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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15 views

How to make the quiz as fair as possible?

This question comes to me after meeting two delegates of my course. Before each class, students have to go through a quiz consisting of $3$ questions. Let $A, B, C$ be the questions in increasing ...
26
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15answers
4k views

Lunch seminars for PhD students

The problem that I would like to ask about is metamathematical, but I hope the question is appropriate. I would like to know if there exist mathematical departments that run a regular seminar for all ...
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1answer
186 views

Geometry book recommendation

Context and mathematical maturity: I have knowledge of the usual engineering math courses, meaning differential+integral+vector calculus, linear algebra, probability and statistics, etc. and some pure ...
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0answers
749 views

A mysterious paper of Stallings that was supposed to appear in the Annals

In Stallings's paper Stallings, John, Groups with infinite products, Bull. Amer. Math. Soc. 68 (1962), 388–389. he briefly discusses how to prove "several generalizations" of Brown's ...
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78 views

Books on limiting properties of matrices with growing size

This question has been posted on Math-Se previously. I am studying asymptotic properties of the Projection Matrix $$ H_n=X'(X'X)^{-1}X $$ By the Gerschgorin disc theorem, the bounds on the ...
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3answers
1k views

Theoretical results on neural networks

With this question I'd like to have a recollection of theoretical rigorous results on neural networks. I'd like to have results that have been settled, as opposed to hypothesis. As an example, this ...
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Thomason's "open letter" to the mathematical community

In the 1989, Bob Thomason left his CNRS position in Orsay and moved to Paris VII. It was during this period that he composed his "Open Letter" to the mathematical community. The letter ...
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1answer
295 views

Are there some interesting propositions independent with ZF+V=L that do not increase consistency strength?

In some MO questions such as this and this, Hamkins gave some examples that is independent with ZF+V=L, however, all of them increase the consistency strength. Are there some propositions P, which is ...
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2answers
419 views

Can the theory of elliptic functions developed from purely geometric considerations?

I always had this question, but was unable to get a definitive answer to it. There is the theorem of division of the arc length of the lemniscate with ruler and compass. So I always wondered, is it ...
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3answers
2k views

How can I seek help in preparing a very long research article for publication?

Some background first. I recently graduated with a master's degree in applied mathematics. During graduate school I began working on a paper, which I continued to work on post-graduation. A complete ...
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0answers
76 views

Self study guide to Hamiltonian Monte Carlo

I was wondering if anybody has a suggested self-study path to understand the mathematical aspects on Hamiltonian Monte Carlo. In this paper The Geometric Foundations of Hamiltonian Monte Carlo it is ...
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3answers
1k views

Should every modern day mathematician care about category theory? [closed]

As far as I know, category theory is used mainly in topology. I have a dislike towards category theory, similar to my dislike of Bourbakism, and want to avoid it as much as I can. However, the head of ...
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Continuous decomposition of permutation-invariant set functions

The seminal machine learning paper Deep Sets (Zaheer et al., 2017) discusses representations of permutation-invariant functions on real tuples, or (multi)set functions. Given a countable set $X$ and a ...
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0answers
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A question on convergence rates of Fourier series and strict convergence

Consider BV functions on a torus. The Fourier partial sum using the first $n$ coefficients will converge to the function at every point of continuity, as $n\to\infty$. The convergence rate is $O(1/n)$....
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1answer
214 views

Sheaf of chain complexs glued by chain homotopy equivalences

Let $(X,\mathcal O_X)$ be a locally ringed space with an open covering $\mathscr U$. Suppose: For any $U\in\mathscr U$, we have a chain complex $(C_U, d_U)$ such that $C_U$ is an $\mathcal O_X(U)$-...
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23answers
6k views

Golden ratio in contemporary mathematics

A (non-mathematical) friend recently asked me the following question: Does the golden ratio play any role in contemporary mathematics? I immediately replied that I never come across any mention of ...
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2answers
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John von Neumann's remark on entropy

According to Claude Shannon, von Neumann gave him very useful advice on what to call his measure of information content [1]: My greatest concern was what to call it. I thought of calling it '...
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2answers
1k views

Math overdose? Professional advise how to cope with it [closed]

I'm a PhD student, currently working on my Thesis. Over the years I have many time encountered a problem. Maybe professional mathematicians know what I'm talking about? When I study a math topic, ...
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0answers
237 views

Does Arnold say that Hardy is responsible for Ramanujan's untimely death? [closed]

Vladimir Arnold in his book Yesterday and Long Ago, Springer (2007) writes: When I resided at Cambridge as a senior fellow of Trinity College, Indian colleagues told me some details of Ramanujan'...
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3answers
1k views

Is computer algebra or symbolic computation an active area of research?

I'm interested in doing PhD in computer algebra or symbolic computation, and was wondering if this is an active area of research? Would this area of research also help me in the transition to ...
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1answer
806 views

Why are W-types called "W"?

Why are W-types called "W"? Probably "W" means either "wellordered" or "wellfounded". (Martin-Löf uses the term "wellorder".) But these are notions ...
4
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2answers
646 views

Research topics in representation theory of algebras [closed]

I was wondering what are some of the hot topics in quiver representation or representation theory of algebras that can lead to good mathematics and is important to many mathematicians and top ...
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0answers
181 views

A new method for processing music scores?

I have developed a method and python script: https://github.com/githubuser1983/algorithmic_python_music which allows the user to input a midi file and then chose a few numbers as parameters, and the ...
6
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1answer
195 views

Comparing Kripke-Joyal semantics of toposes to model-theoretic satisfaction

Let $\mathcal E$ be a topos and $\varphi$ a statement formulating a property of toposes. There are two ways of checking whether $\mathcal E$ satisfies $\varphi$: Consider the first-order language $L$ ...
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3answers
848 views

Role of univalence in homotopy group calculations

This book has a section with proofs of the fact $\pi_1(S^1)=\mathbb Z$ using the univalence axiom. They are a bit too technical for me at the moment to read, but I want to understand the following (...
29
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21answers
5k views

Theorems with many distinct proofs

I was told that whenever one learns a new technique, it is a good idea to see if one can prove a well-known theorem using the new technique as an exercise. I am hoping to build a list of such theorems ...
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0answers
315 views

Context of set theory in which one doesn't have to worry about size issues

In this beautiful talk by Colin McLarty, McLarty quotes Grothendieck: It would be nice to have a context where one doesn't add any real axioms to set theory, and yet one can work with categories ...
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1answer
312 views

Rejection of a paper because of not suitable level of rigor without a single example of a mathematical error/imprecision [closed]

Question 1. A paper was rejected because of 'not suitable level of rigour' without a single example of a mathematical error/imprecision. What can the author do in this situation? It sounds a matter of ...
2
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1answer
144 views

Articles providing informal discussions on major developments on a particular problem?

Some time ago, when I was studying the Kervaire-Milnor paper (Groups of Homotopy Spheres I), I found the following survey article very helpful in guiding my studies: Milnor, Differential Topology 46 ...
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0answers
69 views

Name for the theory of words with equal length, prefix, successors

I've worked with this theory for a while, but I've never been quite sure what to call it: $$(\Sigma^*, =_{el}, \preceq, (S_a)_{a \in \Sigma})$$ Where $\Sigma^*$ is the set of finite words on finite ...
6
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1answer
378 views

Coarsening arXiv subject tags by journal preferences

Sorry if this question is too soft or off-topic, but I wasn't sure where else to get feedback from a broad range of subfields. Short version: How should one condense the 32 existing arXiv math subject ...
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4answers
1k views

Proving a binomial sum identity

QUESTION. Let $x>0$ be a real number or an indeterminate. Is this true? $$\sum_{n=0}^{\infty}\frac{\binom{2n+1}{n+1}}{2^{2n+1}\,(n+x+1)}=\frac{2^{2x}}{x\,\binom{2x}x}-\frac1x.$$ POSTSCRIPT. I like ...
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2answers
302 views

Applications of ZFA-Set Theory

The set theory with atoms (ZFA), is a modified version of set theory, and is characterized by the fact that it admits objects other than sets, atoms. Atoms are objects which do not have any elements. ...
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1answer
3k views

Who is Mrs. Gerber?

This question on a theorem in information theory called Mrs. Gerber's lemma piqued my curiosity. Who is this individual, and why the "mrs." ? A quick Google search was not informative, ...
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0answers
32 views

Length of shorter directed cycle relative to average with oppositely oriented counter-part

let $G(V,E)$ be a symmetric cyclic graph with positively weighted edges; derive now from $G$ a network $N(V,A)$ by replacing each undirected edge $e_{ij}$ of $G$ by a pair of antiparallel arcs $\...
3
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1answer
773 views

What's the point of differential geometry? [closed]

I've been self studying differential geometry for a little while now (4-6 months). I am learning from Lee's Introduction to Smooth Manifolds, and I just don't quite get the point of the subject. Why ...
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0answers
234 views

Quantum mechanics outside $L^{2}$ spaces

To this day, it is known that a satisfying mathematical formulation of quantum field theory is far from sight, even though some noninteracting theories can be described in rigorous mathematical ...
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6answers
4k views

Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be false

What are some examples of serious mathematical theory-building around hypotheses that are believed or known to be false? One interesting example, and the impetus for this question, is work in number ...
20
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0answers
728 views

Connes's Absolute Geometry and Lurie's Spectral Algebraic Geometry

Alain Connes and Caterina Consani seem to be currently working on "absolute algebraic geometry", which is a kind of "algebraic geometry over the sphere spectrum" (https://arxiv.org/...
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1answer
265 views

Does there exist a "citation distance" calculator for papers or authors?

This question is not directly a mathematical question, but I am interested in whether there exists a calculator akin to an Erdős number calculator. The main difference is that I am not interested in ...
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3answers
451 views

Papers on history and philosophy of mathematics suitable for master's students

In the fall, I will give a course called "Perspectives in Mathematics". This is a mandatory course at our master's program in mathematics (including applied mathematics and statistics). The ...
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0answers
155 views

Are there some algorithms which have high consistency strength?

Are there some algorithms, their time complexity is relatively good, for example polynomial time. And the correctness of them has high consistency strength. And these algorithms shouldn't able to ...
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1answer
341 views

Can mathematics help in defining free-will? [closed]

In the celebrated Free Will Theorem of Conway and Kochen it is made use of "free will" without giving a "mathematical definition" of it. The definition of the experimenter is the &...
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0answers
216 views

The most simple proof of projective determinacy

I want to read the proof of projective determinacy. But every proof I could find (martin-steel original, koepke's, the proof in schindler's book, martin's new book) is too long. Are there a simple ...
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0answers
35 views

Choice of splitting in domain decomposition algorithms

When solving a PDE numerically by domain decomposition methods, what is the "optimal way" to split the domain? Are there any results stating that a particular partition of the domain yields &...
3
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1answer
726 views

Summary of ``Almost All Orbits of the Collatz Map Attain Almost Bounded Values"

Terence Tao's 2019 paper ``Almost all Orbits of the Collatz map attain almost bounded values" is pretty famous. However, it's also long and complicated. I think there are useful techniques to ...
6
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1answer
478 views

Practical Benefits of HTT/univalent foundations for assisted proofs

I'm trying to understand what the claimed practical benefits of HTT/univalent foundations are for doing computer assisted proofs and while I've seen a lot of claims of benefits they all seem to be ...
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6answers
5k views

Peer review 2.0

I have an idea for a website that could improve some well-known difficulties around peer review system and "hidden knowledge" in mathematics. It seems like a low hanging fruit that many ...
20
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2answers
1k views

sci.math.research archive?

Does there exist an archive somewhere of posts to the USENET newsgroup sci.math.research? The best approximation I'm aware of is Google Groups. However, despite ...
51
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10answers
6k views

Changes forced by the pandemic

The Covid-19 pandemic has changed our work-lives in ways few of us could have anticipated. These exceptional circumstances have forced each one of us and each one of our institutions to adapt, ...

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