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41

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 ...


29

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 ...


28

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 ...


22

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 ...


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 ...


18

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 ...


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 ...


17

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 ...


17

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 ...


16

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 ...


16

Following on from the Galois theory example of Johannes, one straightforward way to produce an explicit polynomial with non-soluble Galois group over ${\mathbb Q}$ is to use an irreducible quintic with exactly three real roots, which necessarily has Galois group $S_5$. To check that an explicit polynomial (such as $x^5-4x+2$ if I am not mistaken, I am typing ...


14

An interesting application of calculus is the elementary polynomial case of Mason's ABC theorem. This yields, for instance, a completely trivial proof of the polynomial case of FLT (Fermat's Last Theorem). That this works so effectively for polynomials (functions) vs. numbers is due to the fact that for functions we have available the derivative, which ...


14

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 ...


14

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 ...


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

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 ...


12

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 ...


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 ...


12

The theory of iterated integral gives a mixed Hodge structure on rational homotopy of a variety. In the case of the fundamental group, as far as I know, one can only detect the nilpotent completion of the fundamental group. (At least, this is the only part of $\pi_1$ that people work with in a motivic context --- see e.g. Deligne's paper on the thrice ...


12

As was mentioned above, many moduli spaces have a quiver description; one of the most famous example is given by Nakajima quiver varieties, which are defined for any quiver (and they serve as the main example of symplectic complex varieties which are resolutions of an affine variety), but when the quiver is the affine quiver of ADE type, they describe moduli ...


11

As mentioned by Emerton, iterated integrals only work well for unipotent representations of $\pi_1(X,x)$. The reason for this is that differential forms are abelian objects: for paths $\gamma_i$, and a closed 1-form $\alpha \in \Omega^1(X)$, $$ \int_{\gamma_1\gamma_2} \alpha = \int_{\gamma_1} \alpha + \int_{\gamma_2} \alpha = \int_{\gamma_2\gamma_1} ...


11

If you are interested in actual computations using modern algebraic geometry, there are plenty to be had in Gromov-Witten theory and enumerative geometry. For example, Kontsevich's formula counting rational plane curves is a famous example. The proof itself does not use any scheme theory, but it was based on the structure of a very delicate object called ...


10

In general I don't think there's anything easy about nearby and vanishing cycles. However, I tend to find it enlightening to just consider their topology. Namely, if f:X \to C is a function on a complex algebraic (or analytic) variety, then the stalk cohomology of the nearby cycles functor applied to some complex of sheaves F at a point x \in f^{-1}(0) is ...


10

I think the original interest in subfactors probably came about from the search for invariants of factors, which in turn was partly motivated by the free group problem. The free group problem is one of those questions that drives mathematicians crazy; it's straightforward to state, and hard to do anything with. It's described below. In general, von ...


10

If you are interested in representation theory of finite dimensional algebras (including group algebras and their blocks—and everyone is interested in representations of groups, even if they don't know it), then considering quivers (and bound quivers) is a natural thing to do: all algebras (up to the appropriate equivalence relation relevant in the context ...


10

I guess one answer is there's an isomorphism between your group and $$SL_n \mathbb C \times_{\mathbb Z /n\mathbb Z} SO_2$$ My notation means take the product and mod out by the diagonally embedded copy of $\mathbb Z/n \mathbb{Z}$. The embedding of $\mathbb Z/n\mathbb Z$ in $SO_2$ is as the cyclic subgroup of order $n$, and the embedding in $SL_n \mathbb ...


9

A cool application which I can somehow appreciate is Van den Bergh's proof of dimension $3$ case of the Bondal-Orlov conjecture that two birational smooth Calabi-Yau varieties $X,X'$ have equivalent derived category $D(X) \sim D(X')$. Note that since one can construct the pluricanonical ring from $D(X)$ , this is a generalization of the Batyrev's conjecture ...


8

Here are some partial answers to some of the questions posed: Any finite group has an outer action on the hyperfinite $II_1$-factor $R$. This means from an inclusion of finite groups $H\leq G$, we can use the crossed product construction to form the subgroup subfactor $N=R\rtimes H\subseteq R\rtimes G=M$ whose index is $[G\colon H]$. The principal graph of ...


8

Here was my explanation for why I'm interested in Subfactors, which has a different flavor than my other answer for why "people" are interested in them: Subfactor planar algebras are just unitary versions of something very natural. For example, they're unitary versions of 2-categories with 2 objects (tensor categories are 2-categories with 1 object, so this ...


8

Since you asked "Why do people expect this to be true?" and nobody mentioned it so far, part of the reason is "numerical evidence". Drew Sutherland (of MIT) very recently computed a massive amount of new data about images of Galois for over 100 million elliptic curves. See the notes from this talk or contact Sutherland.



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