soft question – Why is the Shafarevich Tate group called "Tate-Shafarevich Group"?

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soft question – How many graduate students in mathematics are there worldwide?

There is already a question about the number of mathematicians, but I was wondering:

Is it possible to estimate the number of mathematics graduates worldwide in 2019? (Note that it differs significantly from the other question, especially since not all GS remain in the academic world.)

Ideally, I'm looking for a graphic since 1950 with a breakdown by country.

For the US, such a graph is found on a fine page of AMS data for the period to 2015-16 (published in March 2018), with the annual number of early school leavers being crucial.

So the main purpose of my question is to find the equivalent for other countries.

soft question – Does a result remain known after all who knew it have died?

When working on a research project,
you try to spend your time answering questions that have not yet been answered.
There is the terminology of "known" versus "unknown" results,
Under that we generally understand whether a problem has already been solved.
On the other hand,
we know that math is always in progress,
including cases of "known" facts that have proven to be false.

The evidence of some results is quite esoteric,
requires extreme specialization on the subject in order to understand.
It is possible that an article will be subject to a review.
accepted by the community,
and his sentences are included in the mathematical canon,
Only for those who are able to follow the arguments and then die
Do not leave behind suitable offspring to gain the knowledge.
My question is whether these results are still considered "known".
The deeper question is how important it is to find new and more accessible evidence for such outcomes.
so that they are better known in the truest sense of the word.

To make the question less subjective,
Let's focus on the etiquette of using this terminology.
So that a mathematician publicly proclaims that something is "known",
they must have read and understood the proof
know someone who has read and understood the proof
and if the latter,
Does this person have to be alive?
On the other hand, does "known" only mean that evidence has been published in a peer-reviewed journal at any point in history, no matter how long ago?

(Soft question) Why should one learn pure mathematics?

I am a math major and enjoy pure math. I do not have a special area of ​​interest yet, but I look. Apart from that, I often ask myself the question "What is the meaning of pure mathematics?" and "what is it good for?" and fight to find answers to these questions. I am able to understand and appreciate the beauty of pure mathematics (and that is very rewarding), but besides, I can not understand why I should do pure math. Do any people in the math community (including those doing math / research for a living) have any idea of ​​how I can look at this question or interesting answers?

Again, I'm not really looking for something like "because math is so beautiful" or "math itself is an art" – I totally agree with that. But Why should I (or anyone) pursue it? Is there nothing else to promote it have Results / solve own problems?

PS – I'm not looking for answers in the sense of "But what makes sport like this?" – Sport is about getting people involved in an activity (you can not agree with that, but that's not important to my question). Tldr – I still can not answer the following question correctly: What is the point of pure mathematics?

soft question – mathematical tricks

For this question, examples of "tricks" are needed, where "trick" is defined as a mathematical statement with the following properties:

  1. It was used in a proof of a main clause (major is somewhat subjective, eg, a sentence that solves a long-standing problem or opens a new direction of research).
  2. It's pretty short / easy to say.
  3. It looks like a red herring (D. Does not look very natural when you first hear about it). It may, for example, contain some arbitrary-looking numbers. Optional: It's really not clear how to generalize it.

In short, Grothendieck would probably be annoyed. Some candidates:

  • Zarhin's trick about abelian varieties (see here).
  • In Weil I there were two statements that could qualify, one is Rankin-style power trick, the other is rational $ l $-adic coefficients (eg here).

P.S. This is not a duplicate of this question (that requires a common characterization) and not a duplicate of that question (ours is less ambitious).

soft question – reference request for Grothendiek's thesis "Integration with Values ​​in a Topological Group"

Disclaimer. This question has already been asked in Mathematics Stack Exchange (see link here). I wanted the question to be migrated here, but I was told by a moderator that an old question should not be migrated (something I did not know before).

I recently read the available part of the second part of W. Scharlau's book on Alexandre Grothendieck (see here). There I found

An anecdote survives as Grothendieck arrived at Nancy: the story of his rude reception by Dieudonné, when he showed him a dense handwritten manuscript on "generalized integrals" on his first contact. He had already mentioned this work in writing to Dieudonné and received a warm and friendly reply in which Dieudonné praised his "passion for mathematics". However, Dieudonné's initial receptivity did not survive the first look at the actual text. Those who remember this incident (or rather Dieudonné's account of it) claim that Dieudonné Grothendieck has a
quite disguised, the work showed a despicable tendency to an unfounded generality.

Later it is also mentioned that (as Schwartz tells in his autobiography),

He first gave Dieudonné an article of about fifty pages about "Integration with Values ​​in a Topological Group". It was right, but absolutely uninteresting. Dieudonné, with the (always temporary) aggressiveness he was capable of, gave him a catchy rant and demanded that one should not work that way just to generalize the pleasure. The problem considered had to be difficult and applicable to the rest of the mathematics
(or other sciences); His results would never be useful to anyone.


  1. Does anyone know how Grothendieck dealt with the problem of integration with values ​​in a topological group that he presented to Dieudonné (I seem to find nothing on the internet)?

  2. Why was Grothendiek's work on "Integration with Values ​​in a Topological Group" by Dieudonné described as "not useful to anyone"?

  3. Was this topic researched in the future?

  4. Where can I find (if at all possible) the original work by Grothendieck on the Internet?

Lighting – Unity Soft Shadows

I'm building a puzzle level in Unity. I have a bullet with cracks and a point light in it. The goal is to project the light coming through the cracks into the walls.

I did it, but the lighting is not soft (it does not scatter on the surface, so it looks very sharp). The walls are simple unity cubes.

Soft shadows

I use & # 39; soft shadows & # 39; for the properties of light, but that does not seem to make a difference. I suspect that the occluder is so close to the light source that the illumination gets very sharp. Is there a way to control the softness?

I tried to mess with the smoothness of the wall material, but it made no difference.

I've also tried to make the cracks into an albedo map with alpha channel (with zero or near zero alpha values ​​where I wanted the cracks) and make the ball a perfect ball rather than the net of the Balls has physical cracks. Although it works in cutoff mode, it gives the same result as before. Another thing I tried was to set the transparent mode, but the transparent shadows are terrible! Is there a way to improve their quality?

I searched for (free) soft shadow assets for point lights, but could not find any.

Is there a solution for this?

Thank you in advance!

How To Encourage Lovebirds To Be Ngekek

Lovebird has another chirping sound, but most fans prefer Lovebird, which is active. Even for Lovebirds who diligently practice, there are not a few who uniquely make their efforts. Lovebird lovers like this bird because of its husky voice.
The longer you are, the better the bird will be. In addition to being used for masters, long nailers with longworms are often included. Therefore, many people try to train and cook their favorite birds. Apart from using master audio from MP3 devices, practicing lovebirds to make their hoaxes work can also be done in a variety of ways.
Quite a few lovers of love birds who use certain items to train Lovebird to be diligent.

Here are some unique ways to catch lovebird birds:
1. Use Styrofoam foam
Styrofoam foam, which is commonly used to wrap electronic items, can be used to lure your beloved Lovebird so hard that it will tackle it. The result is that the bird becomes addicted and sounds loud. When this happens routinely, Lovebird becomes more diligent.
2. Use used beverage cans
Used soft drink cans can be used to make lovesick sounds as small stones can be used in soft drink cans. Then the can is shaken so that it can produce a unique sound, similar to Lovebird.
3. With Citan Cit toy akibat lovebird kurang birahi

Another unique option is to use devices that can be found on the bird market or in the aviation accessories store. Just tell the seller that you are looking for a special parrot toy called Pet-Petan or Cit-Citan. This toy is made of plastic, the way to use it is very simple, by pressing it by hand and creating a sound similar to that of Lovebird ngekek.

These are some unique ways to provoke a nagging voice on Lovebird, so if you train more often, Lovebird can make a loud noise. Lovebird Biar Ngekek Rajin


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