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Results tagged with big-list
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Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
4
votes
Accepted
What are some (popular) references on variants of the classical gambler's ruin problem that ...
Multi-dimensional generalizations (one player against $d$ other players) are explored by P. Lorek in Generalized Gambler's Ruin Problem: explicit formulas via Siegmund duality.
For the analogue on a r …
- 188k
6
votes
Accepted
Current state of the art in geometric complexity theory
As explained by Scott Aaronson, in his 2017 overview of Geometric Complexity Theory, the core problem it tries to solve is to prove Valiant's conjecture VP$\neq$VNP that the permanent requires exponen …
- 188k
21
votes
Who says understanding physics helps mathematicians? (A reference request) [Take the word "w...
Michael Atiyah, On the Work of Edward Witten:
In his hands physics is once again providing a rich source of
inspiration and insight in mathematics. Of course physical insight
does not always lead to …
10
votes
Where do root systems arise in mathematics?
The eigenvalue distribution functions of random matrices in different universality classes are determined by the multiplicities of the restricted roots of the corresponding symmetric spaces, see Rand …
11
votes
What are applications of asymptotic freeness of random matrices?
Here are some applications of free probability of random matrices:
Neural networks: The asymptotic freeness assumption plays a fundamental role in the study of the propagation of spectral distributio …
- 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
1
vote
One-step problems in geometry
Von Neumann's law for two-dimensional cell growth
A soap froth or polycrystalline slab is modeled by a two-dimensional network of piecewise smooth curves, joined at vertices at internal angles of $2\p …
3
votes
What are the applications of spin geometry?
I understand the question as requesting applications of the geometry of spin manifolds outside of pure mathematics. There are many, starting with the Dirac equation for the spinor of a relativistic el …
9
votes
What are some interesting applications/corollaries of Kleene's Recursion theorem?
Kleene’s amazing second recursion theorem by Moschovakis contains:
A list of some (18) of the most significant applications of the
Kleene's Second Recursion Theorem, in a kind of “retrospective exhib …
- 188k
7
votes
PhD dissertations that solve an established open problem
Since the OP mentions Gauss, this entry could be an appropriate addition to the list:
Manjul Bhargava's PhD thesis, Higher composition laws (2001), concerns a problem going back to Gauss. In the ninet …
1
vote
What is the most "informative" Yes/No math question you know?
A general strategy:
Since you state that the yes/no answer will come with a proof, I presume the proof will be understandable by humans, so it will need to contain much background material. I would ar …
27
votes
Modern results that are widely known, yet which at the time were ignored, not accepted or cr...
The classification theorem for three-dimensional convex polyhedra known as Steinitz's theorem first appeared in a 1922 publication of Ernst Steinitz. Because it did not use the language of graphs it r …
27
votes
Interesting and surprising applications of the Ising Model
An application of the Ising model in social sciences is to voter models: The dynamics of the Ising model tries to align neighbouring spins, similarly, perhaps, to humans deciding on their political, r …
- 188k
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 …
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 …