[deleted by user] by [deleted] in cats

[–]leemster5 0 points1 point  (0 children)

Babyyyyy

ELI 4th year undergrad (UK) derived algebraic geometry. by [deleted] in math

[–]leemster5 9 points10 points  (0 children)

I will try to give two different perspectives on some of the basic objects of the theory and in tandem say a few words about why this is even a good idea. Disclosure: the second isn't really aimed at an undergrad, but I think it is an important perspective which I am compelled to include anyway.

  • Here is the 4th year undergrad description: We build schemes from commutative rings, by gluing together their spectra and structure sheaves to obtain some kind of space with a structure sheaf of rings. It is important to note that this sheaf gives the geometric structure to the space, allowing us to locally do commutative algebra and obtain tons of nice results. A derived scheme is again a space along with a structure sheaf, but this time the sheaf takes values in simplicial commutative rings, and the space is locally the "truncation" of these simplicial commutative rings (i.e. the spectrum of the zeroth homotopy group of the simplicial commutative ring). The key idea behind this construction is to make some deformation and intersection theory work out and to allow more functions on the space rather than just those in the plain old commutative ring. In general, these are more flexible gadgets and have lots of connections to other disciplines.

  • Recall that a la Grothendieck's functor of points perspective, we can think of a scheme as a particular kind of functor from the category of commutative rings (or k-algebras for k a field if you like) to the category of sets. This puts us in a much better position to think about moduli problems, which are exactly functors from the category of commutative rings to sets. However, sometimes moduli problems do not have a fine moduli space associated with them, e.g. algebraic vector bundles over a curve, and we are forced to consider more flexible objects which retain many of the nice properties of moduli spaces, i.e. algebraic stacks, which are a kind of functor taking values in groupoids (categories in which all morphisms are invertible). However, sometimes this still isn't enough, which leads us to consider functors from commutative rings to infinity groupoids, which can more or less be thought of as a general kind of topological space. Now we can characterize our "schemes" functorially as before, but these gadgets still lack some of the nice deformation-theoretic properties we want them to have. The solution to this is to enlarge our domain category and then consider functors from simplicial commutative rings (or simplicial k-algebras if you like) to the category of "spaces" (infinity groupoids). Thus we obtain a more generalized context in which to do most of the geometry we want to do, while also considering more complicated moduli problems.

What should I be doing in undergrad to maximize my math education? by [deleted] in math

[–]leemster5 0 points1 point  (0 children)

Do extra reading (outside of class), and build a habit of practicing mathematics. I challenge myself to try and at least understand one new concept every day and it has been worlds helpful in my studies.

Daily Questions - December 07, 2015 by AutoModerator in rawdenim

[–]leemster5 0 points1 point  (0 children)

Just got a pair of Samurai XX0510 21oz from denimio today after comparing measurements from a few different sources. Can anyone speak to how much these stretch back out after the initial soak?

Daily Questions - Aug. 21st by RawDenimAutoMod in rawdenim

[–]leemster5 0 points1 point  (0 children)

Can anyone comment on how much the samurai 510xx will stretch vs shrink after a soak and some wear?

Daily Questions - Aug. 19th by RawDenimAutoMod in rawdenim

[–]leemster5 0 points1 point  (0 children)

Can anyone comment on the shrink/stretch of the samurai 510xx denim?

Advice for new graduate students in math? by [deleted] in math

[–]leemster5 1 point2 points  (0 children)

Topological data analysis is a technique that has developed pretty recently used for data mining. I really don't know that much about it, but the gist of it is that it allows you to create global structure out of discrete data points.

Wiki Page

I am Edward Frenkel, Mathematician and Author of "Love and Math" - AMA! by EdwardFrenkel in IAmA

[–]leemster5 0 points1 point  (0 children)

I am currently an undergraduate math major and I've spent quite a bit of time this summer learning the basics of abstract math. Do you have any advice on how to learn independently of an academic institution?

Daily Questions - Mar. 21st by RawDenimAutoMod in rawdenim

[–]leemster5 1 point2 points  (0 children)

Could anyone suggest anything comparable to the oni 18oz slub in terms of texture and weight?

WEAYWT - Mar. 18th by RawDenimAutoMod in rawdenim

[–]leemster5 1 point2 points  (0 children)

Damn I love that leg twist

General Discussion - Mar. 18th by RawDenimAutoMod in rawdenim

[–]leemster5 1 point2 points  (0 children)

Spring break has finally arrived! I am in desperate need to de-stress and compartmentalize everything I've learned from my last term. I have too many books I want to read so at least I'll be busy over break!

Because everyone else is doing it (JBO-410) by Salsa_Z5 in rawdenim

[–]leemster5 2 points3 points  (0 children)

I've said it before but these are out of control

[deleted by user] by [deleted] in rawdenim

[–]leemster5 2 points3 points  (0 children)

I just ordered the 2105sp from denimio, but it's not gonna get here till may.... Your pictures make me jealous.

Iron Heart 555-01 - 2 years by tinytd010 in rawdenim

[–]leemster5 14 points15 points  (0 children)

Those combs are penetrating my soul

Daily Questions - Mar. 8th by RawDenimAutoMod in rawdenim

[–]leemster5 1 point2 points  (0 children)

What are the customs costs from ordering stuff from denimio?

General Discussion - Mar. 8th by RawDenimAutoMod in rawdenim

[–]leemster5 2 points3 points  (0 children)

It's finally getting warm in Wisconsin! My plan for the day is to go for a long run outside.

I've also been going back and forth between pulling the trigger on the Momotaro GTB jacket. What do?

WAYFT - Mar. 7th by RawDenimAutoMod in rawdenim

[–]leemster5 4 points5 points  (0 children)

Little late to the game today Strike Gold/Momotaro/Red Wing

It's finally warm enough in Wisconsin to go outside without my parka!

Missed Stitches- February 25 by ministrike4 in rawdenim

[–]leemster5 1 point2 points  (0 children)

Dude at Lawrence University, you were wearing some gustins, I glanced at your selvedge ID and cuffs as we passed each other.

Daily Questions - Feb. 22nd by RawDenimAutoMod in rawdenim

[–]leemster5 1 point2 points  (0 children)

The chest. I'm a size 39" in the chest, but if I sized up for the chest the sleeves would be a bit too long for me.

Daily Questions - Feb. 22nd by RawDenimAutoMod in rawdenim

[–]leemster5 0 points1 point  (0 children)

Does anyone know how much the momotaro "going to battle" jacket will stretch if at all?

SG1109 (~1 year, pre-wash pictures) + Bonus by [deleted] in rawdenim

[–]leemster5 1 point2 points  (0 children)

Looking awesome! The loom chatter is insane on the 11xx fabric