Questions tagged [big-list]
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.
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What is your favorite ADE-style classification? [duplicate]
Possible Duplicate:
ADE type Dynkin diagrams
What is your favorite ADE-style classification?
Here ADE style is to be understood in a very broad sense. A classification which is not precisely ...
56
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13
answers
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Cardinalities larger than the continuum in areas besides set theory
It seems that in most theorems outside of set theory where the size of some set is used in the proof, there are three possibilities: either the set is finite, countably infinite, or uncountably ...
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undergraduate logic textbook
I am going to teach the standard undergraduate Logic course for math and engineering majors. What are good (bad) text-books and why. I have not taught that course for a while and wonder if there are ...
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6
answers
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Still Difficult After All These Years
I think we all secretly hope that in the long run mathematics becomes easier, in that with advances of perspective, today's difficult results will seem easier to future mathematicians. If I were ...
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127
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Most memorable titles
Given the vast number of new papers / preprints that hit the internet everyday, one factor that may help papers stand out for a broader, though possibly more casual, audience is their title. This view ...
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What recent discoveries have amateur mathematicians made?
E.T. Bell called Fermat the Prince of Amateurs. One hundred years ago Ramanujan amazed the mathematical world. In between were many important amateurs and mathematicians off the beaten path, but what ...
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What is a good introductory text for moduli theory?
Hi,everyone. I am looking for an introductory textbook on moduli theory,about the background on algebraic geometry,I have read Hartshorne chapter1~4. could you please show some good books or roadmap ...
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Examples where the analogy between number theory and geometry fails
The analogy between $O_K$ ($K$ a number field) and affine curves over a field has been very fruitful. It also knows many variations: the field over which the curve is defined may have positive or zero ...
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Proof synopsis collection
I hate to keep going with the big lists, but the question about one-sentence summaries of topics/areas spurred this question...and I just can't help myself!
Definition (Fraleigh): A proof synopsis ...
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31
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Extremely messy proofs
Currently in my undergraduate courses I am being taught how to set up various machinery using slick, short proofs and then how to apply that machinery. What I am not being taught, largely, is what ...
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1
answer
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Classification Problems [closed]
I was thinking about the famous question in philosophy of mathematics: "When are two proofs the same?" and I was wondering if we could somehow "classify" proofs by establishing some sort of functorial ...
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Examples of categorification
What is your favorite example of categorification?
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Wonderful applications of the Vandermonde determinant
This semester I am assisting my mentor teaching a first-year undergraduate course on linear algebra in Peking University, China. And now we have come to the famous Vandermonde determinant, which has ...
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Examples of prime numbers in nature [closed]
Finding primes in signals is seen as a sign of some kind of intelligence - see e.g. the role of primes in the search for extraterrestrial life (see e.g. here). This is because there are relatively few ...
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How and how much do the notations and diagrams influence our understanding of mathematical concepts?
How and how much do the notations and diagrams influence
our understanding of mathematical
concepts?
This question was stimulated by the MathOverflow questions Thinking and Explaining and ...
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81
answers
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Suggestions for good notation
I occasionally come across a new piece of notation so good that it makes life easier by giving a better way to look at something. Some examples:
Iverson introduced the notation [X] to mean 1 if X is ...
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More upper/lower semi-continuous functions in (algebraic) geometry?
The notion of upper/lower semi-continuity is sometimes encountered in algebraic geometry.
Here by upper semi-continuity one means a function on a topological space $f:X\rightarrow S$ with value in ...
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Piece of a sequence
Suppose we are given a representation of a finite series of natural numbers:
$\sum_{i=0}^N{c_i x^i}$
The representation is essentially an expression that is a rational function of two polynomials.
...
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3
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Concavity of $\det^{1/n}$ over $HPD_n$.
One of my beloved theorems in matrix analysis is the fact that the map $H\mapsto (\det H)^{1/n}$, defined over the convex cone $HPD_n$ of Hermitian positive definite matrices, is concave. This is ...
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Examples of ZFC theorems proved via forcing
This is an old suggestion of Joel David Hamkins at the end of his answer to this question: Forcing as a tool to prove theorems
I just noticed it while trying to understand his answer. But indeed it ...
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67
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Awfully sophisticated proof for simple facts [closed]
It is sometimes the case that one can produce proofs of simple facts that are of disproportionate sophistication which, however, do not involve any circularity. For example, (I think) I gave an ...
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applications of Tate-Poitou duality
What are nice applications of Tate-Poitou duality?
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Applications of Stacks
I've been aware of stacks since grad school, and I can usually follow in rough lines a discussion about stacks, but I've often wondered what particular (purely!) scheme-theoretic argument or theorem ...
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Link Repository of International Dissertations
This question (cry for help?) grew out of Colin Tan's question: does anyone have a copy of schmid’s effective work on hilbert 17th? which was a request for a copy of an Habilitationsschrift.
We've ...
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31
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Errata for Atiyah–Macdonald
Is there a good list of errata for Atiyah–Macdonald available? A cursory Google search reveals a laughably short list here, with just a few typos. Is there any source available online which lists ...
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PDEs as a tool in other domains in mathematics
According to the large number of paper cited in MathSciNet database, Partial Differential Equations (PDEs) is an important topic of its own. Needless to say, it is an extremely useful tool for natural ...
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Hecke-algebras in your field of mathematics
(How) do Hecke-algebras arise naturally in your field of mathematics and why are they important?
How would you define them and how do you think about them?
e.g. generators and relations, functions ...
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Generalized notions of solutions in various areas of mathematics
In many areas of mathematics (PDE, Algebra, combinatorics, geometry) when we have difficulty in coming with a solution to a problem we consider various notions of "generalized solutions". (There are ...
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Brownian motion, martingales, Markov Chains - Rosetta Stone
What are the most
fundamental/useful/interesting ways in
which the concepts of Brownian motion,
martingales and markov chains are
related?
I'm a graduate student doing a crash course in ...
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Ingenuity in mathematics
[This is just the kind of vague community-wiki question that I would almost certainly turn my nose up at if it were asked by someone else, so I apologise in advance, but these sorts of questions do ...
158
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48
answers
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Generalizing a problem to make it easier
One of the many articles on the Tricki that was planned but has never been written was about making it easier to solve a problem by generalizing it (which initially seems paradoxical because if you ...
125
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23
answers
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Collection of equivalent forms of Riemann Hypothesis
This forum brings together a broad enough base of mathematicians to collect a "big list" of equivalent forms of the Riemann Hypothesis...just for fun. Also, perhaps, this collection could include ...
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Number of ways to construct mathematical objects
This question stems from this other one mentioning 7 ways of constructing smooth manifolds. I quote:
At the 2010 Clay Research Conference, Gromov explained that we know of only 7 different methods ...
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Theorems first published in textbooks?
According to Wikipedia, the Bohr-Mollerup Theorem (discussed previously on MO here) was first published in a textbook. It says the authors did that instead of writing a paper because they didn't think ...
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Book on symplectic geometry
Can someone please tell me some introductory book on symplectic geometry? I have no prior idea of the subject but I do know about Lagrangian and Hamiltonian dynamics (at the level of Landau-Lifshitz ...
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answers
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Jokes in the sense of Littlewood: examples? [closed]
First, let me make it clear that I do not mean jokes of the
"abelian grape" variety. I take my cue from the following
passage in A Mathematician's Miscellany by J.E. Littlewood
(Methuen 1953, p. 79):
...
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How common is it for universities to create new positions for dual hires?
Let's say you happen to be a mathematician, and your spouse is also an academic and in a humanities field in which there are very few jobs advertised. Assuming that you can convince universities to ...
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Online math history lectures
This question is somewhat similar to this: Best online mathematics videos?
I'm using the word "history" loosely here. What I'm looking for are those lectures that put various mathematical ...
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23
answers
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Thinking and Explaining
How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words differ, ...
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Singularities of space curves: Open question lists?
For plane curve singularities most questions have been answered,
in large part due to the Newton-Puiseux expansion.
I've heard that there are a number of open problems regarding
space curve ...
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Demonstrating that rigour is important
Any pure mathematician will from time to time discuss, or think about, the question of why we care about proofs, or to put the question in a more precise form, why we seem to be so much happier with ...
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What is your favorite isomorphism? [closed]
The other day I was trying to figure out how to explain why isomorphisms are important. I pulled Boyer's A History of Mathematics off the bookshelf and was surprised to find that isomorphism isn't ...
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Undergraduate Probability Topics
I am teaching undergraduate probability this semester, and I am looking for some suggestions about inspiring applications that could be reasonably covered over the course of two one-hour lectures or ...
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1
answer
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What are the topological properties of the metric space retained (inherited) for its completion
Let $(X,d)$ be a metric space and $(\bar{X},\bar{d})$ its completion.
There is a list of topological properties
Wikipedia - Topological property
Does anybody know list which of them are retained (...
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11
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What are some open problems in algebraic geometry?
What are the open big problems in algebraic geometry and vector bundles?
More specifically, I would like to know what are interesting problems related to moduli spaces of vector bundles over ...
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13
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What are the big problems in probability theory?
Most branches of mathematics have big, sexy famous open problems. Number theory has the Riemann hypothesis and the Langlands program, among many others. Geometry had the Poincaré conjecture for a long ...
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Journals for undergraduates
Are there math journals that are aimed for undergraduates? I don't mean here journals where students can publish their papers, but journals that publish introductory articles that an undergraduate can ...
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What category without initial object do you care about?
Recently I have been listening to some constructions that have been designed to accommodate categories without an initial object. The speaker has given some idea of a category or two that he cares ...
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4
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Are there extensive tables of Fourier transforms available online?
I hope this is suitable for MO... I was wondering if someone can suggest a website (or some online document) containing an $extensive$ table of Fourier transforms? When I try obvious Google searches, ...
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Facts from algebraic geometry that are useful to non-algebraic geometers
A professor of mine (a geometric topologist, I believe) once criticized the core graduate curriculum at my institution because it teaches all sorts of esoteric algebra, but does not include basic ...