Tagged Questions

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,...

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Why certain diophantine equations are interesting (and others are not) ?

It is quite clear why certain differential equations, among the jungle of possible diff equations that is possible to conceive, are studied: some come from physical problems, or from "spontaneous" ...
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Why do we want to have orthogonal bases in decompositions?

In the decompositions I encountered so far, we all had orthogonal set of bases. For example in Singular Value Decomposition, we had orthogonal singular right and left vectors, in [discrete] cosine ...
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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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Quantized Calculus, the Hilbert Transform and the Upper Half-Plane

Given a *-algebra $\mathcal{A}$ over $\mathbb{C}$, a Fredholm module is a *-representation $\pi$ of $\mathcal{A}$ as operators on a Hilbert space $\mathcal{H}$ along with a self-adjoint operator $F$ ...
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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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How do you decide whether a question in abstract algebra is worth studying?

Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my ...
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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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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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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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Integrals from a non-analytic point of view

I've mentioned before that I'm using this forum to expand my knowledge on things I know very little about. I've learnt integrals like everyone else: there is the Riemann integral, then the Lebesgue ...
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Morava's “Motives and cell bundles”?

Hello, do you know more about, or some exposition of Morava's talk?
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Applications of Math: Theory vs. Practice

I have a problem: I learned about a lot of the applications of mathematics from academics. Neither they nor I have had much contact with the "real world" to go and see for ourselves how mathematics ...
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Linear/Non-linear sigma model

This is slightly an open-ended invitation to discuss references and reasons for excitement about the linear and non-linear sigma model. I gauge from some other interactions that it has considerable ...
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Graphical representation of mathematical structures (in the spirit of unified modeling language)

In software engineering the unified modeling language ("UML") is a well established technique for providing overview of complex systems and an efficient means of communicating about them. There are ...
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P not eq. NP news?: [closed]

"Vinay Deolalikar. P is not equal to NP. 6th August, 2010 (66 pages 10pt, 102 pages 12pt). Manuscript sent on 6th August to several leading researchers in various areas. Confirmations began arriving ...
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What is Realistic Mathematics?

This post is partially about opinions and partially about more precise mathematical questions. Most of this post is not as formal as a precise mathematical question. However, I hope that most readers ...
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Analytic tools in algebraic geometry

This is not a very precise question, but I hope it will get some good answers. As someone with a background in smooth manifold theory, I have experienced algebraic geometry as a beautiful but foreign ...
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The unprecedented success of the “intersection” operator

You might think that the title is an overstatement of a well-known fact but it is the best title I can come up with for the wonders the intersection operator does in some fields of math. Recently,(...
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Is there a common genesis for ADE classifications?

Recall that a certain type of object admits an ADE classification if there is a notion of equivalence relative to which equivalence classes of objects of the given type can be placed in one-to-one ...
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Is the dual notion of a presheaf useful?

It seems that there is a common theme in mathematics where, if we want to find out about a category C, then we look at $\hat{C}$ (the category of contravariant functors from $C$ to $Set$). There are ...
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Intuitions/connections/examples for “eigen-*”

There are many concepts in mathematics that begin with the German word "eigen": eigenvector, eigenvalue, eigenspace, eigenstate, eigenfunction, eigensystem etc. (to name just the most important (?) ...
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Modern algebraic geometry vs. classical algebraic geometry

Can anyone offer advice on roughly how much commutative algebra, homological algebra etc. one needs to know to do research in (or to learn) modern algebraic geometry. Would you need to be familiar ...
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Idea of presheaf cohomology vs. sheaf cohomology

Let $X$ be a topological space and $U$ an open cover of $X$. In this thread Angelo explained beautifully how presheaf cohomology (Cech cohomology) relates to sheaf cohomology: The zeroth Cech ...
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Intuition and/or visualisation of Ito integral/Ito's lemma

Riemann-sums can e.g. be very intuitively visualized by rectangles that approximate the area under the curve. See e.g. Wikipedia:Riemann sum The Ito integral has due to the unbounded total variation ...
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Defining variable, symbol, indeterminate and parameter

Are there precise definitions for what a variable, a symbol, a name, an indeterminate, a meta-variable, and a parameter are? In informal mathematics, they are used in a variety of ways, and often in ...
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What's wrong with the surreals?

Of all the constructions of the reals, the construction of the surreals seems the most elegant to me. It seems to immediately capture the total ordering and precision of Dedekind cuts at a ...
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Intuitive and/or philosophical explanation for set theory paradoxes

Every student of set theory knows that the early axiomatization of the theory had to deal with spectacular paradoxes such as Russel's, Burali-Forti's etc. This is why the (self-contradictory) ...
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Geometric interpretation of the fundamental groupoid

Motivation The common functors from topological spaces to other categories have geometric interpretations. For example, the fundamental group is how loops behave in the space, and higher homotopy ...
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To what extent is it true that “number theory = mathematics”? [closed]

In a thought-provoking answer to this MO question, Kevin Buzzard and several commentators have described a multitude of ways in which number theory is related to other parts of mathematics. It seems ...
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Intuitive crutches for higher dimensional thinking

I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows- An engineer, a physicist, and a mathematician are discussing how to visualise four ...
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A Sketch of “Esquisse d'un Programme”

I'm refering, of course, to Grothendieck's ambitious program available fully here: http://www.math.jussieu.fr/~leila/grothendieckcircle/EsquisseFr.pdf The text is, as described in its title, a sketch,...
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What do you use categorical glueing/sconing/Freyd covers for?

In the theory of programming languages and structural proof theory, one of the handiest techniques we have available is a method called "logical relations", in which you can prove properties of ...
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Why are modular forms interesting?

Well, I'm aware that this question may seem very naive to the several experts on this topic that populate this site: feel free to add the "soft question" tag if you want... So, knowing nothing about ...
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What is the origin of the formula for the Lie derivative along a Killing vector?

Background Let $(M,g)$ be an $n$-dimensioal riemannian manifold. A vector field $X$ on $M$ is said to be a Killing vector if the flow it generates is an isometry; that is, it preserves the metric $g$...
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Equality vs. isomorphism vs. specific isomorphism

This question prompted a reformulation: What is a really good example of a situation where keeping track of isomorphisms leads to tangible benefit? I believe this to be a serious question because ...
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Big picture concerning Ito integral, Stratonovich integral and standard results in probability theory

I am confused and don't get the big picture concerning the connection between Ito integral Stratonovich integral Standard results in probability theory concerning skewed distributions. Example: ...
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Natural setting for characteristic classes?

In my mind, algebraic topology is comprised of two components: Chain complex information, which is intrinsic information concerning how your object may be built up out of simple "lego blocks". ...
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Are there any “homotopical spaces”?

This is a somewhat vague question; I don't know how "soft" it is, and even if it makes sense. [Edit: in the light of the comments, we can state my question in a formally precise way, that is: "Is ...
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Is it true that the only interesting topologies are metric topologies and weak topologies?

In "Infinite dimensional analysis, A hitchhikers guide" by Aliprantis and Border, they write that these 2 classes of topologies "by and large include everything of interest". @Pete Clarke: I was ...
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Are rings really more fundamental objects than semi-rings?

The discovery (or invention) of negatives, which happened several centuries ago by the Chinese, Indians and Arabs, has of course be of fundamental importance to mathematics. From then on, it seems ...
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Path integrals outside QFT

The main application of Feynman path integrals (and the primary motivation behind them) is in Quantum Field Theory - currently this is something standard for physicists, if even the mathematical ...
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How to find ICM talks?

I am very interested in reading some and skimming through the list of invited talks at the International Congress of Mathematicians. Since the proceedings contain talks supposedly by top experts in ...
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Is the ABC conjecture known to imply the Riemann hypothesis?

I once heard from a graduate student that the ABC conjecture implies the Riemann hypothesis. I can't find a reference for this, but given the department the student is from I tend to believe he might ...
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Mathematics of path integral: state of the art

I was told that one of the most efficient tools (e.g. in terms of computations relevant to physics, but also in terms of guessing heuristically mathematical facts) that physicists use is the so called ...
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Doing geometry using Feynman Path Integral?

I have often heard in the folk-lore that Feynman Path Integral can be used to compute geometric invariants of a space. Coming from a background of studying Quantum Field Theory from the books like ...
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Does the presence of cocycle conditions indicate the existence of an underlying cohomology theory?

Motivation: We have two examples: (Abelian) Kummer theory (resp. Artin-Schreier theory) has a hidden cohomology theory given by Galois cohomology. The cocycle conditions become clear when you look ...
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Why are tensors a generalization of scalars, vectors, and matrices?

Take two vector spaces $V$ and $W$ over a field $F$. One may form the tensor product $V\otimes W$ and it fulfills an universal property. Elements of $V\otimes W$ are called tensors and they are linear ...
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What are some fundamental “sources” for the appearance of pi in mathematics?

I thought it might be fun to ask this question as a way of celebrating Pi Day. One way in which people popularize pi is that they say that even though it's defined in terms of properties of a circle, ...