I am asked to obtain (117) from (21) but I am stuck and do not know how to go about this.
https://imgur.com/a/f4jgbpt
Many undergraduate linear algebra classes motivate the introduction of bilinear forms via inner products. Suppose the instructor at the end wanted the students to study alternating bilinear forms and needed some motivation. What could the students be given that would make sense at their level?
I happened to come across some controversy in the field of symplectic geometry while I was searching for some related resources. I know nothing of this, but after reading through it and noticing that nothing has been updated, and that there are many more accounts of the dispute still present on A. Zinger's website, I wanted to ask if anyone on here knew any more of this.
From AZ's account, it seems as if experts in the field have just brushed aside major flaws in their peers work, these types of controversies are usually a big story (I know these are pretty different scenarios, but seems like this should be bigger than it is?)
Thanks!
Pretty much what the title says. I have been learning a bit about Symplectic geometry and I was wondering about this.
Would you change anything in my definition? Is there something mathematically wrong? This was quite tough to formulate.
Thank you
https://preview.redd.it/8ywozlufh1951.png?width=2223&format=png&auto=webp&s=02cb4e315e8f4a646203515b39622d3b0eb56bc5
Hello, I'm taking an advanced computational physics class and we missed out on a lot of lectures before due professor's schedule. Now due to Coronavirus, the professor decided to change the class format to a project based and that's all we are graded on. So, I'm looking for a project in symplectic geometry. It can be anything as long as it's not too short. Let me know if you have any suggestions
Today's topic is Symplectic geometry.
This recurring thread will be a place to ask questions and discuss famous/well-known/surprising results, clever and elegant proofs, or interesting open problems related to the topic of the week.
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Next week's topics will be Mathematical finance
Would someone be kind enough to share a connection between those subjects? I’ve been studying algebraic topology, cohomology theories in particular, and recently I heard there’s a connection to symplectic geometry.
- What is SG?
- Is there a useful connection between the two?
When reading about Wiles's proof of Fermat's Last Theorem, it seems that "Ihara's Lemma" played a big role in producing congruences between modular forms. Here the version of Ihara's Lemma is the one used by Ribet to show that certain maps on Jacobians (or cohomology) are surjective. On the other hand, when proving the Sato-Tate conjecture, Richard Taylor managed to avoid Ihara's Lemma in the context of representations to GL(n) (though the statement of Ihara's Lemma seems quite different in these papers). Does Taylor's argument also work for other groups, like GSp(4)?
The ICM 2018 plenary lectures are finally available on YouTube. I thought Michael Jordan's plenary lecture would be relevant here since it discusses some interesting work on acceleration in gradient descent methods in optimization.
More details are also provided in Jordan's proceedings article as well as his earlier paper here.
Hi,
I have a BS in physics and in my senior year I was introduced to Hamiltonians and symplectic area, but didn't dive too deep. If I wanted to learn about sypmlectic geometry more formally what are some pre-reqs you'd recommend?
I have a strong background in calculus and linear algebra. I know the very basics of tensor stuff and non-Riemannian differential geometry for context.
Thanks.
Ready, set, go!
P.S. - Thanks in advance!
I will be applying to PhD programs soon. I am particularly interested in differential geometry (things like geometric flows, minimal surfaces, applications to mathematical physics). I see that lots of academics focus on symplectic geometry and would like a good introduction for someone with a pretty good grasp with DG at the graduate level. I hear that it seems to be the "natural setting" for classical mechanics, but do not know what that really means yet or why people claim that. I like sources to be more thorough.
To the symplectic geometers on here: tell me what your field of study is. Why do you like it? Any applications (to pure mathematics or otherwise) that other mathematicians can appreciate? Do you like physics or do you find the applications to physics interesting?
Hi,
I know this may be a little difficult to find some help on, but I am not entirely sure where to post this to get help. However, I am attempting to understand a couple of journal articles that use the same mathematics to solve boundary problems in 3-D elasticity. I wasn't sure how to attach the full article, however, I attached the two relevant pages, I think. Specifically, my problem is coming from the application of equations 35 and 36. When I see equation 35, it makes me think that I am supposed to set the term in the brackets to 0 and find a system of linear equations, however, one source makes it sound like I am supposed to integrate and then set the expression to 0, as well as, the section shown in the attached image is titled "boundary integration". However, that wouldn't make sense to me because wouldn't that just be picking out one of the coefficients due to the orthogonality conditions that are reviewed in the section before, for example, picking out A_i once distributing through the brackets which is making it 0, making the problem trivial? If someone could help point me in that correct direction, that would be great. Thank you.
https://preview.redd.it/kz1jgfsg50911.jpg?width=585&format=pjpg&auto=webp&s=5e6ca3b5dc1ffa24f04554f539a456b2e48a9602
https://preview.redd.it/9uahggsg50911.jpg?width=557&format=pjpg&auto=webp&s=c744bed6061c5ec628019480da62f70d590e6c0b
So I was reading about the 'affine-symplectic non-squeezing' theorem which states that the image of a ball under a symplectic map can only fit into the symplectic image of a cylinder if it's radius is smaller than that of the cylinder.
That got me thinking, take a sphere and act on each point of it (considered as an element of R^(3)) with a symplectic matrix. How twisted can the image of the sphere look like?? Is there some software to visualise this already out there?
There is probably some topological argument giving some sort of possibilities of winding numbers of these maps on the sphere?
Can anyone point me to any sources on this? Thanks.
