Quick Questions: June 15, 2022

Does some analogue of flat base-change hold for singular cohomology? Say we have a fiber product diagram of smooth complex varieties (projective, even); then is there a similar statement of the compatibility between the pushforwards and the pullbacks in this diagram?

Edit: I think I should be more precise: given flat f : X -> Z and proper g : Y -> Z completed to a Cartesian square with f' and g' the obvious maps, flat base-change tells us that f\^\*g\_\* = g'\_\*f'\^\* on all algebraic cycles. Does it hold on all singular cohomology classes?
How do you schedule and track your math research? I am used to tools like Jira from software development, but that's a little too team-based and Gantt-chart oriented for my needs as an individual grad student.
I am currently trying to understand the exterior derivative which in my case is derived from axioms (dd = 0, signed Leibniz rule and so on)

My issue is that in one step we write  for a 1-form d(wdxi)= d(w ^ xi) where the ^ is supposed to denote the wedge and then apply the signed Leibniz rule and everything falls into place. I thought I understood differential forms but I don't quite understand how wdxi=w ^ dxi.

The left side takes a point  p in M and yields an alternating linear Form on TpM, the right side yields an alternating bilinear Form. How can you equate those things ?
It's common in modern algebraic topology to not distinguish between homology and cohomology theories (which are of course equivalent by Spanier-Whitehead duality). For example, the Landweber exact functor theorem technically yields a homology theory, but people usually refer to the associated cohomology theory as being produced by LEFT. But is there an easy way to actually compute cohomology from homology or vice versa? In general, I mean—obviously this can be done for certain theories (e.g. ordinary cohomology over a PID) quite easily.
Book Recommendations:

Hello all, I am searching for a rather niche topic; I want to learn more about interpretability in first order logic. That is, translating one first order language to another.

I learned a bit about it from a logic book where it embeds arithmetic in set theory to deduce its incompleteness, but I want a more detailed treatment. Please let me know of anything you recommend!
Is there a difference between A-stability and Lax–Richtmyer stability for PDE?
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What is the most efficient way to prove 3 statements connected by iff? I recall my analysis teacher mentioning a very neat trick for these types of proofs, but can't recall it. Just to be clear I am attempting to prove

p ⇔ q ⇔ r
Im reading "discrete mathematics with application" by Sussanne epp and there is a definition of functions based on sets, it is as follow:

"A function F from a set A to a set B is a relation with domain A and a co-domain B that satisfies the following two properties:
1. For Every element x in A, there is an element y in B such that (x,y) E F [ (x,y) belongs to F] .
2. For all elements x in A and y and z in B, if (x,y) E F and (x,z) E F, then y=z."

I understand the first property but i have a doubt regarding the second. What if the function F(x) =√x?
In that case doesn't F(x) have two values for positive real numbers? And so if x = 4 by property 2 we would have -2 = 2.

What am i missing? What am i not understanding?
Are functions different for sets?
What are analysis requirements for J lee's smooth manifolds. By that, I mean which chapters of Rudin would I need to study? Do I need chapter 9\~10 or do I just need 1\~8?
Where is the best place to go for help with differential equations and Matlab/Python? I need help writing a code for optimizing parameters in a system of diff eqs. I'm looking for 1 on 1 and willing to pay

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