## Hot answers tagged motivation

168

I would have preferred not to comment seriously on Mochizuki's work before much more thought had gone into the very basics, but judging from the internet activity, there appears to be much interest in this subject, especially from young people. It would obviously be very nice if they were to engage with this circle of ideas, regardless of the eventual ...

139

I'll take a stab at answering this controversial question in a way that might satisfy the OP and benefit the mathematical community. I also want to give some opinions that contrast with or at least complement grp. Like others, I must give the caveats: I do not understand Mochizuki's claimed proof, his other work, and I make no claims about the veracity of ...

74

Last revision: 10/20. (Probably the last for at least some time to come: until Mochizuki uploads his revisions of IUTT-III and IUTT-IV. My apology for the multiple revisions. )
Completely rewritten. (9/26)
It seems indeed that nothing like Theorem 1.10 from Mochizuki's IUTT-IV could hold.
Here is an infinite set of counterexamples, assuming for ...

48

Algebraic K-theory originated in classical materials that connected class groups, unit groups and determinants, Brauer groups, and related things for rings of integers, fields, etc, and includes a lot of local-to-global principles. But that's the original motivation and not the way the work in the field is currently going - from your question it seems like ...

42

[The answer below is a response to an earlier version of the question that was rather different in certain respects. Minhyong Kim's answer gives excellent insight into ideas that Mochizuki had back in 2000 and that provide essential building blocks for the more recent work. But I still believe that it is too premature for a non-expert to seek insight into ...

33

Let me also try to give, in a modest complement to Minhyong Kim's great post, some additional remarks on Mochizuki's strategy. The idea that has led to the development of "Inter-universal Teichmuller theory for number fields" is certainly very beautiful, and was known to Mochizuki, along with the nature of the final estimate, already in 2000. (But let us ...

33

Let's say you have a resolution $0\to A\to J^0\to J^1\to\dots$ (of a module, a sheaf, etc.) If $J^n$ are acyclic (meaning, have trivial higher cohomology, resp. derived functors $R^nF$), you can use this resolution to compute the cohomologies of $A$ (resp. $R^nF(A)$). If $J^n$ are not acyclic, you get a spectral sequence instead, and that's the best you can ...

31

I'm going to interpret your question in the language of Gowers's "two cultures" essay as follows:
How does one get good at theory-building?
The process of developing a good theory can seem deceptively simple. One takes some definitions, perhaps by generalizing some known definitions, and deduces simple consequences of them. In comparison with the ...

30

I want to point out a bibliographical information that perhaps is not very well-known and can be taken as "evidence" for the possibility of applying anabelian geometry to the ABC conjecture successfully. However, I am not claiming that this is related in any sort of way to Mochizuki's work.
Here is the fact: There is a $\pi_1$ proof of the function field ...

29

Qiaochu links to a really nice article by Timothy Chow that says a lot about the mechanics of how to go from a filtered complex to its spectral sequence. Two questions that remain are, (1) why do filtered complexes show up so much, and (2) is there anything that you could do with a filtered complex other than compute its spectral sequence?
VA gives a very ...

23

A combinatorial motivation is the n! conjecture, whose proof by Haiman uses Hilbert schemes. An account of this work written by Haiman for the Current Developments in Mathematics conference in 2002 is at math.berkeley.edu/~mhaiman/ftp/cdm/cdm.pdf. Haiman emphasizes at the start of the paper that the main geometric results which had to be proved were ...

22

My personal story with this question is that, sometime in 2007, I wanted to find a project for a student I was mentoring at RSI (a program for high school students which produces real research) and thought some variant of the question "how can you visualize all the different geometric structures on a topological torus (elliptic curve/$\mathbb{C}$)?" would be ...

21

It may be helpful to say how I got into groupoids.
In the 1960s, I was writing a topology text and wanted to do the fundamental group of a cell complex, which required the van Kampen Theorem (I have now been persuaded to call this the Seifert-van Kampen theorem, as on wikipedia, so I call it SvKT). I was kind of irritated that this did not as then ...

19

Charles,
a couple of reasons why a complex algebraic geometer (certainly someone who is
interested in moduli spaces of vector bundles, as your profile tells me) might
at least keep
an open verdict on the stuff NC-algebraic geometers (NCAGers from now on) are trying to do.
in recent years ,a lot of progress has been made towards understanding moduli ...

19

I think a key point is that algebraic K-theory is defined not only for rings, but also for schemes (and other kinds of "generalized spaces" in algebraic geometry). If you believe that generalized (Eilenberg-Steenrod) cohomology theories are useful/interesting in algebraic topology, then it is also reasonable to think that they might be interesting in ...

18

Positivity of Kazhdan--Lusztig polynomials (and all the other positivity results in Kazhdan--Lusztig theory in general).
Consider the Hecke algebra $H_n(q)$. It is a particular deformation of the group algebra of the symmetric group (or some other Coxeter group). As such, it has a basis $T_w$ indexed by permutations, and multiplication is given by
...

18

In addition to being a nice example for abelian, $A_{\infty}$ and Calabi-Yau categories, and being a prototypical example for Generalized Donaldson - Thomas Invariants and the Wall Crossing Phenomenon, the quivers have a lot of applications in variours different fields. Since the question is applications in addition to representation theory, I'm listing a ...

17

Let me first try to answer a simpler question:
Why are long exact sequences so ubiquitous?
Almost anything that is written as a capital letter, followed by a subscript i or superscript -i, i an integer, and finally some stuff in parentheses, can be interpreted as πi of some spectrum (or sometimes space, as in nonabelian group cohomology, or maybe a ...

17

Dear Alex,
It seems to me that the general question in the background of your query on algebra really is the better one to focus on, in that we can forget about irrelevant details. That is, as you've mentioned, one could be asking the question about motivation and decision in any kind of mathematics, or maybe even life in general. In that form, I can't see ...

17

Alexander realized they were useful, then Conway. However, Jones clearly was the one
who really made a big bang with a skein relation. This allowed him to see a connection between the Jones polynomial and state sums in statistical mechanics. This was followed by HOMFLYPT, which might be the first time a skein relation was used to define an invariant rather ...

16

"How much would you subscribe to the statement that studying questions one finds interesting is something established mathematicians do, while younger ones are better off studying questions that the rest of the community finds interesting?"
Not at all. I don't think anyone, young or old, will find success by working on questions other than those they find ...

16

For a striking example of a classical result in algebraic geometry given a tropical proof, see A tropical proof of the Brill-Noether Theorem by Cools, Draisma, Payne, and Robeva. The original proof of this theorem (by Griffiths and Harris) involves subtle transversality arguments, which they are able to circumvent in this "combinatorial" proof. The new ...

15

As Pace Nielsen already posted, the strength of quiver theory is to provide easy examples and counterexamples.
The first applications are of course inside representation theory and ring theory, because Gabriel's Theorem states, that if you have a property of a finite dimensional algebra over an algebraically closed field that can be detected in the module ...

15

I think it's important to take a historical perspective. There was a time not so long ago when computers as we know them now did not exist. At that stage, coming up with a precise definition of an algorithm or of a Turing machine was a major advance, allowing one to build the earliest modern computers and begin the revolution that we take for granted ...

15

A student of mine asking for a motivation unmotivated by applications?
Haven't I taught you anything, Mike? (Joking of course.) However, perhaps one way to avoid
talking about future applications is to reflect on implicit past
applications and the explicitly stated original motivations. This is
not to take away from your answer, Craig, you know I agree ...

14

I suggest looking at the introduction to Waldhausen's original paper on algebraic K-theory (Algebraic K-theory of generalized free products, Part I, Ann. Math., 108 (1978) 135-204).
Waldhausen started out as a 3-manifold theorist, and he realized that certain phenomena in the topology of 3-manifolds would be explained if the Whitehead groups of classical ...

14

NEW !! (2013-02-21)
A Panoramic Overview of Inter-universal Teichmüller
Theory
By
Shinichi Mochizuki
http://www.kurims.kyoto-u.ac.jp/~motizuki/Panoramic%20Overview%20of%20Inter-universal%20Teichmuller%20Theory.pdf

13

First, recall the slogan:
Small constructions are good for making calculations, but large constructions are good for proving theorems.
K-theory is certainly a large construction.
In general, K-theory seems to turn up in topology when the following slogan holds:
Chain compex good; homology bad.
You can often construct exactly the same invariant using ...

13

If you want to understand some collection of objects, naturally you should also want to understand all maps between them. Since factors have no two-sided ideals, every map between factors is an inclusion. So to understand maps between factors is the same thing as understand subfactors. (This is a way in which factors are noncommutative analogues of fields, ...

12

As a particular application of algebraic K-theory, let me mention the intersection product on regular schemes. Let X be a regular scheme over spec Z. Then, one can use the Quillen spectral sequence and Adam's operations on K-theory to produce an intersection product on the Chow groups tensored with Q. To my knowledge, this is the first definition of an ...

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