlying geometric property of the manifold.
James Cox
>why do you like topology I like shapes, I like to play with in my head. How the abstract-looking definitions of concepts like connectedness and such correspond to what you would intuitively think of, that I find very nice too. >where do you use it It is applicable to a lot of things. Analysis is the obvious answer, but it can be used to determine stuff about rings, and also
Ethan Gutierrez
thank you for using an anime image
Carter Hernandez
bump
Adam Barnes
Hey Animenon!
Life is great. I recently celebrated my twenty-first birthday, and my research is going very well. I've been making some mad progress on tying F1 into linear algebra and homotopy theory; right now I am developing a general nerve-realization setup to show that homotopy theory is the "proper" notion of geometry locally modeled on F1 vector spaces, just as the theory of manifolds is locally modeled on real vector spaces. I'm hoping to get a better understanding of the situation over finite fields as well.
That localic homotopy theory sounds pretty interesting. I'm wondering if the adjunction relating Top to Locale is actually a Quillen adjunction relating these homotopy theories? Unfortunately I'm working a pretty long shift today, but I may look into that further tomorrow. Thanks for sharing! How is your thesis coming along?
Oh, sorry about that. A stack is just an infinity presheaf satisfying descent conditions, right? I've always just conflated stacks with algebraic stacks in my head, I guess. How do algebraic stacks differ in their construction? Thanks for pointing this out to me!
Haha, sorry for letting my autism spill out with so little restraint, but it's not every day that I get to share the beauty of homotopy with you anons!
Landon Harris
Feels like everything can be reduced to a topology problem
Oliver Barnes
We quite a bit of work ahead of us to fully generalize and collate all of the results that relate geometry of spaces to their topology (we have Atiya-Singer, de Rham's theorem, and Grothendieck-Riemann-Roch relating algebraic data to topological data, et cetera), but it does seem that some of the most powerful theorems out there are either of the flavour relating these two sorts of information or theorems in model theory that prove powerful existence results based on pretty mild criteria (such as the compactness theorem and Loewenheim-Skolem). It would be quite lovely if we could find nice proofs of this latter breed in terms of topology... imagine a dictionary letting us translate between model theory and homotopty type theory that shows some of these finishing moves are in fact equivalent statements in different languages! The consequences would be quite far-reaching.
Logan Perez
We have* Atiyah* homotopy* (sorry, I'm under time pressure)
Charles Butler
>What is ZFC Set Theory
We have had a grand unified theory for a long time pal.
Liking category theory or other alternative foundations is just the usual rebellious phase juniors get in to feel smart before they have to do research to earn their degrees and remember that before all this bullshit about arrows that means nothing, there was a functioning theory of mathematics they left behind and better get back to before it is too late.
I see no link going to a source that says that category theory is 100% reliable, no paradoxes nor independent statements.
Nathaniel Brooks
>what are the incompleteness theorems
Colton Turner
Exactly. Which means that the exact same shit is going to happen in category theory, just with a different flavor.
John Lewis
The foundations (ZFC and some topoi) and often mutually consistent. Not that I defend either of these shitty (because unpractical) constructs. It's good they exist, and then you can go back to math. imho sets that represent function spaces (the reason afaik that the power set axiom is even a thing) are a drag
Chase Carter
>In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics.
Jacob Adams
This is well understood. Only retards think ZFC is a "grand unifying theory" of math. It's like saying chemistry is the grand unifying theory of biology.
Adrian Lee
Okay, so then why did you call category theory a grand unifying theory?
Ryder Barnes
Nice, happy birthday! Are you talking about some sort of categorical geometry or something? That would hurt some butts, to defile the holiest part of math with the blasphemous theory of categories!
On those locales, there is, atleast, an equivalence between the category of sober spaces and that of spatial locales. This is both good and bad. I'm interested in seeing homotopy on locales, and the easiest way would be to take the (topological) homotopy applied on their spectra, but this would give no advantage: most of the interesting spaces are sober, and thus homeomorphic to their own spectra. No simplification, nothing! Why this interests me most is that some things can be done in locales with a lot less machinery than they would require in topological spaces, for example, the Tychonoff theorem and the axiom of choice. If this can be done, what else can be? Could this affect homotopy, too? An idea of what could be (if there was a homotopy theory for locales) would be that topological homotopy would imply localic homotopy, which would be comparable to how one proves stuff like that an (n-1)-sphere is not a retract of a closed n-ball, etc., to use a weaker notion to disprove claims about the stronger. Weaker because non-sober spaces can have isomorphic frames even if they are not homeomorphic.
On my own stuff, it's almost done. I'm going to include a corollary that there is a ring for each essentially small abelian category is equivalent to a subcategory of R-mod, etc. Nothing too deep, but stuff I've derived myself.
Pls be patient
Ian Jones
is there any paper on homotopy of the fetuses? For example, every fetus starts as a torus
Tyler Lee
That's awesome dude! Pointset topology is very different than the flavour of topology that I do, so I cannot speak much about locales. They are most interesting to me as they fit in to the various periodic tables of higher categories; perhaps a homotopy theory of locales will tell us how to generalize to a homotopy theory of (n,r)-categories. Maybe we can finally figure out what the "right" definition is for an (infinity, infinity)-category.
Category theory is more of a structural meta-theory than a theory itself. We prove theorems for increasingly broad structures so that they apply to any suitable category theory; this is really what started driving the march into higher category theory.
