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Anonymous at Tue, 29 Oct 2024 05:10:06 UTC No. 16454471
What are some profound scientific or mathematical truths
Anonymous at Tue, 29 Oct 2024 05:23:09 UTC No. 16454476
>>16454471
Reality is bigger than rationality, formally and colloquially
Anonymous at Tue, 29 Oct 2024 09:13:55 UTC No. 16454603
>>16454471
https://en.wikipedia.org/wiki/Groth
Anonymous at Tue, 29 Oct 2024 10:13:32 UTC No. 16454623
>>16454471
/sci/ is for midwits
Anonymous at Tue, 29 Oct 2024 13:16:22 UTC No. 16454716
>>16454471
The state of a system of many things that can interact in many different ways, will on average require more information to describe than what it started with, because there are more states that require more information to specify than states that require less information to specify.
Anonymous at Tue, 29 Oct 2024 13:41:55 UTC No. 16454730
>>16454603
Literally how do you even understand its implications without having a PhD in maths?
Stop guessing start learning at Tue, 29 Oct 2024 14:25:29 UTC No. 16454775
>>16454476
What does this even mean? Be honest you just put a bunch of big words together to make you sound smart.
Anonymous at Tue, 29 Oct 2024 20:06:26 UTC No. 16455164
>>16454730
The Grothendieck–Riemann–Roch (GRR) theorem has profound implications in several areas of mathematics, particularly algebraic geometry, topology, and complex analysis. Here are some of the key implications:
Bridge Between Geometry and Algebraic Topology: The GRR theorem connects algebraic and topological invariants of coherent sheaves on complex manifolds, enabling complex analytic and algebraic information (like sheaf cohomology) to be translated into topological data (like Chern classes and characteristic classes). This creates a pathway between algebraic geometry and topological K-theory, broadening the scope for deeper insights in both areas.
Foundation for Index Theory: The GRR theorem serves as a conceptual foundation for the Atiyah-Singer Index Theorem, which calculates the index of elliptic operators on compact manifolds. By relating the Euler characteristic (a topological invariant) of sheaves with integrals of Chern classes, GRR provides tools to handle complex calculations in index theory and differential geometry.
Tool for Computing Intersection Numbers: GRR allows for the computation of intersection numbers and dimensions in a more straightforward manner through the use of Chern classes and K-theory. This has practical applications in enumerative geometry, where the aim is often to count the number of solutions to geometric problems.
Unified Framework for Riemann–Roch Type Theorems: The classical Riemann–Roch theorem is a specific case of the broader GRR framework. By generalizing Riemann–Roch to higher-dimensional varieties and more general morphisms, it gives a unified theory for computing Euler characteristics across different contexts, from curves to complex varieties.
These implications make the GRR theorem a central result in modern geometry and have led to significant advances in the study of moduli spaces, mirror symmetry, and string theory, where connections between geometry, physics, and topology are explored.
Garrote at Tue, 29 Oct 2024 20:13:04 UTC No. 16455168
a+b=b+a
Anonymous at Tue, 29 Oct 2024 20:32:42 UTC No. 16455188
>>16455164
Fuck this ChatGPT output
Anonymous at Wed, 30 Oct 2024 01:08:11 UTC No. 16455450
>>16454603
>https://en.wikipedia.org/wiki/Grot
i hope to understand what the hell any of this means one day
Anonymous at Wed, 30 Oct 2024 02:17:56 UTC No. 16455491
>>16454716
That sounds like Gödel's incompleteness theorem with extra steps.
Anonymous at Wed, 30 Oct 2024 02:44:45 UTC No. 16455519
>>16454471
The profoundedest thing is that none of it is all that profound, no matter how deep you go. For instance, perhaps nature is le quantum fields? So what? It makes no difference on a philosophical level. Physics cannot at this time differentiate between QM interpretations, but that could be a little more interesting.
Anonymous at Thu, 31 Oct 2024 17:45:52 UTC No. 16457219
>>16454471
my benis is the biggest object in the universe
Anonymous at Thu, 31 Oct 2024 17:48:31 UTC No. 16457222
>>16454471
E=mc^2 * AI
Anonymous at Thu, 31 Oct 2024 17:54:51 UTC No. 16457234
>>16457222
AI still isn't invented, all current models are advanced documents, they're not actually intelligent, it's just ordered information.
raphael at Sat, 2 Nov 2024 02:43:13 UTC No. 16459214
>>16455188
>
agreed op must be a 105 FSIQ nigger trying to act smart
Anonymous at Sat, 2 Nov 2024 06:58:16 UTC No. 16459377
>>16454471
the constants pi and e. pi is the ratio of a circle's diameter to its circumference and holds true no matter how big or small the circle. e is special because mathematicians really wanted a function where f(x) = f'(x). e^x was their answer. I'd say those are my two favorites, and they're quite profound especially given how straightforward they are.
Anonymous at Sat, 2 Nov 2024 07:20:39 UTC No. 16459386
by construction, all mathematical theorems are tautologies
Anonymous at Sat, 2 Nov 2024 07:52:31 UTC No. 16459414
>>16454471
As our estimation of objective reality can never be 100% correct due to the limitations of the mind, we must always remain vigilant of errors. Neglecting sources of error cause catastrophes all the time, especially in space flight.
Anonymous at Sun, 3 Nov 2024 06:41:43 UTC No. 16460618
>>16454471
Yoneda's lemma