Questions tagged [big-picture]

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.

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(Non?)-linearity of the consistency strength ordering in ZF

Much of the research taking place in set theory, is related to the classification of sentences according to their consistency strength relative to ZF. In order to be more specific, we say that for all ...
Marios Koulakis's user avatar
11 votes
5 answers
2k views

what can be said about the choice of a prior in Bayesian statistics?

When reading about the Bayesian approach to statistics, priors are an important component of the whole methodology. Yet, it seems like priors are chosen without any specific theoretical motivation. ...
bayesianlearner's user avatar
11 votes
2 answers
1k views

What are important examples of filtered/graded rings in physics?

Hi, what comes to the mind of a physicist, when he hears words like filtered ring and associated graded? What do these guys describe? What are basic/typical/illuminating examples in physics? Of ...
Jan Weidner's user avatar
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11 votes
2 answers
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Motivation for birational geometry

I'm interested in how do people that work in birational geometry view their field — specifically, what are the kinds of geometric questions (as opposed to commutative-algebraic questions) that ...
roymend's user avatar
  • 221
11 votes
4 answers
1k views

Surreal numbers, ultrapowers of $\Bbb R$, ordinal-valued functions and the slow-growing hierarchy

Philip Ehrlich's paper “The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small”, The Bulletin of Symbolic Logic 18 (1) 2012, pp. 1-45. claims as a theorem that, in NBG, $\...
Mike Battaglia's user avatar
11 votes
1 answer
633 views

Poincare duality spaces vs. manifolds via lifting maps, the obstruction theory and the role of simply connectedness

Suppose that we are given a topological space $X$: assume for simplicity that $X$ is compact we want to adress the following question: Is it true that one can find a manifold $M$ which is homotopy ...
truebaran's user avatar
  • 9,140
11 votes
1 answer
288 views

Why do convex polytope options constrict with dimension, rather than expand?

There are an infinite number of regular polygons in the plane, five regular polyhedra, six regular polytopes in $\mathbb{R}^4$, and then three regular polytopes in every dimension $d > 4$. There ...
Joseph O'Rourke's user avatar
11 votes
5 answers
4k views

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 ...
11 votes
0 answers
402 views

What is the motivation for a Frobenius manifold?

A Frobenius manifold is a type of manifolds with extra structure. The main examples are quantum cohomology (viewed as a space itself), GBV algebras, the ``Saito'' examples arising from singularities (...
Pulcinella's user avatar
  • 5,507
11 votes
0 answers
314 views

Hausdorff dimension and von Neumann dimension

There are two subjects in which non-integral dimensions appear: fractal geometry: consider the well-known Hausdorff dimension of fractals. von Neumann algebra: consider a type ${\rm II_1}$ ...
Sebastien Palcoux's user avatar
11 votes
0 answers
508 views

Floer cohomology from mapping spaces of $\infty$ categories

There's a meta-observation (of Urs Schreiber, who attributes it to Ken Brown and Lurie) that 'cohomology theories come from mapping spaces of $(\infty,1)$ categories'. This is described in detail at ...
zzz's user avatar
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10 votes
5 answers
1k views

Visual representation of mathematical research interrelationships

I remember seeing a visualization in the form of a 2d (nodal) graph of all areas of academia, with math, physics and engineering over in one section, connecting in an arc to the central area of ...
DoubleJay's user avatar
  • 2,353
10 votes
2 answers
841 views

Why is modular forms applicable to packing density bounds from linear programming at $n\in\{8,24\}$?

Sphere packing problem in $\mathbb R^n$ asks for the densest arrangement of non-overlapping spheres within $\mathbb R^n$. It is now know that the problem is solved at $n=8$ and $n=24$ using modular ...
VS.'s user avatar
  • 1,816
10 votes
2 answers
1k views

Integrable dynamical system - relation to elliptic curves

From seminar on kdV equation I know that for integrable dynamical system its trajectory in phase space lays on tori. In wikipedia article You may read (http://en.wikipedia.org/wiki/Integrable_system): ...
kakaz's user avatar
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10 votes
1 answer
362 views

Can $N!$ be computed in less than $\mathcal{O}(N)$ operations?

The standard algorithm to compute the factorial function $N!$ via repeated multiplications has complexity $\mathcal{O}(N)$, in the model in which each operation costs 1, no matter how many digits the ...
user6873235's user avatar
10 votes
1 answer
905 views

Analogue to Serre spectral sequence for cofiber sequences and homotopy

(This is a follow-up question to this one). As it is nicely outlined in an answer to this question, homotopy groups behave well with respect to (Serre)-fibrations and (co)homology groups behave well ...
roger123's user avatar
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10 votes
0 answers
2k views

How much of modern algebraic geometry is there in modern complex(algebraic, analytic, differential) geometry?

Good day to you, people of mathoverflow. I'll get to the point. I wonder how much of modern abstract algebraic geometry is there in modern complex geometry? What do I mean by complex geometry? ...
Seidarkman's user avatar
9 votes
5 answers
5k views

Advice on choosing an area of specialization [closed]

I'm not sure if this is an appropriate question for MO, but I figured it couldn't hurt to ask. I'm a second year graduate student, currently gearing up to construct a committee and syllabus for my ...
9 votes
4 answers
2k views

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 ...
9 votes
2 answers
1k views

Did Joseph Doob prove that random sequences don't exist?

In the book "The Mathematical Experience" it says: "An infinite [binary] sequence $x_1, x_2, \ldots$ is called random in the sense of von Mises if every infinite sequence $x_{n_1}, x_{n_2}, \ldots$...
teil's user avatar
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9 votes
2 answers
1k views

What is the relation of the Kuznetsov-Bruggeman trace formula and the Selberg trace formula?

I have read that there is an elementary way to show that the above mentioned trace fromulas are equivalent in the sense, that each of them can be derived directly from the other. There should exist a ...
Marc Palm's user avatar
  • 11.1k
9 votes
2 answers
1k views

Differentiation of functions over graphs

In short: There are various ways to define differentiation over a graph. I am trying to get the big picture, like a more complete and structured bestiary. Definitions. Let $G=(V,E)$ be a directed ...
Matthieu Latapy's user avatar
9 votes
2 answers
501 views

categorification of q-series

In his talk, S. Gukov asked two questions: What is the categorification of a $q$-serie ? How to associate to a 3-manifold a $q$-serie ? As far as I understand, he was looking for a bigarded ...
mathphys's user avatar
  • 1,609
9 votes
1 answer
628 views

Three theorems on the number of nonzero coefficients of a polynomial

The number of positive real roots of a polynomial with real coefficients is strictly smaller than the number of nonzero coefficients of the polynomial. This is an immediate corollary of Descartes' ...
Seva's user avatar
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9 votes
0 answers
736 views

Counting Lattice Points in Real Polytopes

Suppose one did have an exact formula for the number of $\mathbb{Z}^n$-lattice points intersecting an arbitrary dilate of a (not necessarily rational) finite, closed and convex $n$-polytope. As a ...
9 votes
0 answers
860 views

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 $g$...
José Figueroa-O'Farrill's user avatar
8 votes
1 answer
1k views

Geometric intuition behind this chain homotopy

My question has to do with the chain homotopy that appears in Lee's Introduction to Topological Manifols and Rotman's Introduction to Algebraic Topology proofs that the inclusion $$C_\bullet^\mathcal{...
Akerbeltz's user avatar
  • 484
8 votes
1 answer
922 views

Steps in Geometric Complexity Theory

GCT purports to provide a program to show that $NP \not \subset P/poly$. At the high level what are the steps involved in the program and what stage is each step in? What difficulties currently are ...
Turbo's user avatar
  • 13.6k
8 votes
1 answer
901 views

Motives over the complex numbers versus mixed Hodge structures

Let $\mathsf{MM}(\mathbf C)$ be the hypothetical category of mixed motives over the complex numbers, and consider the realization functor $\Phi : \mathsf{MM}( \mathbf C) \to \mathsf{MHS}$ to integral ...
Dan Petersen's user avatar
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8 votes
3 answers
2k views

What are Penrose Tilings, and how do they relate to Quasicrystals?

The question is in the title, but let me elaborate a little. Background Penrose Tilings are really pretty and satisfy some remarkable properties. For instance, I believe the following is true: even ...
Vidit Nanda's user avatar
  • 15.4k
8 votes
1 answer
6k views

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: Take ...
vonjd's user avatar
  • 5,855
8 votes
2 answers
1k views

visualizing what's going on in based homotopy theory, et al.

I'm reading J.P. May's Concise Course in Algebraic Topology, and I'm having a lot of trouble visualizing how things work in Chapter 8, "Based cofiber and fiber sequences". Of course this is pretty ...
Aaron Mazel-Gee's user avatar
8 votes
3 answers
406 views

How does one identify properties of objects with good "inheritance"?

When you are dealing with a very general object like a topological space or a ring, usually you impose an additional condition (such as compact Hausdorff or Noetherian) with the property that the ...
Qiaochu Yuan's user avatar
8 votes
1 answer
1k views

Recursion theory from the standoint of category theory

It is (I believe) a very easy exercise to prove that the general recursive functions over the natural number object $N$ form a category. But what sort of category is it? From the fact that one can ...
Thomas Benjamin's user avatar
8 votes
2 answers
807 views

Categorical Invariants

I apologize in advance if this question seems too vague. In many topology courses, concepts like the fundamental group and homology groups are introduced as a means of distinguishing non-...
Daniel Miller's user avatar
8 votes
1 answer
2k views

Why is the Arthur trace formula so powerful?

Considering the Arthur trace formula, why are the sort of convolution operators, whose "normalized traces" are given in geometric terms and spectral terms, actually able to distinguish all ...
Marc Palm's user avatar
  • 11.1k
8 votes
3 answers
1k views

"Right" Way of Introducing Modular Forms to Undergraduate Audience?

I am giving an extremely short talk (~12 minutes) in a few weeks in which I need to be able to introduce and motivate the idea of a classical modular form to an audience of undergraduates in as short ...
Nick Salter's user avatar
  • 2,770
8 votes
1 answer
752 views

What are the uses of coefficient systems for arithmetic cohomology theories?

In topology when studying a space with non-trivial fundamental group it becomes important to consider homology and cohomology with coefficients in representations of the fundamental group, i.e. local ...
user50948's user avatar
  • 183
8 votes
1 answer
1k views

Getting the story of Dynkin and Satake diagrams straight

I've been trying to teach myself the theory of Lie groups. The sources I've been reading reference Lie algebras in the context of Dynkin and Satake diagrams, but not Lie groups (which I am more ...
Sam D.'s user avatar
  • 89
8 votes
1 answer
324 views

Existence of Randomized polynomial time algorithm and some arithmetic analog of $ACC^0$ circuits for Factoring of primitive polynomials before LLL?

Before LLL came along in $1982$ there was no deterministic polynomial (in degree and number of bits in coefficients) way to factor square free primitive polynomials in $\Bbb Z[x]$. However was there ...
user avatar
8 votes
1 answer
225 views

Classifying two-faces of four-polytopes

Motivation: This question is related to my study of hyperbolic Coxeter polytopes. In general, if one put some restrictions on the type of their dihedral angles (say, all dihedral angles are equal to $\...
SashaKolpakov's user avatar
8 votes
1 answer
1k views

Motivation for $C^*$-algebras

I just gave a presentation on exotic group $C^*$-algebras and someone asked why these are studied. I could answer that they can be used to construct $C^*$-algebras with certain properties. However, I ...
Emiel Lanckriet's user avatar
8 votes
0 answers
361 views

What is the relationship (if any) between constructivism, finitism and predicativism?

The terms “constructivism”, “finitism” and “predicativism” refer to ideas / currents in the philosophy of mathematics (or loosely defined conditions on a system of logic) that I think I understand ...
Gro-Tsen's user avatar
  • 29.8k
8 votes
0 answers
337 views

Canonical Time Evolution for Type $II_{1}$-Factors?

This question was spurred by the answer of Steve Huntsman to the MO question here. The Tomita-Takesaki modular automorphism group gives rise to a canonical time evolution on a type $III$ factor (...
Jon Bannon's user avatar
  • 6,977
7 votes
5 answers
2k views

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,(on ...
Unknown's user avatar
  • 2,817
7 votes
4 answers
3k views

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 ...
Josh's user avatar
  • 1,402
7 votes
1 answer
3k views

Why we study Geometric invariant theory?

I am trying to learn Geometric invariant theory like it was introduced by Mumford. But I do not have a strong motivation and so I want to know the reason of studying Geometric invariant theory. I just ...
riu_ss's user avatar
  • 87
7 votes
2 answers
1k views

Mazur's torsion theorem on elliptic curves and its generalisations

I want to study Mazur's torsion theorem for elliptic curves over $Q$ and its generalizations for number fields, i.e., papers by Kamienny, Kenku & Momose, Filip Najman. So please suggest to me what ...
7 votes
3 answers
547 views

Abstract Relation between Presehaves and Simplicial Sets

Every presheaf (let's say on a topological space) comes with restriction maps. The open sets of a topological space are ordered by inclusion and these inclusions yield the restrictions. Now a sheaf ...
Konrad Voelkel's user avatar
7 votes
3 answers
1k views

spectacular applications of functional analysis in resolutions of apparently unrelated problems

What are some of the spectacular applications of functional analysis to apparently unrelated problems.One that comes to my mind is Per Enflo's resolution of Hilbert's 5th problem.There are also ...

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