O Level Geometry

[Trinh Livingston]

Inalgebraic geometry, a level structure on a space X is an extra structure attached to X that shrinks or eliminates the automorphism group of X, by demanding automorphisms to preserve the level structure; attaching a level structure is often phrased as rigidifying the geometry of X.[1][2]

In applications, a level structure is used in the construction of moduli spaces; a moduli space is often constructed as a quotient. The presence of automorphisms poses a difficulty to forming a quotient; thus introducing level structures helps overcome this difficulty.

There is no single definition of a level structure; rather, depending on the space X, one introduces the notion of a level structure. The classic one is that on an elliptic curve (see #Example: an abelian scheme). There is a level structure attached to a formal group called a Drinfeld level structure, introduced in (Drinfeld 1974).[3]

This may be a stupid question, but it's always bugged me and I hope it's a super simple answer that I have just never learned.

I'm working with files that were drawn poorly in the first place. There are multiple levels featuring very similar (if not identical) geometry. These files have many different tool paths with geometry that is on tons of different levels. When I want to make a change to the tool path geometry it's a real pain in the xxxx to find it through like 50+ different levels. From the geometry tab under various tool paths, is there any way to determine what level the geometry is on? I can "show" the geometry, but it only shows if the level is active which is useless when I dont know the level.

Analize like JLW Said above, analyze was also improved upon in 2020s release to include some new cool features so you can now even see what toolpaths, planes, etc. is associated to the geometry. Vid below, pretty useful in my opinion

My biggest concern is geometry. While I think I'm doing pretty well in number theory, algebra, combinatorics etc., I still can't say I really understood Olympiad geometry (and bashing is not always possible). Obviously, there are always problems one can and one can't solve, regardless of the training, but my question is:

I must say, I'd prefer ones that don't use overly sophisticated notations and don't go into super-advanced theorems you happen to use once or never. And I'm aware that a month is not really a long time and this question may seem a bit like: "quickly, what should I learn to win an Olympiad". I don't mean it. I'm just asking - what to study Olympiad geometry from to greatest benefit while certainly greatly determined to?

EDIT: I'd like to address some comments regarding taking Olympiad too seriously. While I agree to great extent with this reasoning and I appreciate your concern, I'd like to clarify: it's not that I'm going to be working $\frac247$ on maths only. I'd just like to work with the best resources available and thus not waste my time on exercises that aren't of much benefit to my geometry skills.

"Euclidean Geometry in Mathematical Olympiads" by Evan Chen: this is a problem-solving book focused on Euclidean geometry, suitable and specifically written as a preparation for mathematical olympiads;

"Problems in Plane Geometry" by Igor F. Sharygin: it has several "non-standard" problems with increasing levels of difficulty, so that it is useful to understand some issues of plane geometry not often described in standard books, such as what additional constructions can be made, which "alternative" pathways could be used to arrive at the solution, and so on;

"Geometry Unbound" by Kiran S. Kedlaya, a very good paper structured in the form of a textbook, which starts from rudiments and arrives to the most modern areas of geometry, including inversion and projective geometry;

lastly, a complete book is "Problems in plane and solid geometry" by Viktor Prasolov (already cited in one of the comments), a comprehensive 600-page text with thousands of problems and detailed solutions covering all areas of plane and solid geometry.

Geometry is not all of the Math Olympiad. Almost all coaches say that you may not solve Algebra, you may not solve Combinatorics or a number theory problem, But you should be able to solve the Geometry one.

Here are solving books that you should try, for getting better place in Olympiad-

Note: You should see more and more SOLUTIONS even if you solved them. You need to see the techniques. Seeing solutions has a bad effect but I think not in this area. Weather or not you solve the problem, see the solution / technique.

My goal is to study derived algebraic geometry, where derived schemes are built out of simplicial commutative rings rather than ordinary commutative rings as in algebraic geometry (there's also a variant using commutative ring spectra, which I don't know anything about). Anyways, since the category of simplicial rings form a model category, we can apply homotopy theoretic methods to study derived schemes.

I thought the first thing I should do is study simplicial homotopy theory, in order to learn about model categories and simplicial objects. So I started reading Simplicial Homotopy Theory by Goerss and Jardine. How should I study this book? There are very few exercises, unlike standard graduate textbooks like Hartshorne, and a lot of the proofs are simplex/diagram chasing, so I decided to skip a lot of the proofs and read the book casually.

A big disadvantage to this method is that I don't understand anything at a deep level and I'm only familiar with a few buzzwords. But I feel overwhelmed by the amount of prerequisite material I need to understand to learn DAG, because most of it is written in the language of $\infty$-categories. So what should I do? How can I get to "research level mathematics"?

I propose the following plan, assuming a basic background in scheme theory and algebraic topology. I assume that you are interested in derived algebraic geometry from the point of view of applications in algebraic geometry. (If you are interested in applications to topology, you should replace part 2) of the plan by Lurie's Higher algebra.) The plan is based on what worked best for myself, and it's certainly possible that you may prefer to jump into Higher Topos Theory as Yonatan suggested.

0) First of all, make sure you have a solid grounding in basic category theory. For this, read the first two chapters of the excellent lecture notes of Schapira. I would strongly recommend reading chapters 3 and 4 as well, but these can be skipped for now.

Then read about stable $\infty$-categories and symmetric monoidal $\infty$-categories in these notes from a mini-course by Cisinski. (By the way, these ones are in English and also summarize very briefly some of the material from the longer course notes). These notes are very brief, so you will have to supplement them with the notes of Joyal. It may also be helpful to have a look at the first chapter of Lurie's Higher algebra and the notes of Moritz Groth.

Read lecture 4 of part II of Moerdijk-Toen, Simplicial Methods for Operads and Algebraic Geometry together with section 3 of Lurie's thesis. Supplement this with section 2.2.2 of Toen-Vezzosi's HAG II, referring to chapter 1.2 when necessary. This material is at the heart of derived algebraic geometry: the cotangent complex, infinitesimal extensions, Postnikov towers of simplicial commutative rings, etc.

3) Before learning about derived stacks, I would strongly recommend working through these notes of Toen about classical algebraic stacks, from a homotopy theoretic perspective. There are also these notes of Preygel. This will make it a lot easier to understand what comes next.

4) Finally, read about derived stacks in lecture 5 of Moerdijk-Toen and section 5 of Lurie's thesis. Again, chapters 1.3, 1.4, and 2.2 of HAG II will be very helpful references. See also Gaitsgory's notes (he works with commutative connective dg-algebras instead of simplicial commutative rings, but this makes little difference). His notes on quasi-coherent sheaves in DAG are also very good.

In my opinion the best foundations to any modern topic in homotopy theory, and derived algebraic geometry in particular, is "Higher topos theory" of Lurie. The scope covers all the required ($\infty$-)categorical framework, and every chapter starts with a very conceptual motivation. In addition, the book also contains appendices which explain classical material (such as model categories) in a very readable way. You might find in the beginning some proofs which involve technical combinatorics of simplices. Don't be discouraged. Feeling comfortable with simplices is essential and this requires working out some details. The proofs in the book do become increasingly conceptual with each chapter, as the concepts themselves get built and acquire depth.

I'm going to take a dissenting view, here. I think the best way to assimilate concepts in derived algebraic geometry (for finite fields, $\mathbbR$ or $\mathbbC$), is to understand where and why they are used. Then, work backwards when the need arises. Personally, I found it formidable to read through any section of Toen-Vezzosi's homotopical algebraic geometry series straight through. I'd first recommend reading and understanding the content of Vezzosi's AMS notice, here: Once you begin digesting the need for replacing the source category for Grothendieck's functor of points approach to algebraic geometry with derived commutative algebras, browse through the literature and find instances where this becomes necessary. From my perspective, the most striking application is here: , where one sees (sloppily speaking here), that even replacing the source category with truncated derived objects goes a very long way in recovering classical results. Feel free to let me know if you'd like me to explicate further.

Maybe it was my unclear question. My intention is to assign the elements an existing Revit level parameter, to be more specific, the same one that is used for assigning Revit element like walls, door, panel and so on. Because if I add new shared parameter that means I will have another level parameter if I understand correctly. Is it because of Level parameter belongs to Revit system parameter so that RiR is not capable to edit or touch? However, from your suggestion I can see that, possibly I can use the level parameter from the Revit as a input for the name slot of shared parameter definition right?

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