Most Popular
1500 questions
92
votes
1 answer
The mathematical theory of Feynman integrals
It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly.
Arguably, they are the most important such tool. Briefly, the question I'd like to ask is this: what are the existing proposals for…
algori
- 23,231
92
votes
13 answers
Deep learning / Deep neural nets for mathematician
I am interested in finding out the math ideas behind the technologies that are under the umbrella of "Deep Learning" or "Deep neural nets".
Most of the papers/books that are often quoted in papers/online as references are not written in a very…
aa12
- 591
- 1
- 5
- 4
92
votes
7 answers
Lost soul: loneliness in pursing math. Advice needed.
This is a atypical question for the forum. I'd like to get some advice on whether I should keep pursuing Math in the traditional route, i.e. get a PhD, do research & teach, etc.
Due to financial constraints, I worked almost full time and did not…
Flora
- 351
91
votes
4 answers
Can every manifold be given an analytic structure?
Let $M$ be a (real) manifold. Recall that an analytic structure on $M$ is an atlas such that all transition maps are real-analytic (and maximal with respect to this property). (There's also a sheafy definition.) So in particular being analytic is…
Theo Johnson-Freyd
- 52,873
91
votes
8 answers
Is there a natural random process that is rigorously known to produce Zipf's law?
Zipf's law is the empirical observation that in many real-life populations of $n$ objects, the $k^\text{th}$ largest object has size proportional to $1/k$, at least for $k$ significantly smaller than $n$ (and one also sometimes needs to assume $k$…
Terry Tao
- 108,865
- 31
- 432
- 517
91
votes
10 answers
Reflection principle vs universes
In category-theoretic discussions, there is often the temptation to look at the category of all abelian groups, or of all categories, etc., which quickly leads to the usual set-theoretic problems. These are often avoided by using Grothendieck…
Peter Scholze
- 19,800
- 3
- 98
- 118
91
votes
8 answers
Has incorrect notation ever led to a mistaken proof?
In mathematics we introduce many different kinds of notation, and sometimes even a single object or construction can be represented by many different notations. To take two very different examples, the derivative of a function $y = f(x)$ can be…
Mike Shulman
- 65,064
91
votes
6 answers
Why didn't Vladimir Arnold get the Fields Medal in 1974?
As you all probably know, Vladimir I. Arnold passed away yesterday. In the obituaries, I found the following statement (AFP)
In 1974 the Soviet Union opposed Arnold's award of the Fields Medal, the most prestigious recognition in work in…
Harun Šiljak
- 1,514
91
votes
4 answers
How do you select an interesting and reasonable problem for a student?
I am interested in how to select interesting yet reasonable problems for students to work on, either at Honours (that is, a research-based single year immediately after a degree) or PhD.
By this I mean a problem that is unsolved but for which there…
Gordon Royle
- 12,288
91
votes
2 answers
$A$ is isomorphic to $A \oplus \mathbb{Z}^2$, but not to $A \oplus \mathbb{Z}$
Are there abelian groups $A$ with $A \cong A \oplus \mathbb{Z}^2$ and $A \not\cong A \oplus \mathbb{Z}$?
Martin Brandenburg
- 61,443
91
votes
2 answers
Is it possible to express $\int\sqrt{x+\sqrt{x+\sqrt{x+1}}}dx$ in elementary functions?
I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here:
Is it possible to express the following indefinite integral in elementary…
Vladimir Reshetnikov
- 6,709
91
votes
70 answers
Old books still used
It's a commonplace to state that while other sciences (like biology) may always need the newest books, we mathematicians also use to use older books. While this is a qualitative remark, I would like to get a quantitative result. So what are some…
Lennart Meier
- 11,877
90
votes
3 answers
Is every sigma-algebra the Borel algebra of a topology?
This question arises from the excellent question posed on math.SE
by Salvo Tringali, namely, Correspondence
between Borel algebras and topology.
Since the question was not answered there after some time, I am
bringing it up here on mathoverflow in…
Joel David Hamkins
- 224,022
90
votes
10 answers
How do I check if a functor has a (left/right) adjoint?
Because adjoint functors are just cool, and knowing that a pair of functors is an adjoint pair gives you a bunch of information from generalized abstract nonsense, I often find myself saying, "Hey, cool functor. Wonder if it has an adjoint?" The…
Harrison Brown
- 12,543
90
votes
11 answers
What are possible applications of deep learning to research mathematics?
With no doubt everyone here has heard of deep learning, even if they don't know what it is or what it is good for. I myself am a former mathematician turned data scientist who is quite interested in deep learning and its applications to mathematics…
Jason Rute
- 6,247