Questions tagged [soft-question]

Questions that are about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. Do not use this tag for easy or supposedly easy mathematical questions.

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Research directions in complex differential geometry

Not sure if my question makes sense. Is there an area in complex geometry that is as analytic as possible? Actually what I wanted to ask is an area in complex geometry that is as non-algebraic as ...
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2 votes
0 answers
157 views

Could there be a better classification of finite simple groups?

The current classification of finite simple groups puts every finite simiple group in one of a few categories. There are the "nicely" behaved infinite categories (cyclic, alternating, Lie-...
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10 votes
1 answer
221 views

Fourier's proof of reality of all roots of Bessel function $J_0(x)$

In his "Théorie de chaleur" Fourier proves that the zeros of Bessel function $J_0(x)$ are all real. I want to ask if there is a modern version of this proof exist in literature? If someone ...
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-8 votes
0 answers
153 views

Is there a more descriptive term for Characteristic Classes? [closed]

Recently I went along to a conference on Communication in Mathematics where a number of people, me amongst them, complained about talks where one only understood 5% of what was going on. A part of ...
-10 votes
0 answers
248 views

Is there a simple example of why homotopy type theory is useful? [closed]

Feynman said that when he was attempting to understand a complex physical theory he would hold in his mind the simplest physical example to which it would apply. This is why he was able to demolish ...
1 vote
1 answer
79 views

Notation for infinite cartesian products

This is a soft question, feel free to delete it if deemed inappropriate for the site. What is the best notation for the cartesian product of an infinite number of copies of the same set $E$? Maybe one ...
2 votes
1 answer
454 views

Most efficient way of getting a brief overview of the current active research areas in Algebraic Topology

I'd be applying for a Ph.D. at various grad schools in the U.S. in the coming months and while I know I'd like to pursue research in the field of Algebraic Topology, I am not knowledgeable enough yet ...
3 votes
0 answers
145 views

What does a character of a scheme mean?

Here is a soft question I met in the book Introduction to Grothendieck Duality Theory by Altman and Kleiman. In Chapter I the proposition 2.1 uses a term called "a character of $X$" where $X$...
2 votes
1 answer
247 views

(Dis)prove : if every function with closed graph are continuous then the target space is compact

$(X, \tau_X) $ and $(Y, \tau_Y) $ be two topological spaces. $\forall f\in Y^X$ with $\text{Gr}(f) $ is closed implies $f\in C(X, Y) $. Question : Does this implies $(Y, \tau_Y) $ is compact? ...
6 votes
2 answers
1k views

Poincaré recurrence and its implications for statistical physics and the arrow of time

A very important theorem in mathematical physics is Poincaré’s recurrence theorem. As you recall, this theorem states that given a dynamical system $(M , \phi , \mu)$ with $ \mu M < +\infty$, for ...
3 votes
0 answers
173 views

Resources for PhD students to guide through the research [duplicate]

I am living in a poor country but I have managed to get admission for a PhD program in Western Europe. This was a daunting task for me due to my family background and also social conditions. ( I don't ...
12 votes
4 answers
2k views

How about a statement without proof?

Consider a statement without proof in a paper, with the following assumptions: it is unknown, it is unused in the paper, it is not written as a theorem (or proposition, or lemma…), but just a free ...
0 votes
0 answers
52 views

Relevance of the deduction of similar theorems than Maier's theorem for other constellations of primes

A year ago I asked this question on Mathematics Stack Exchange with identifier 4245823 and same title Relevance of the deduction of similar theorems than Maier's theorem for other constellations of ...
14 votes
0 answers
490 views

Where did the military money go?

In older papers, one sometimes finds references to sources of funding directly linked to or overseen by military agencies. For example, I have memories of seeing acknowledgments to DARPA funding in a ...
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7 votes
0 answers
135 views

Intuition for branch uniqueness in inner model theory

In inner model theory, what is the intuition behind the expectation that under appropriate conditions, we should have a single preferred branch to continue an iteration at a limit stage? At the level ...
1 vote
0 answers
85 views

Formalizing intuition of search hardness

Basically, this is a search problem of an object that is promised to exist. Suppose we have an object that can be described completely and uniquely by $m$ properties (each n bits). Suppose a search ...
32 votes
1 answer
5k views

The editor wrote the paper for me

I submitted a short paper and received a positive review and a negative review. The editor (he) briefly wrote the following things: He thinks my original result could be mistaken because of XYZ He ...
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1 vote
1 answer
291 views

Adjunctions in the real world

What concepts in the real world can be described by adjunctions? For example, parents and children are adjoint to one another. Specifically, work in $ZFC$ plus a finite class of atoms $\mathscr{X}$ (...
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1 vote
0 answers
116 views

Translation of an article by Šapirovskiĭ

The AMS has a book, Fourteen Papers Translated from the Russian, containing an article by Šapirovskiĭ, "Cardinal invariants in compact Hausdorff spaces", that I would very much like to ...
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4 votes
1 answer
438 views

Supervision numbers in pure mathematics

My faculty imposes some numerical "recommendations" for promotions. Instead of arguing this sort of recommendation is ridiculous, I think it is wiser to provide some evidence the recommended ...
1 vote
0 answers
47 views

Asymptotic behaviors of different types of equations with capillarity or viscous terms

Let us consider $$ \begin{align} u_t + \left(\frac{u^2}{2}\right)_{\!\!x} = 0 \\ v_t + \left(\frac{v^2}{2}\right)_{\!\!x} - v_{xx} = 0 \\ \tilde v_t -\tilde v_{xx} = 0 \\ w_t + \left(\frac{w^2}{2}\...
7 votes
1 answer
291 views

Are large cardinals about more than just consistency?

The other day, I was reading the preface of Kanamori's The Higher Infinite and noticed that he says large cardinals provide a useful 'measuring stick' for consistency. That raised the question of ...
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4 votes
0 answers
295 views

Is this result of Hajnal and Juhász correct?

I am having some trouble with the following result presented here: Obviously I'm missing something, but I think from that result it could be shown that if $X$ is an infinite topological space, then $...
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2 votes
0 answers
147 views

Illustration of Liouville theorem

In a class, I'll teach the Liouville theorem for harmonic functions with finite Dirichlet integral. What kind of illustrations can I use to elucidate the meaning and proof of the theorem? Note that a ...
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0 votes
1 answer
120 views

Relationship between elliptic and parabolic problems and their discretizations

Let us consider the fully nonlinear problem $$ \begin{cases} F(x,u,Du,D^2 u) = 0 & \text{ in } \Omega \\ u=0 & \text{ in } \partial \Omega \end{cases} $$ Suppose that we know that the ...
6 votes
0 answers
2k views

Information theory, a categorical perspective [closed]

Note: B-variables were called streams in a previous version -> you won't understand the comments otherwise Definition of $B$-variables Theorem: Let $l_1\leq \dots\leq l_n$ be the lengths of a set ...
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8 votes
1 answer
2k views

Why does Arnold put Hardy on the same level as Bourbakists?

In the preface to his book "Lectures On Partial Differential Equations" Arnold writes: The effort to destroy this unnecessary scholastic pseudoscience is a natural and proper reaction of ...
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8 votes
1 answer
271 views

Formalisation of intuitive concepts in the language leading to mathematical progress

In his work, Albert Lautman thinks the genesis of some mathematical works as a dialectic that takes place between opposite notions, like between global and local. He argues that while those notions, ...
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1 vote
0 answers
37 views

Prove $u(t,\cdot) \in L^\infty(\mathbb R)$ for $u_t + f(u)_x = k u_{xx} + g(t,x)$ with $g \in L^\infty(0,T; L^2(\mathbb R))$

Let us consider the PDE $$u_t + f(u)_x = k u_{xx} + g(t,x),$$ with $k>0$, $g \in L^\infty(0,T; L^2(\mathbb R))$ and $f\in C^2(\mathbb R$) and $f$ strictly convex. Assume $u(0,\cdot) = u_0(\cdot) \...
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3 votes
0 answers
171 views

The etymologies behind certain topologies on the category of schemes

Certain topologies on the category of schemes (or perhaps certain appropriate subcategories thereof) are named rather aptly, e.g. Zariski, étale, fppf, fpqc, syntomic, smooth, v(aluation), etc., but ...
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1 vote
0 answers
310 views

Mathematical technicalities that few people know [closed]

I am looking for a list of mathematical technicalities that are not so well-known, even in the mathematical community. What I mean is, I am looking for examples of phenomenon where it is important to ...
21 votes
0 answers
469 views

Grothendieck's in-spirit-category-theoretic functional analysis?

I heard several times (for instance in these general lectures) that Grothendieck did functional analysis before he started doing algebraic geometry and category theory. It is said that at the time he ...
0 votes
0 answers
78 views

Structures preserve the existence of pushouts and pullbacks?

I'm interested in Ross Tate's idea as discussed on Examples of algorithms that came from category theory?. Where can I find stuff showing what kind of structures preserve the existence of pushouts and ...
-5 votes
1 answer
265 views

Soft question around the equivalent forms of Riemann hypothesis: an overview of the role of these equivalent forms [closed]

Along decades, in the course of investigations around the Riemann hypothesis (I add [1] as general reference) professional mathematicians discovered/stated many much (several hundred of) equivalent ...
4 votes
0 answers
140 views

What's the point of fine sheaves? (As opposed to soft ones)

Why do people care to define fine sheaves? What useful property do they have for which softness is not sufficient? some observations (because I feel guilty about a the one-line question): The point ...
37 votes
3 answers
4k views

How to read an article and make it actually useful?

I've been wondering for a while: how should mathematicians read an article in order to "take most" from it? For example, when I did my Master's thesis I based it on an article (I'm into ...
3 votes
2 answers
341 views

What do you call a scaled orthogonal map?

What do you call a linear map of the form $\alpha X$, where $\alpha\in\Bbb R$ and $X\in\mathrm O(V)$ is an orthogonal map ($V$ being some linear space with inner product)? Are there established names, ...
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1 vote
1 answer
40 views

Symmetry of optimal solutions to symmetric assignment problems

Is there a sound proof of or a counter example to the following conjecture: if $\boldsymbol{A}^T=\boldsymbol{A}$ is the cost-matrix of a bipartite assignment problem with unique optimal assignment, ...
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38 votes
5 answers
7k views

Is spherical trigonometry a dead research area?

When I was an undergrad, the field of spherical trigonometry was cited as a once-popular area of math that has since died. Is this true? Are the results from spherical trigonometry relevant for ...
16 votes
4 answers
6k views

Mathematicians learning from applications to other fields

Once upon a time a speaker at the weekly Applied Mathematics Colloquium at MIT (one of two weekly colloquia in the math department (but the other one is not called "pure")) said researchers ...
0 votes
0 answers
46 views

Critical points of function-curvature

As a side effect of the COVID-19 pandemic exponential growth became a buzz word that was "copy-pasted" a lot in public discussion. It may be assumed that the general public can't make sense ...
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55 votes
8 answers
5k views

Techniques for debugging proofs

After writing many proofs, most of which contained errors in their initial form, I have developed some simple techniques for "debugging" my proofs. Of course, a good way to detect errors in ...
1 vote
0 answers
129 views

examples of function difficult to prove to be $\geq0$?

I have often wondered whether there has ever come a point in your research, when you were confronted with an explicit real function $f(x_1,x_2,\ldots,x_n)$ and an explicitly defined compact set $S\...
15 votes
5 answers
2k views

Resources where I can find open problems in number theory along with their level of difficulty

NOTE: I will not accept an answer because a lot of answers are really good and if anyone want to post under this question later then they are most welcome to post as comment or answer because it will ...
30 votes
5 answers
5k views

How to pass on research posthumously

I've been independently researching math, for a while in the homeless community. While I am generally safe, situational realities (weather, equipment stress, health and especially how cities can ...
29 votes
10 answers
6k views

Is a come back to mathematical research possible?

Living in France, I am sometimes asked about Cédric Villani, a very popular figure here. Will he come back to mathematics ? The question becomes more relevant with the coming parliament elections (he ...
2 votes
1 answer
487 views

Is time spent without a result enough for authorship, in some cases?

Some time ago I had a chat with a friend (and colleague) about some statement I wanted to prove. I was (and am) sure the statement is true, but couldn't prove it. I described some of my attempts and ...
4 votes
0 answers
696 views

Algebraizing topology and analysis via condensed mathematics

I asked this question on Mathematics Stackexchange, but one of the users suggested that I ask this question at MathOverflow. I've just come across a Twitter thread by Laurent Fargues explaining a work ...
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45 votes
7 answers
7k views

Swimming against the tide in the past century: remarkable achievements that arose in contrast to the general view of mathematicians

I would like to ask a question inspired by the title of a book by Sir Roger Penrose ([1]). The germ of this is to ask about the role, if any, of the fashion in research of pure and applied mathematics....
2 votes
1 answer
158 views

Sieve theory through variational principles

Disclaimer: I'm just starting to read Sieve Methods by Halberstam and Richert, so my present knowledge of the subject is close to zero, but it made me wonder if some connection to physics could exist, ...

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