Serre asked whether algebraic vector bundles over affine space are all trivial or not. Quillen and Suslin proved independently that they are, in fact, all trivial. This is some kind of analogue to the topological situation, where all vector bundles over n-dimensional complex affine space (or even n-dimensional real affine space) are trivial.

In the topological case, one classical proof goes like this:
Denote by Ψ the functor of rank r vector bundles up to isomorphism on topological spaces. This functor is representable in the homotopy category, in particular Ψ(X) ≅ [X,Gr(r)], where Gr(r) is the infinite Grassmannian of r-planes in an arbitraily sized vector space and the parentheses [,] mean "homotopy classes of maps". To see this, embed any rank r vector bundle in a trivial one to get a map X->Gr(r), where different embeddings yield homotopic maps. Now observe that the pullback of the tautological bundle over Gr(r) along such a map gives a bundle isomorphic to the one you started with.

If X is contractible, i.e. homotopy equivalent to a point, [X,Gr(r)]=[pt,Gr(r)]={pt}, so there is nothing but one isomorphy-class of vector bundles on X, which must be the class of the trivial bundle. If X is affine, it is in particular contractible.

In the algebraic case, one might try to use this proof idea to proceed similarly. The first question arising here would be: which homotopy classes? Since the easiest example is affine space, one should choose a homotopy theory where affine space is contractible, which is the case for A¹-local simplicial homotopy theory, e.g. motivic homotopy theory.
The next question is now, whether we can represent the functor as in the topological setting.
This is certainly not the case: if G would be a classifying space (a representing object) for the functor Ψ, in the motivic homotopy theory, we'd have Ψ(X) ≅ [X,G], but the right-hand side is A¹-invariant by construction, while the left hand side is not! This means, [XxA¹,G]=[X,G] (along the projection morphism) but Ψ(XxA¹)≠Ψ(X) (this fails for example even in the case X=P¹).

One could stop here, given that there are nice proofs of the Quillen-Suslin theorem. But wait, Quillen-Suslin is only about affine space and we just tried to prove something about all schemes. Let's try to look at something weaker. What about (A¹-)contractible spaces?

There is a nice paper of Asok and Doran on vector bundles on contracible schemes which explains that there are lots of A¹-contractible schemes (over a field) with lots of vector bundles that are not trivial. These vector bundles are somewhat invisible, since they are indistinguishable by cohomology or K-theory: since the base is A¹-contractible and motivic cohomology as well as K-theory are representable in the A¹-homotopy category, cohomology and K-theory of these schemes are those of a point.

Well, it just seems the approach to Quillen-Suslin via A¹-homotopy theory is doomed. Now let me tell you that I think that is the case, but nevertheless, using the Quillen-Suslin theorem (and stronger results on the more general Bass-Quillen conjecture), Morel claims that the representability works like in the topological case, as long as we only look at affine schemes:

Let X be a smooth affine k-scheme, then Ψ(X) ≅ [X,Gr(r)].

For now, the proof didn't appear in any journal, but you can take a look at Morel's book-in-progess "A¹-algebraic topology over a field" here. The parts related to this discussion are mostly in section 7 and 8 and these sections were previously contained in an earlier paper draft called "A¹-homotopy classification of vector bundles over smooth affine schemes".

For some more info about the history and various approaches to Serre's problem I like to recommend Lam's excellent book "Serre's Problem on projective modules".

An arrow (A --> B) roughly means "you could try to prove B by using A and maybe something else, although A might not be necessary to prove B". I tried to remove redundant arrows, so keep in mind you can prove most of the theorems from the Cauchy-Riemann differential equations and some calculus ...

Another hint on the interpretation of my mind-map:
"existence of anti-derivatives" in the context of the mind-map really means "holomorphic functions have anti-derivatives on every simply connected domain", and "homotopy-invariance of path integral" really means "the path integral over holomorphic functions is homotopy-invariant on every simply connected domain". While you can do complex analysis without homotopy theory (even without homology theory), I don't like that approach very much. At least by using homology, some statement are easier to state. With homotopy, they become easier to prove (assuming you are allowed to use algebraic topology).

I created the mind-map (on the occasion of my oral exams) to keep it as a reminder and to sort out which statements could be generalized to harmonic maps or even to analytic maps (that was fun!).

Here it is, and of course you can download it as a PDF or as a SVG (vector graphics) as well (click on the image to enlarge it):

The license is CC-BY-NC-SA (if you redistribute, put my name on it, don't make profit, share alike).

An incomplete list of theorems contained:

• Goursat's Lemma
• Cauchy's integral formula
• Liouville's Theorem
• Maximum Modulus Principle
• Open Mapping Theorem
• Identity Theorem
• Riemann's Removable Singularity Theorem
• Casorati-Weierstraß
• Residue Theorem
• Fundamental Theorem of Algebra
• Montel's Theorem
• Schwarz's Lemma
• Riemann Mapping Theorem

It's a nice exercise to try to prove these on your own.

Enjoy learning, repeating or staring at confusing arrows!

You can get a PDF version of the image or click on it to get a readable version.

I'm still in the process of writing down examples and counter-examples to these properties, maybe that list will be online some day (another kind of "counterexamples in algebraic geometry").

As always, I'm happy to hear any comments (did I miss an important arrow, did I get anything wrong) -- but I should stress that the diagram works in Hartshorne-world, not in EGA-terms (this kind of confusion cost me almost one entire day trying to prove wrong statements..)

UPDATE (2011-11-18): improved diagram (more information, less colour) and higher quality PNG file.

Definition
A (connected closed orientable topological) n-manifold $M$ is called essential, if there exists a continuous map $f : M \to K(\pi_1(M,\ast),1)$ such that the induced morphism on the top homology $f_\ast : H_n(M,\mathbb{Z}) \to H_n(K(\pi_1(M,\ast),1),\mathbb{Z})$ maps the fundamental class $[M] \in H_n(M,\mathbb{Z})$ to some non-zero element $f_\ast([M]) \neq 0 \in H_n(K(\pi_1(M,\ast),1),\mathbb{Z})$.

To have a very explicit example, take a n-torus $M$, that is a manifold of dimension n which is homotopy equivalent to a product of n copies of $S^1$. Each such $S^1$ yields a different non-contractible loop on $M$, so there are n non-homotopic loops $\gamma_1,...,\gamma_n$ and the fundamental group is just $\pi_1(M,\ast) = \mathbb{Z}[\gamma_1,...,\gamma_n]$, the free abelian group generated by the $\gamma_i$. The homology is the exterior algebra over the fundamental group. The cohomology is the exterior algebra over the dual of the fundamental group, i.e. $H^\bullet(M,\mathbb{Z}) = \mathbb{Z}[\gamma_1^\ast,...,\gamma_n^\ast]$. The fundamental class is just $\gamma_1 \wedge ... \wedge \gamma_n \in H_n(M,\mathbb{Z})$. The universal cover of a n-torus is n-dimensional euclidean space, which is contractible, so $M$ has a contractible universal cover, thus it is acyclic, in other words, a $K(\pi_1(M,\ast),1)$. Taking the identity map $f := id_M$, this induces on top homology the identity map (since homology is functorial) and thus maps the fundamental class to itself, a non-zero element. So we have seen that any torus is essential. Note that we haven't looked at metric properties at all, because essentialness is a purely homotopy theoretic notion.

If you look closer, you see that we haven't actually used that the space $M$ was a torus - we just used that it is an aspherical space, so every aspherical manifold is essential.

The Borel conjecture predicts that closed aspherical manifolds are topologically rigid. The most common examples of non-topologically rigid spaces are lens spaces - there are many non-homeomorphic lens spaces of the same homotopy type. Lens spaces are closed, and they are good examples of non-aspherical essential manifolds, so they don't disprove the Borel conjecture.

Definition
Let $p$ and $q_1,...,q_n$ be integers (for some $n \geq 2$), with $q_i$ coprime to $p$ for each $i$. Define $\ell_k := 2\pi i q_k/p$. Take the unit sphere in $\mathbb{C}^n$, which is a $S^{2n-1}$ and let $\mathbb{Z}/p$ act on it by

$[1].(z_1,...,z_n) := (e^{\ell_1}z,...,e^{\ell_n}z).$

This is a $(2n-1)$-dimensional closed manifold with fundamental group $\mathbb{Z}/p$. The universal cover is given by the quotient map $S^{2n-1} \to L(p;q_1,...,q_n)$, so the universal cover is clearly non-contractible and in fact very spherical. This shows that lens spaces are never aspherical.

In the literature on homology and homotopy, you'll often find 3-dimensional lens spaces $L(p,q) := L(p;1,q)$. For these, there exists a nice classification of homeomorphism types via Reidemeister torsion (or: simple homotopy type), ultimately boiling down the question to arithmetic relation between different $q$, modulo $p$.

To see that lens spaces are essential, we have to produce a map $f : L(p;q_1,...,q_n) \to K(\mathbb{Z}/p,1)$ which on top homology maps the fundamental class to a non-zero element. The homology of $K(\mathbb{Z}/p,1)$ is well-known, it is

$H_k(\mathbb{Z}/p,\mathbb{Z}) = \begin{cases} \mathbb{Z} & k=0,\\ \mathbb{Z}/p & k \text{ odd},\\ 0 & k \text{ even}. \end{cases}$

Now we need an explicit model for $K(\mathbb{Z}/p,1)$. One such model is given by the infinite lens space $L^\infty(p) := S^\infty/_{\mathbb{Z}/p}$, where $S^\infty := \lim S^n$ is seen as the union of spheres where the n-sphere sits inside the (n+1)-sphere as equator. The group $\mathbb{Z}/p$ acts by multiplication with p-th roots of unity in each coordinate, which is possible by putting the $S^\infty$ in a $\mathbb{C}^\infty := \lim \mathbb{C}^n$ by taking the limit over the embeddings $S^{2n-1} \to \mathbb{C}^n$.
We can modify this construction slightly, by starting with the lens space $L(p;q_1,...,q_n)$ and taking the limit over all $L(p;q_1,...,q_n,q'_1,...,q'_k)$ for $k \to \infty$ and $q'_i = q_n$ for all i. This yields the same $L^\infty(p)$ up to homotopy and even better, it admits an inclusion map from $L(p;q_1,...,q_n)$. On homology, the inclusion map maps the fundamental form to a generator of $\mathbb{Z}/p$, which is non-zero. Therefore, lens spaces are essential.

With a very similar idea, one can prove that real projective spaces $\mathbb{R}P^n$ are essential, by looking at the inclusion into $\mathbb{R}P^\infty = \lim \mathbb{R}P^k$, which is aspherical with the same fundamental group $\mathbb{Z}/2$.

In general, it suffices to find a continuous map of non-zero degree from a manifold $M$ onto an essential manifold to deduce that $M$ is essential.

To give a counter-example, look at the spherical space $S^n$ (for $n \geq 2$) with trivial fundamental group. It is certainly not aspherical (its higher homotopy groups are quite interesting) but there is an inclusion map $S^n \to S^\infty$ (as above). This inclusion map has to be the zero map on top degree homology, since $H_n(S^\infty,\mathbb{Z}) = 0$ for all $n \geq 1$ (because $S^\infty$ is contractible). This shows that spheres are never essential.

Finally, you might ask
What are essential manifolds good for?
In his 1983 paper "Filling Riemannian Manifolds", Gromov defined essential manifolds the first time, to state (and prove) his "main isosystolic inequality".
To formulate it, we have to say what a systole is first:

Definition
Let $M$ be a Riemannian manifold. Then the systole of $M$ is $sys_1(M) := \inf_{\gamma} length(\gamma)$, where the infimum goes over all non-contractible loops $\gamma$ in $M$ (in fact it is a minimum).

Theorem (Gromov)
Let $M$ be a closed essential Riemannian manifold of dimension $n$. Then

$sys_1(M) \leq C_n \sqrt[n]{Vol(M)}$

$0 < C_n < 6(n+1) n \sqrt[n]{(n+1)!}.$

So the job of essential manifolds is to be the domain where Gromov's theorem holds. As far as I know, it is not so clear whether there exist larger classes of manifolds that satisfy such a systolic inequality.

The theorem is a generalisation of a theorem on tori:
Theorem (Loewner)
Let $\gamma$ be a shortest closed geodesic in a flat torus $T^n$. Then

$sys_1T^n = length(\gamma) \leq C_n \sqrt[n]{Vol(T^n)}.$

$sys_1M \leq C_2 \sqrt{Area(M)}$

Pu proved a similar systolic inequality on $\mathbb{R}P^2$, so it is very reasonable to look for a class of closed manifolds that contain tori and real projective space and furthermore allow systolic inequalities.

Well, that's enough for today!

Definition
A topological space $M$ is called aspherical if all higher homotopy groups vanish, i.e. $\pi_n(M,m_0) = 0 \quad \forall n > 1$ where $m_0 \in M$ is an arbitrary basepoint and $M$ is assumed to be connected.

Since manifolds admit universal covers, you could equivalently define a manifold to be aspherical if and only if its universal cover is contractible.

Just one example illustrating how rich this class of spaces is:
Metric spaces that are of non-positive curvature (i.e. locally CAT(0)-spaces), for example the Bruhat-Tits building of a simple algebraic group over a field with a discrete valuation, are aspherical.

A good survey on aspherical manifolds was given by Wolfgang Lück.

Definition
A connected topological space $X$ is called Eilenberg-MacLane-space for a group $G$ and a natural number n if its nth homotopy group is exactly $G$ and all other homotopy groups vanish, i.e.
$\pi_k(X,x_0) = \begin{cases} G & k=n \\ 0 & else.\end{cases}$
Then one calls $X$ also $K(G,n)$.

The standard examples of $K(G,1)$ spaces are $S^1$, which is a $K(\mathbb{Z},1)$ and $\mathbb{R}P^\infty$, which is a $K(\mathbb{Z}/2,1)$.
Of course, every $K(G,1)$ is aspherical and every aspherical space is a $K(G,1)$ for $G$ being its fundamental group.

One can also define a functorial construction of a $K(G,1)$ which gives a CW-complex model for every group $G$ and transforms group homomorphisms into continuous maps of spaces.

For this, we need the functorial nerve construction.
Definition
The nerve $N(G)$ of a (discrete) group $G$ is the simplicial $G$-set with n-simplices being the (n+1)-fold cartesian product of sets $G \times G \times \cdots \times G$, face maps just omitting one factor in the cartesian product, degeneracies adding the identity element of $G$ in one factor.
By construction, seen as a discrete simplicial group, $G$ embeds into $N(G)$ as the 0-skeleton. Observe that $N(G)$ is contractible, since every n-simplex $(g_0,...,g_n) \in N(G)_n$ is the face of $(e,g_0,...,g_n) \in N(G)_{n+1}$ which also has the face $(e,g_1,...,g_n) \in N(G)_n$, thus allowing to move every point to the identity $(e,...,e) \in N(G)_m$ which is just a degeneracy of $e \in N(G)_0 = G$.
The group $G$ acts diagonally on $N(G)$, i.e. it acts on an n-simplex by the formula $(g,(g_0,...,g_n)) \mapsto (gg_0,...,gg_n) \in N(G)_n$. This action is compatible with face and degeneracy maps, thus making $N(G)$ into a simplicial $G$-set. The action is free, i.e. no two elements of $G$ operate in the same way.

Using the nerve construction, we now define the classifying space:
Definition
The classifying space $BG$ of a group $G$ is the quotient $BG := |N(G)|/G$ of the geometric realisation $|N(G)$ of the nerve construction by the group action described above. It turns out that $G$ operates on $|N(G)|$ like a deck transformation group, thus giving $BG$ the structure of a CW-complex with universal cover $|N(G)|$ and fundamental group $G$.
A group homomorphism $\phi : G \to H$ gives rise to a morphism of simplicial sets $N(\phi) : N(G) \to N(H)$ by pointwise application. Geometric realisation is also functorial, and due to $\phi$ being a homomorphism, the continuous map $|N(\phi)| : |N(G)| \to |N(H)|$ descends to a continuous map of classifying spaces $B\phi : BG \to BH$.

If you are not into simplicial sets and geometric realisation, you can look for a more hands-on approach in Hatcher's book "Algebraic Topology", on page 87, chapter 1.B, more specifically Example 1B.7 on page 89.

Now back to our first definitions: An aspherical manifold is just a manifold which happens to be a $K(G,1)$ for $G$ being its fundamental group. The classifying space is just an explicit (functorial!) construction which gives a $K(G,1)$ for every group $G$ (although most authors would call our $BG$ just one explicit model for $BG$...).

One would like to work only with CW-complexes, if possible, since they allow induction over the skeleton and cell-by-cell arguments. Is every manifold homeomorphic to a CW-complex - long time ago there was the "Hauptvermutung" (main conjecture) which asked this, but it's wrong. While compact manifolds admit a homotopy equivalent CW-model (by Kirby and Siebenmann), this is not true for topological manifolds in general. Let us look what one could do with a CW-model:

Proposition
Let $X$ be a connected CW complex and $Y$ be a $K(G,1)$ (for example, your favourite aspherical manifold). Let $\phi : \pi_1(X,x_0) \to \pi_1(Y,y_0) = G$ be a homomorphism of groups. Then there is a continuous map $\Phi : X \to Y$ mapping $x_0$ to $y_0$ which induces $\phi$ on fundamental groups; furthermore, the map $\Phi$ is unique up to homotopy relative $x_0$.

The proof of this proposition goes roughly like that: First, let $\Phi$ map $x_0$ to $y_0$. Now, for each 1-cell $\gamma$, take a representative of $\phi([\overline{\gamma}]) \in \pi_1(Y,y_0)$ to define $\Phi$ on $\gamma$. Then one has to extend the map given on the 1-skeleton to $X$, using the fact that $Y$ has no higher homotopy.

Corollary
Every two CW-complexes $X,Y$ which are both $K(G,1)$-spaces are homotopy equivalent ("of the same homotopy type").

To prove this, just take isomorphisms $f : \pi_1(X,x_0) \to G$ and $g : \pi_1(Y,y_0) \to G$ and define $\phi := f \circ g^{-1}$ which gives $\Phi : Y \to X$ with inverse up to homotopy given by $\Psi : X \to Y$ induced by $\psi := g \circ f^{-1}$.

This justifies that every invariant of $BG$ that depends only on the homotopy type, is actually an invariant of $G$ - a very useful idea. One can define group homology with integer coefficients of $G$ by the formula $H_n(G,\mathbb{Z}) := H_n(BG,\mathbb{Z})$.

One drawback of the classifying space via the nerve construction is that it is usually very large - there are simplices in arbitrary high dimensions. For example, the circle $S^1$, given as example of a $K(\mathbb{Z},1)$, is much more efficient than $B\mathbb{Z}$.

Of course, talking about aspherical manifolds, we don't want to forget the manifold structure. Given a group $G$, one could expect that many non-homeomorphic aspherical manifolds with fundamental group $G$ exist - even many non-homotopy equivalent ones. At least we can say that such non-homotopy equivalent aspherical manifolds are not of CW homotopy type. There is an old conjecture on this theme:

Conjecture (Borel)
Let M and N be closed aspherical manifolds, and let $f : M \to N$ be a homotopy equivalence. Then $f$ is homotopic to a homeomorphism.

Together with the result of Kirby and Siebenmann (that every closed manifold is of CW homotopy type), this would imply that closed aspherical manifolds are classified by their fundamental group up to homeomorphism.

The property that every homotopy equivalence is homotopic to a homeomorphism is called topological rigidity.

You can read something about the content in this blog post, containing an extended abstract in english.

For the german-speaking mathematicians, here is some abstract:

Matrizengruppen über den reellen oder komplexen Zahlen sind topologische Gruppen und ihre Fundamentalgruppen sind interessante Invarianten. Während die Fundamentalgruppe über stetige Schleifen am Basispunkt und Homotopien definiert ist, können wir uns fragen, welche Gruppe man erhält, wenn man nur polynomiale Schleifen am Basispunkt und polynomiale Homotopien zulässt. Während die Fundamentalgruppe eine Klassifikation der Überlagerungen erlaubt, so erhalten wir durch die “polynomiale Fundamentalgruppe” eine Klassifikation der zentralen Gruppenerweiterungen - ein homologietheoretisches Analogon von Überlagerungen.

Für eine Überlagerung X → X mit Faser F über dem Basispunkt und punktiertem Schleifenraum ΩX gibt es eine Liftungsabbildung L : ΩX → F, die im Fall einer topologischen Gruppe X = G ein Gruppenhomomorphismus ist. Ist X die universelle Überlagerung, so liefert L unter π_0 einen Isomorphismus π_1(X,∗) → F.
Jede Überlagerung topologischer Gruppen ist auch eine zentrale Erweiterung - jedoch nicht umgekehrt. Die zentralen Erweiterungen einer perfekten Gruppe G werden klassifiziert durch den Schur-Multiplikator H_2(G,Z); das ist der Kern der universellen zentralen Erweiterung. Wir betrachten den Fall X = G(k), wobei G(k) die k-rationalen Punkte einer einfach zusammenhängenden Chevalley-Gruppe G mit Wurzelsystem Φ für einen unendlichen Körper k ist. Dann ist der Schur-Multiplikator H_2(G(k),Z) =: K_2(Φ,k), die zweite instabile K-Theorie bezüglich Φ und die universelle zentrale Erweiterung ist die Steinberg-Gruppe St(Φ, k).

Wir definieren eine simpliziale Gruppe SingG(k), deren n-Simplizes genau die Matrizen aus G(k[t1,...,tn]) sind und zeigen, dass ihre simpliziale Fundamentalgruppe genau die instabile K-Theorie ist, indem wir (unter einer gewissen Regularitätsvoraussetzung an K_2(Φ,k)) die simpliziale Überlagerung
K_2(Φ,k) → SingSt(Φ,k) → SingG(k)
und ihren Liftungshomomorphismus L studieren. Wir geben eine explizite Umkehrabbildung zu L an und können somit alle Schleifen in SingG(k) bis auf Homotopie explizit beschreiben.
Die genannte Regularitätsvoraussetzung ist Homotopieinvarianz von K_2(Φ,·) in einer Variablen über einem Körper k, d.h. K_2(Φ,k[t])=K_2(Φ,k).
Für Φ = A_n mit n ≥ 3 ist die Aussage bereits durch Sätze von Quillen und van der Kallen bekannt. Für rk Φ ≥ 3 ist dies eine Vermutung von Wendt.
Wir arbeiten mit der Steinberg-Präsentation der universellen zentralen Erweiterung St(Φ,k) von G(k) und mit der Matsumoto-Präsentation ihres Kerns, der instabilen zweiten K-Theorie von k:
K_2(Φ,k)=KSp(k), falls Φ symplektisch, K_2(Φ,k)=KM(k) (Milnor-K-Theorie) sonst.

Matsumotos Beweis dieser Präsentation wird in dieser Arbeit ausführlich nachgerechnet. Die Ergebnisse über die Fundamentalgruppe von SingG(k) liefern eine Verallgemeinerung eines Satzes von Jardine auf instabile K-Theorie und unendliche Körper, die nicht notwendig algebraisch abgeschlossen sind:
π_1(SingG(k))=K_2(Φ,k).

Unter Verwendung von Resultaten von Morel und Wendt
π_1^A¹(G)(k)=π_1(SingG(k))
erhalten wir schließlich eine Aussage über π_1^A¹, die motivische Fundamentalgruppe im Sinne von Morel und Voevodsky.
Damit ist die ursprüngliche Fragestellung der Arbeit beantwortet:
“Wie sehen die Schleifen in der A¹-Homotopietheorie von G(k) aus?”.

...und weil beim herauskopieren des Abstracts aus der PDF-Datei bestimmt das ein oder andere schief gegangen ist, sollte man lieber gleich das PDF lesen, wenn man denn überhaupt etwas lesen möchte.

Classical quantum theory is about modelling purely quantum effects, i.e. without considering gravity, or at least without considering relativistic effects. There, a separable Hilbert space as a state space is appropriate. The bounded linear operators on the state space form a certain kind of normed algebra with compatible involution (taking the adjoint) called C*-algebra and measurements correspond to self-adjoint operators.

Quantum field theory tries to incorporate quantum mechanics into electromagnetic field theory (or vice versa) and gravitational field theory. So far, no such theory-of-everything has been developed with falsifiable predictions, although there are some promising candidates.

As a starting point for mathematicians, I recommend the book by Folland, Quantum Field Theory: A Tourist Guide for Mathematicians. Another useful source might be the work of Zeidler, which I haven't had the time to look at yet.

Now AQFT, or axiomatic quantum field theory, is one approach to develop a QFT. The first input is to think about a quantum system not as a certain Hilbert space but as a certain C*-algebra of observables. One can always find a Hilbert space representation of a C*-algebra, which means that the two approaches are mathematically equivalent (by the GNS construction). The second input is to take some spacetime manifold (like flat Minkowski space or something more curved) and attach a C*-algebra to each open subset in a compatible way. This amounts to saying that quantum fields are defined as copresheaves of C*-algebras whose co-restriction morphisms are monomorphisms (see where I have borrowed this phrasing). In some sense, this means to attach a presheaf of non-commutative spaces to the spacetime, since C*-algebras are a model for noncommutative geometry. AQFT people call such a copresheaf of C*-algebras a local net of observables.

One important aspect is locality, which means that effects at some point do not influence some other distant point - it takes time. More specifically, there is spacelike locality, which forbids two spacelike separated regions to influence each other at all, which means that the corresponding observable C*-algebras commute with each other (i.e. a measurement in one subsystem does not affect the other subsystem, they are independent).

The third axiom is about spacetime covariance, which means that the elements S of the symmetry group (like the Poincaré group or a subgroup) transforms the observables of one open subset U to the observables of the subset S(U). This transformation is required to be an epimorphism of C*-algebras.

The fourth axiom is positivity of energy. A mathematical axiomatisation of this would be that translation operators have spectral support in the closed forward light cone.

This approach was developed in 1964 by Rudolf Haag and Daniel Kastler in "An algebraic approach to quantum field theory", Journal of Mathematical Physics, Bd.5, p.848-861.

In AQFT, the observables depend on time (since they are associated to certain spacetime regions) and the "state", i.e. the spacetime and the copresheaf of C*-algebras are fixed. This corresponds to the classical Heisenberg picture of quantum mechanics.

One of the big successes of AQFT are the CPT-symmetry theorem. C stands for charge, P for parity and T for time. The CPT-symmetry conjecture states that the laws of nature are invariant if time goes backwards, all charges are conjugate and all parities are reversed (i.e. all chiral properties). In the AQFT framework, there is a mathematically precise formulation of this conjecture in terms of operators and there also is a proof.

The approach to quantum mechanics via C*-algebras (or some other kind of observable algebras) is philosophically the most satisfying: a physical model should predict only measurable data, so it is most natural to let the model consist of every measurement which can be performed. A state is somehow just a set of outcomes of all measurements (more precise, a positive linear functional on the observable algebra). Via the GNS-construction, elements of a Hilbert space form such states for the C*-algebra of linear bounded operators on the Hilbert space.

To be even more precise, the most satisfying approach to observables would be to model them as projection-valued measures on a separable Hilbert space. Via the spectral theorem, these correspond to self-adjoint operators. Let me elaborate on the justification for a separable Hilbert space. As one sees in the two-slit experiment, superposition of states is possible, so one has to allow convex combination of states, therefore the Hilbert space (where one is actually working in the projective space over the Hilbert space). The Hilbert space has to be separable (which means having a countable Hilbert base, so every vector can be written as limit of countable sums of Hilbert basis vectors), since one can only perform finitely many measurements on any system, but an arbitrary large number of measurements. So one would actually like to use just finite dimensional Hilbert spaces, which would make the maths much easier, but then one would impose an arbitrary limit on the number of possible measurement processes. Maybe nature itself consists of a highly infinite-dimensional Hilbert-space, not admitting any Hilbert basis of low cardinality at all - but this we will never be able to falsify and thus must discard such models. It is in fact meaningless to talk about non-measurable physical realities.

Now to projection-valued measures: A single yes-or-no question can be modelled as a projector, since asking whether a state satisfies a certain property will result in a positive answer if the state was an eigenstate of the projector to the subspace of all states satisfying that property. It becomes clear that such a projector would also project all superpositions of yes-or-no states w.r.t. that property onto the yes-subspace, thus changing the state. This is also an important ingredient of all quantum theories: the observer changes the system, as we can also see in the double-slit experiment.
Any property of states which consists of something more complicated than just yes-or-no questions can be written as a logical expression in yes-or-no questions. This logical expression in "quantum logic" translates to union, intersection and complement of Hilbert space subspaces, so to projections onto these subspaces.

As already said, observables correspond to self-adjoint operators (by the spectral theorem). If you're into Lie algebras you already noticed that skew-hermitian matrices form the Lie algebra of the Lie group of unitary matrices. Physicists like to drop some -i and take hermitian matrices as the infinitesimal generators of unitary time evolution (that's their slang, not mine).

Let me finish this sketch on AQFT by mentioning two other nice aspects of this approach:
One the one hand, C*-algebras naturally admit a commutator which one may think of some quantum version of the Poisson bracket in Lagrangian mechanics. The correct way of doing this is called deformation quantization.
On the other hand, C*-algebras have interesting representation theory and this is also connected to physics. One example would be super-selection rules, which can be easily explained by telling you that a super-selection sector is an isotypic component of a C*-representation. Another example (which I think is really nice) is the Aharonov-Bohm effect, where the quantum system behaves differently if a magnetic field is enabled somewhere where particles are unable to enter at all. This corresponds to two systems with the same observable algebra but different representations (since the measurements you can do are the same, but the physics isn't).

One shouldn't mention the Heisenberg picture without the Schrödinger picture, where we have to relate AQFT to the so-called FQFT, or functorial quantum field theory. In the Schrödinger picture, observables are fixed and the state has time-evolution. In FQFT, the path integral is formulated axiomatically as a functor. There are various kinds of FQFT and the kind I like most is TQFT, which stands for topological quantum field theory. There, the path integral is a functor from a cobordism category to C*-algebras, associating to each object of the cobordism category (i.e. each manifold) an operator-algebra for that specific space and to each morphism in the cobordism category (i.e. each cobordism) a morphism of operator-algebras that encodes time evolution. A cobordism is considered to model a spacetime segment in this formalism.

I didn't put so many links to read further in this text, so if you need more resources to learn about this stuff, just comment here and I will put links to some other introductory material here.

As a teaser, here is the abstract of the abstract:

In classical covering space theory we have an isomorphism of the fundamental group with the fibre of the universal cover over the basepoint. Covering spaces of topological groups are group extensions, but not every group extension is a covering space. Perfect groups admit a universal central extension and the kernel of this extension is also called fundamental group. For simply connected Chevalley-groups over a perfect field, this fundamental group, classically called second unstable K-Theory, is exactly the fundamental group of a simplicial resolution. The loops are described explicitly by matrices.

Questions part V - Life and Metaphysics [Sch68]

1. Is nature deterministic?
2. Can causality be defined without reference to time? [BLMS87] [Sua01]
3. How is it possible that semantic information emerges from purely syntactic information? [BLHL+ 01]
4. Is there an inherent tendency in evolution to accumulate relevant information on the real world?
   Is there an inherent tendency in evolution to increase the complexity of organisms and the biosphere as a whole?

   “Humanity is now experiencing history’s most difficult evolutionary transformation.” – Buckminster Fuller, 1983

5. Why are robustness and simplicity good and applicable criteria to describe nature (with causal networks)? [Jen03]
6. Should we re-define “life”, using information-theoretic terms?
7. What do Gödel’s theorems imply for information and complexity theory? [Cha82]
   Is there an analogy between emergence and true but unprovable statements? [Bin08]
8. Are there limits of self-prediction in individuals and societies?
   

   “The human brain is incapable of creating anything which is really complex.” – Andrey Nikolaevich Kolmogorov 1990

9. What is the answer to life, the universe and everything?
   

   “There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another which states that this has already happened.” – Douglas Adams, 1980

References

• [Bin08] P.M. Binder, Theories of almost everything, Nature 455 (2008), 884–885.
• [BLHL+ 01] T. Berners-Lee, J. Hendler, O. Lassila, et al., The semantic web, Scientific american 284 (2001), no. 5, 28–37.
• [BLMS87] Bombelli, Lee, Meyer, and Sorkin, space-time as a causal set, Physical Review Letters 59 (1987).
• [Cha82] Gregory J. Chaitin, Gödel’s theorem and information, International Journal of Theoretical Physics 21 (1982), 941–954, 10.1007/BF02084159.
• [Jen03] Erica Jen, Essays & commentaries: stable or robust? what’s the difference?, Complex. 8 (2003), no. 3, 12–18.
• [Sch68] E. Schrödinger, What is life?, Cambridge University Press Cambridge, 1968.
• [Sua01] A. Suarez, Is there a real time ordering behind the nonlocal correlations?, 2001.

Questions part IV - Philosophy of Science [Pop34] [Kuh62] [Fey75] [Mil09]

1. Does the point of view of information theory provide anything new in the sciences? [GM94]
   Does information theory provide a new paradigm in the sciences? [Sei07]
2. Is quantum information the key to unify general relativity and quantum theory?
   Is information theory a guiding principle for a “theory of everything”?

   “I think there is a need for something completely new. Something that is too different, too unexpected, to be accepted as yet.” – Anton Zeilinger, 2004

3. (Why) are real discoveries possible in mathematics and other structural/formal sciences? [Bor07]
4. Can we create or measure truly random numbers in nature?
   How would we recognize random numbers?
   What is a random number (or a random string of digits)?

   “Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such thing as a random number — there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method.” – John von
   Neumann, 1951

5. What is semantic information, what is meaning in science?
   What do we expect from an “explanation”?

   “The Tao that can be told is not the eternal Tao.” – Lăozı, 4th century B.C.

6. How do the concepts “truth” and “laws of nature” fit together? [Dav01] [Car94]
7. Does is make sense to use linguistic terminology in natural sciences? [Gad75]
8. Should physicists try to interpret quantum physics at all? [Dir42]
9. Would it make sense to adapt the notion of real numbers to a limited amount of memory?
   Can we build a theory of physics upon intuitionist logics?

References

• [Bor07] Borovik, Mathematics under the microscope, 2007.
• [Car94] J.W. Carroll, Laws of nature, Cambridge Univ Pr, 1994.
• [Dav01] D. Davidson, Inquiries into truth and interpretation, Oxford University Press, USA, 2001.
• [Dir42] PAM Dirac, Bakerian lecture. the physical interpretation of quantum mechanics, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 180 (1942), no. 980, 1–40.
• [Fey75] Paul Feyerabend, Against method, 1975.
• [Gad75] H.G. Gadamer, Truth and method (g. barden & j. cumming, trans.), 1975.
• [GM94] M. Gell-Mann, The quark and the jaguar, Freeman New York, 1994.
• [Kuh62] Thomas Kuhn, The structure of scientific revolutions, 1962.
• [Mil09] David Miller, Hard questions for critical rationalism, 2009.
• [Pop34] Karl Popper, The logic of scientific discovery, 1934.
• [Sei07] C. Seife, Decoding the universe: how the new science of information is explaining everything in the cosmos, from our brains to black holes, Penguin Group USA, 2007.