**4**

votes

**1**answer

251 views

### Alexandrov angles in Riemannian manifolds

Dear all, I am teaching a course in Riemannian geometry, and I would like to prove some comparison theorems in the next lessons, building on the well-known theory of Jacobi fields, and of Rauch ...

**13**

votes

**4**answers

2k views

### motivating geometric representation theory

I am wondering if there is a good motivation for geometric representation theory from within the questions of classical representation theory.
In other words, I'd be curious to see something using ...

**8**

votes

**6**answers

1k views

### Intuitionistic logic as quantization of classical logic?

A classically trained mathematician is more likely to be familiar (at least anecdotally) with an area of mathematical physics such as deformation quantization than with Intuitionistic logic. It is ...

**5**

votes

**5**answers

2k views

### What does a mathematician expect from mathematics education? [closed]

Consider that my question is not a personal and/or subjective question. Perhaps, you have hired a mathematics educator in your department and you are interested in finding a way to communicate with ...

**2**

votes

**2**answers

247 views

### Stronger theorem not resulting from proof analysis

Suppose that we proved $\varphi$ from a theory $T$. Often we ask whether or not we could have proved $\varphi$ with a weaker theory, to find out we usually analyze the proof and try to figure out ...

**3**

votes

**2**answers

212 views

### Equivalent definitions of ample bundles

M. Atiyah in "VECTOR BUNDLES OVER AN ELLIPTIC CURVE" defined ample line bundle $E$ on $X$ as satisfying the following conditions:
Canonical map $H^0(X, E)\to E_x$ is surjective for any $x\in X$.
...

**28**

votes

**13**answers

2k views

### Great mathematics books by pre-modern authors

Last summer, I read Euclid's Elements, and it was an eye-opening experience; I had assumed that three thousand years' difference would make the notation incomprehensible and the reasoning alien, but ...

**16**

votes

**2**answers

721 views

### Age of Stochasticity?

One user on MSE made an interesting question, which was unanswered so I suggested him to post it here but he refused for personal reasons and said I could ask it here.
The question is this:
Today ...

**7**

votes

**7**answers

975 views

### Gelfand representation and functional calculus applications beyond Functional Analysis

I think it is fair to say that the fields of Operator Algebras, Operator Theory, and Banach Algebras rely on Gelfand representation and functional calculus in a crucial way.
I am curious about ...

**4**

votes

**5**answers

394 views

### What is “Data” involved in a mathematical construction?

What exactly do mathematicians mean when they refer to "the data" involved in a construction?
I've encountered this many times and I can usually figure out what's going on, but I am curious about the ...

**3**

votes

**1**answer

429 views

### The average number of people that can sit on a bench of a given length.

Let me explain what I mean:
The width of the average person varies, perhaps with a normal distribution.
Given a specific variance, how many people (on average) can sit side-by-side on a bench of a ...

**9**

votes

**3**answers

814 views

### Is there an observer dependent mathematics? [closed]

Is there any field of mathematics that deals with the role of the observer? E.g., some formulation in which a set is changed, in some unspecified way, when it is observed? Or maybe some philosophy of ...

**3**

votes

**1**answer

713 views

### The shortest mathematical paper [duplicate]

I was looking at the paper Zum Hilbertschen Nullstellensatz [1] and wondered if there was a shorter mathematical paper than this one. A colleague of mine rumored about a number-theoretic paper where ...

**13**

votes

**1**answer

1k views

### Euler's mathematics in terms of modern theories?

Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "one can see in ...

**10**

votes

**4**answers

1k views

### How to refer to a theorem that you have shown to be wrong

I am writing a paper about a flaw that I found in a published paper. There, the statement is called “Theorem 2”. In my paper, I am reproducing the other paper’s definitions, and steps leading towards ...

**7**

votes

**2**answers

2k views

### Two questions about combinatorics journals

Hello,
I have two questions regarding combinatorics journals. I hope that this is the right place for such questions.
Which combinatorics/DM journals would you consider as the "top tier"?
I tried ...

**3**

votes

**2**answers

2k views

### On mentioning recommenders' names in cover letter for postdoctoral applications

If I want to apply for a postdoctoral job, can I mention the name of my recommenders in my cover letter just to bolster my application, particularly when I am sure that the people who will read my ...

**0**

votes

**2**answers

384 views

### Correct definition of the sequence of natural numbers with set theory, but without counting or measuring size [closed]

This question may appear banal, but there seems to be more than meets the eye; a common glitch is to explain numbers by the "size" of sets without saying how to measure or compare the size of sets.
...

**14**

votes

**6**answers

1k views

### Mathematical Paper That Just Links Two Different Fields of Sciences

I have a soft question that is interesting for me in some aspects. I appreciate your answers and comments about it.
Four years ago, one of my friends in MIT, in the biology lab, had working on ...

**20**

votes

**8**answers

1k views

### Self-containing structures

(This question is partly inspired by What is inter-universal geometry?.)
I have absolutely no background in Teichmuller theory or any related subject, but what I can follow of Mochizuki's description ...

**7**

votes

**4**answers

817 views

### Coboundaries and Gluing in Cech Cohomology - Intuition?

I'm trying to develop an intuition for Cech cohomology geometrically, but am currently failing. A lot of people seem to say that the groups $H^n$ measure obstructions to gluing local sections to get ...

**10**

votes

**1**answer

1k views

### How many proofs of the Weil conjectures are there?

I hope this this is not seen as too much as jumping on the band-wagon, but here goes.
Deligne's proof of the last of the Weil conjectures is well-known and just part of a huge body of work that has ...

**29**

votes

**0**answers

2k views

### The Work of Pierre Deligne

In this biography of Pierre Deligne, there is a quote of Jacques Tits which says that "quite a few of his best ideas have never been written!".
What are some of his best ideas that you have heard of ...

**1**

vote

**1**answer

871 views

### PhD in operator algebras and non-commutative geometry [closed]

I do not know whether it is a good place to ask this question or not.
I want to PhD in operator algebras and non-commutative geometry. What are the best places in the world for that? I want a good ...

**8**

votes

**1**answer

685 views

### Topology, the board game

Edit: I am reposting this question fom math.stackexchange.com; there may be some professors here who have more experience teaching topology.
This is a math education question that I've been thinking ...

**9**

votes

**1**answer

414 views

### New research on coding in reverse mathematics?

Coding is obviously a fundamental tool in reverse mathematics, and practitioners take care to both demonstrate the correctness of their coding mechanisms and point out their limitations. Harvey ...

**4**

votes

**2**answers

1k views

### How long should one wait for a report before asking about its status? [closed]

I apologize if this question is too soft or if its answers would be too subjective for this site. However, I would find it highly useful to have such a question answered on this site, and I believe ...

**1**

vote

**1**answer

273 views

### If X is a Haussdorf topological space and R and equivalence relation on X, when is X/R Haussdorf?

I was wondering if there are some necessary and sufficient conditions for the quotient space to be Haussdorf. I have been trying a little for a while, but I only got very restrictive sufficient ...

**13**

votes

**3**answers

615 views

### Is there an editors checklist for mathematics?

I always have trouble with editing math papers for publication. I know there are plenty of checklists for English exposition but is there one for math specific exposition errors?

**9**

votes

**4**answers

1k views

### Role of applications in modern mathematics [closed]

Older days scientists were universalists and philosophy, physics and mathematics were a part the same question - understanding the world.
Nowadays one may get feeling that the role of applications ...

**7**

votes

**3**answers

620 views

### Formal writing: numbers under 10

I've been tasked with proofreading an Engineering/Mathematics thesis paper. I was always told that numbers under 10 should be spelled out (one, two, three, ...) but I was wondering if this rule holds ...

**5**

votes

**2**answers

652 views

### Faculty Handbook: Mentoring Undergraduates in Research and Scholarship

A few days ago I was asked by the director of the Center for Undergraduate Research and Scholarship at Georgia Regents University (formerly known as MCG and Augusta State) to contribute an article for ...

**5**

votes

**0**answers

773 views

### “Must read ”papers on analytic number theory

Question: What would be some must-read
papers for an aspiring analytic number
theorist? In other words, what are the papers that any analytic number theorist would have read? (Background: ...

**22**

votes

**7**answers

2k views

### Should one attack hard problems?

When I applied for a PhD student position I had an interview with two professors. Somehow we touched the problem if $P$ is $NP$ and, once we got there, for some reason both professors made it clear ...

**7**

votes

**2**answers

1k views

### How should a professor feel peace of mind when a student leaves academia? [closed]

I apologize if this is the wrong forum for this question -- if so, could somebody please point me to the correct one?
I'm a professor who recently started advising graduate students, and I'm trying ...

**7**

votes

**0**answers

157 views

### Tangent space, metrics etc. on simplicial sets

Is there a way to attach some sensible notion of tangent space to a simplicial set? If yes, is it possible to transfer typical local data from differential geometry such as metrics to this setting?
...

**49**

votes

**5**answers

2k views

### Math Annotate Platform?

Suppose most mathematical research papers were freely accessible online.
Suppose a well-organized platform existed where responsible users could write comments on any paper (linking to its doi, ...

**15**

votes

**1**answer

576 views

### Mendelson's *Mathematical Logic* and the missing Appendix on the consistency of PA

A very soft question, but I hope not out of order here.
In the first edition of Elliott Mendelson's classic Introduction to Mathematical Logic (1964) there is an appendix, giving a version of ...

**3**

votes

**1**answer

618 views

### Math major at 36 [closed]

I decided to go for math at 36. Is this idea possible? I studied literature, political science and international relations and still I am not really sure what I am doing.
Since I was kid, I was not ...

**38**

votes

**19**answers

6k views

### Mathematicians whose works were criticized by contemporaries but became widely accepted later

Gauss famously discarded Abel's proof that an algebraic equation of degree five or more cannot have a general solution (Abel himself had rejected divergent series as the work of the devil). Cantor's ...

**5**

votes

**3**answers

875 views

### group of diffeomorphisms of a manifold

How much has been the group of diffeomorphisms of a manifold " been studied.
I got this information from wiki.
" Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra ...

**3**

votes

**2**answers

415 views

### To what extent should a pure mathematician care about the meaning of things? [closed]

I have asked this on SE before, But i DO hope that you kindly give me more insight about this question since it's very important to me
Having come from engineering background, I am kinda obsessed ...

**1**

vote

**1**answer

217 views

### Bounds for conjugacy classes of subgroups

My question is rather general and the reason for this is that I am primarily interested in finding sources where I can read more about this type of problems. So here goes:
To begin with, I want to ...

**49**

votes

**11**answers

3k views

### Why is Set, and not Rel, so ubiquitous in mathematics?

The concept of relation in the history of mathematics, either consciously or not, has always been important: think of order relations or equivalence relations.
Why was there the necessity of singling ...

**32**

votes

**31**answers

6k views

### Trichotomies in mathematics

Added. Thanks to all who participated! Let me humbly apologize to those who were annoyed (quite understandably) by this thread, deeming it nothing more than an exercise in futility. If you thought the ...

**23**

votes

**9**answers

2k views

### Is the empty graph a tree?

This is a boring, technical question that I stumbled upon while making a contribution to Sage. I would still like to hear a constructive answer so hopefully the question does not get closed.
The ...

**6**

votes

**4**answers

863 views

### fourier analytic proofs

While searching through Mathoverflow, I found out a fourier analytic proof of the Isoperimetric Inequality.Also, by google search I found a fourier analytic proof of Quadratic Reciprocity theorem.I ...

**3**

votes

**1**answer

461 views

### Integral representation of the modified Bessel functions of the second kind and asymptotic expansion

The modified Bessel function (Macdonald function) $K_\alpha(z)$ is known to have the following asymptotic expansion for large positive $z$:
$$
K_\alpha(z)=\sqrt{\frac{\pi}{2z}}e^{-z}\sum_{k=0}^\infty ...

**25**

votes

**14**answers

1k views

### Discovering and selecting conferences

Last summer, there were several excellent summer schools in my field that I learned of only after the application date. The events I did attend were chosen without too much care. I'm planning for the ...

**5**

votes

**5**answers

2k views

### Financial Mathematics Books

Hi, I am looking for a good reference to start reading about Mathematical Finance/Financial Engineering. I have a good background in Math but have no idea at all in Finance.