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Results tagged with big-picture
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Questions designed to get an overview of a specific subject or body of results or to understand the relations among similar definitions, techniques or concepts appearing in different sub-fields of mathematics. While such questions by their very nature sometimes cannot be made very narrow and focused, it can be helpful to keep in mind that the design of MathOverflow does not make it a good fit for questions that are too broad.
52
votes
Breakthroughs in mathematics in 2021
Advancing mathematics by guiding human intuition with AI, Nature 600, 70 (2021), stands out because it represents the first significant advance in pure mathematics generated by artificial intelligence …
31
votes
Interpretations and models of permanent
In quantum physics, the state of non-interacting particles that are fermions is represented by a determinant, while for bosons it is a permanent. The computational complexity of the evaluation of a pe …
- 188k
26
votes
What is a cumulant really?
It might help to take a broader perspective: in some contexts (notably quantum optics) the emphasis is not on cumulants but on factorial cumulants, with generating function $h(t)=\log E(t^X)$. While c …
- 188k
13
votes
What are the "hot" topics in mathematical QFT at the time?
I am not sure that "hot topic" is an advisable criterion for a Ph.D. research project, since this will typically mean that easy/doable questions have been done and only the hard/intractable questions …
12
votes
Meaning of a quantum field given by an operator-valued distribution
Q: Why do we need operator-valued distributions instead of operator-valued functions?
Consider the commutator of two canonically conjugate quantum fields, for example, for $\mathbf{p},\mathbf{p}'\in\m …
- 188k
10
votes
Where can square roots come from when they are not distances?
The square root of SWAP (which is the only two-qubit operation needed to realize a universal quantum computation) has no "distance" interpretation I can think of.
9
votes
Accepted
Monte Carlo integration
this is perhaps more a comment than an answer, but here is one expert opinion on whether one can base the foundations of integration theory on Monte Carlo integration:
Another way to obtain contin …
- 188k
7
votes
Dyson's invitation: Opportunities in juxtaposition of incompatibles
My favourite "juxtaposition of a pair of incompatible concepts that is acknowledged but unexplained": the Riemann hypothesis and spectral theory, as phrased in the Hilbert-Pólya conjecture that the im …
6
votes
What are some interesting relationships between pi and phi?
Q: Is there any way to use the golden ratio $\Phi$ to define $\pi$ ?
as proven by John Baez.
- 188k
3
votes
Accepted
What are Penrose Tilings, and how do they relate to Quasicrystals?
As explained here, there is an infinite number of distinct tilings that can be constructed using the three sets of tiles introduced by Roger Penrose (rhomb, kite-dart, boat-star). The distinction betw …
- 188k
3
votes
How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to hi...
Ramanujan's reasoning is analysed in detail by Berndt, in his discussion of chapter 6 of Ramanujan's notebooks. Ramanujan starts from the Euler-MacLaurin summation formula,
$$\sum_{k=1}^x f(k)=C +\int …
- 188k
2
votes
A geometric interpretation of the fractional Fourier transform
For the record, to answer Q2, let me write down the Mehler kernel for the square root of the Fourier transform,
$$K(x,y)=(2\pi)^{-1/2}\sqrt{1-i}\exp\left[{\tfrac{1}{2} i \left(x^2+y^2-2 \sqrt{2} x y\r …
- 188k
2
votes
Where do these divergent integrals appear in mathematics and physics?
Perhaps the most "famous" divergent integral in physics is the Casimir vacuum energy of the electromagnetic field,
$$\int_0^\infty x^2\sqrt{1+x^2}\,dx"="-\frac{1}{10},$$
and the related class of integ …
- 188k