Fabian Haiden

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@nisamhuligan @michael_nielsen Huge improvement!

@fabhaiden @michael_nielsen Knowing the limitations of the model (the boundary where it starts to hallucinate) is a big part of efficiently using it. The free model doesn't fare well in technical topics unfortunately. As a data point, here's GPT4's answer to the same prompt: chat.openai.com/share/d17bbdb5…

University responses to ChatGPT often seem based primarily on whether it's convenient for professors ("Oh my god, we might have to change how we assess"), rather than: "These are powerful new tools, how can we learn to use them well, and help students learn to use them well?"

@michael_nielsen Here, ChatGPT tells me that there are exactly two integers. Bad prompt or limitation of the model? ChatGPT "knows" that pi_1(RP^2)=Z/2 and that |Z/2|=2, but cannot put these two things together.

One tell: lots of people yelling that LLMs are useless / hallucinate etc. I mentally translate that as: those people are bad at using LLMs. No shame in that - no-one masters every cognitive tool - but not especially interesting, either

@Francis16833887 I found this useful for learning the big picture: arxiv.org/abs/math/00100… Local systems are objects in the Fukaya category of the cotangent bundle. Since Fukaya categories are Calabi-Yau, there is a shifted symp structure on the moduli of objects.

@fabhaiden Thanks! I think that makes sense, but I need to read more about it. Do you know a good place to start? Also, do you know how this relates to the shifted symplectic structures on the character stacks?

I've just written a blog post about character varieties, Skein algebras, and String topology: francisrlb.wordpress.com/2022/11/14/cha… 1/4

@Francis16833887 Great blog post! The modern/futuristic point of view on this is to consider symplectic homology, which in the case of cotangent bundles reduces to homology of the loop space. Quantization amounts to turning on higher genus corrections.

I've recently been trying to learn a bit more about these subjects, and the relations between them. I've written down the small bit I think I understand. There are probably many mistakes, and I still have many unanswered questions. 4/4

Hopf algebras are groups (in the category of vector spaces) Lie algebras are examples of Hopf algebras (via universal enveloping algebra) Lie "algebras" are really groups 🤯

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@JadeMasterMath A set is a tree where all branches look different and any leaf is a finite distance from the trunk.

What is a set?

@NoahJSnyder Back to the roots of algebra.

"This may be a graduate class, but it's still called Algebra, so we're doing the quadratic formula on the first day!" --me, apparently

@kareem_carr And here is Google, for comparison.

Finally tried out the GPT-3 model from OpenAI. The green text is unaltered output generated by the AI.

@dzackgarza What is a moduli space? In my mind, it means points are isomorphism classes of something.

I'm so 𝘧𝘰𝘳𝘨𝘦𝘵𝘧𝘶𝘭 - I can't remember any example of a functor with a left adjoint.

@TimHenke9 If a.b is associative in the usual sense, then a*b:=(-1)^{deg(a)}a.b satisfies (a*b)*c=(-1)^{deg(a)}a*(b*c). This comes up when constructing the bar complex.

Is graded associativity a thing? Like (a*b)*c = (-1)^{deg b} a*(b*c) or something?

@TimHenke9 "following the uncertainty principle..."

mathematical physicists will open a scheduling email with "following the principle of least action,"

0.99999... = 1 ...99999.0 = - 1

@TimHenke9 From the functor of points perspective, schemes are certain functors from rings to sets. If we "homotopify" the target category instead (considering functors to spaces) we get higher stacks. Or we could do both and get "derived higher stacks"!

@TimHenke9 I think of this as part of the general theme of replacing sets by spaces, the linear version of which is replacing abelian groups by chain complexes (by Dold-Kan), thus rings by dg rings. Why this is useful is another question, which has many answers!

What bothers me is I only ever hear "problem X goes away in derived geometry" Of course, it's wonderful when problems go away, but that feels like a post-hoc motivation I wanna know why they're natural objects. Why do they capture what I find interesting & forget what I don't?

@TimHenke9 (this is the affine version of your question)

@TimHenke9 Do you think of (super-) commutative differential graded rings as a natural generalization of commutative rings?