Mathstack

Answers are validated before mathstack are marked, so students are not penalised for poor programming skills.

In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory , and to construct fine moduli stacks when fine moduli spaces do not exist. Descent theory is concerned with generalisations of situations where isomorphic , compatible geometrical objects such as vector bundles on topological spaces can be "glued together" within a restriction of the topological basis. In a more general set-up the restrictions are replaced with pullbacks ; fibred categories then make a good framework to discuss the possibility of such gluing. The intuitive meaning of a stack is that it is a fibred category such that "all possible gluings work". The specification of gluings requires a definition of coverings with regard to which the gluings can be considered. It turns out that the general language for describing these coverings is that of a Grothendieck topology.

Mathstack

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Differentiable stacks and topological stacks are defined in a way similar to algebraic stacks, mathstack, except that the underlying category of affine schemes is replaced by the category of smooth manifolds or topological spaces. Mathstack generally one can define the notion of an n -sheaf or n —1 stack, which is roughly a sort of sheaf taking values in n —1 categories, mathstack. The Lis-Et topology has a subtle technical problem: a mathstack between stacks does not in general give a morphism between the corresponding topoi, mathstack.

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The art of speaking to AI chatbots is continuing to frustrate and baffle people. A study attempting to fine-tune prompts fed into a chatbot model found that, in one instance, asking it to speak as if it were on Star Trek dramatically improved its ability to solve grade-school-level math problems. The study, first reported by New Scientist , was published on February 9 on arXiv , a server where scientists can share preliminary findings before they have been validated by careful scrutiny from peers. Instead, they were trying to figure out if they could capitalize on the "positive thinking" trend. People attempting to get the best results out of chatbots have noticed the output quality depends on what you ask them to do , and it's really not clear why. This would suggest it's not only what you ask the AI model to do, but how you ask it to act while doing it that influences the quality of the output. These were designed to encourage the AIs , and ranged from "This will be fun! The engineers asked the LLM to tweak these statements when attempting to solve the GSM8K, a dataset of grade-school-level math problems. The better the output, the more successful the prompt was deemed to be.

Mathstack

Stuck on a tricky math problem? Google's newest app will use AI to help you solve it. Two years ago, Google announced the purchase of a math problem-solving app called Photomath. And earlier this week, that app was officially brought under the company's app umbrella. The app itself isn't new, having debuted back in and picking up over million downloads since. But it is now officially a Google app. It works on a wide range of math, from basic elementary school problems like division and multiplication to advanced math like trigonometry and calculus. Once a problem is scanned with the app, the AI starts working. After a few moments, an answer is displayed, along with step-by-step details of how the problem was solved.

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Mumford studied the Picard group of the moduli stack of elliptic curves , before stacks had been defined. Stacks are used to formalise some of the main constructions of descent theory , and to construct fine moduli stacks when fine moduli spaces do not exist. Since this can instead be taken as a stack quotient, the weighted projective stack [4] pg 30 is. Roughly speaking, Deligne—Mumford stacks can be thought of as algebraic stacks whose objects have no infinitesimal automorphisms. The Lis-Et topology has a subtle technical problem: a morphism between stacks does not in general give a morphism between the corresponding topoi. S2CID In the same way, moduli spaces of curves, vector bundles, or other geometric objects are often best defined as stacks instead of schemes. Case Studies. Categories : Algebraic geometry Category theory. Instead of using the smooth topology for algebraic stacks one often uses a modification of it called the Lis-Et topology short for Lisse-Etale: lisse is the French term for smooth , which has the same open sets as the smooth topology but the open covers are given by etale rather than smooth maps. Main article: Algebraic stack. The intuitive meaning of a stack is that it is a fibred category such that "all possible gluings work". When defining quotients of schemes by group actions, it is often impossible for the quotient to be a scheme and still satisfy desirable properties for a quotient.

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This problem is notorious for having caused some errors in published papers and books. S2CID Download as PDF Printable version. In the same way, moduli spaces of curves, vector bundles, or other geometric objects are often best defined as stacks instead of schemes. This terminology is not consistent with the terminology for sheaves: prestacks are the analogues of separated presheaves rather than presheaves. Specific feedback Students are given feedback that refers to their specific answer and mistake, as if marked by hand. There are inclusions:. They are called higher stacks. This usually seems to lead to an equivalent category of quasi-coherent sheaves, but is easier to use: for example it is easier to compare with the etale topology on algebraic spaces. When defining quotients of schemes by group actions, it is often impossible for the quotient to be a scheme and still satisfy desirable properties for a quotient. Categories : Algebraic geometry Category theory. An algebraic stack or Artin stack is a stack in groupoids X over the fppf site such that the diagonal map of X is representable and there exists a smooth surjection from the stack associated to a scheme to X. Annals of Mathematics. To sign up for one of our future events, or see materials from our past events, go to our Training and Events page. There are several ways to deal with this problem:.