If you have any recommendations on what to add to this list, please leave a comment below.

Reference & Bibliography tools

Doing research (and even preparing seminar talks), you need to check the available literature. There are many tools to do this, most notably MathSciNet to get correct bibliographic data, reviews of articles and links to journals to download articles, if possible.

• MathSciNet Mathematical Reviews (MR), paper reviews (subscription required).
• MathSciNet Lookup, bibliographic information (no subscription required).
• Zentralblatt MATH (Zbl), paper reviews.
• Google Scholar, a citation search engine.
• Eigenfactor, a website with tools to work with "new" metrics.
• Mendeley's mathematics section - Mendely is a reference management tool.
• CiteSeer, a citation search engine.
• RetractionWatch mathematics section - RetractionWatch is tracking retractions as a window into the scientific process.
• Allyn Jackson: From Preprints to E-prints (AMS Notices 2002, PDF)
• JabRef, an open source bibliography manager (written in Java).
• Nature Special: Science Metrics, like citation count, eigenfactor and h-index.
• Bookworm arXiv, search for trends in the arXiv.
• Grocko.org is a system to recommend scientific publications to academics.
• Google Books NGram Viewer, to see when a specific term was used a lot.

Open Access Journals & Preprint Archives

Nowadays many publications are discussed as soon as they reach a preprint archive, or an e-print archive. It can take years until journal publication of a preprint, so you will most likely work with preprints anyway.

• ArXiV | How to use ArXiV RSS Feeds
• K-Theory Preprint Archive
• NUMDAM, digitalised math articles, mostly from France.
• Publications of the AMS at JSTOR, open access
• The full JSTOR list, many open access journals
• Project EUCLID, offers access to many open access journals as well as many subscription-only journals.
• Publications on the nLab, a web-based journal for peer-reviewed publication of original research and expository writing.
• DOAJ - directory of open access journals, mathematics section.

Open Access Textbooks & Videos

Sometimes there is no need to buy that expensive textbook.

• Allen Hatcher: Algebraic Topology
• Charles Weibel: The K-book: An introduction to algebraic K-theory, a graduate textbook in progress
• EGA at NUMDAM, Grothendieck's Elements de Géometrie Algébrique
• WikiBooks on University-level Mathematics
• J. S. Milne's Course Notes, mostly on algebraic topics.
• Trillia: Alexandre Stefanov's list of textbooks
• The Catsters, a classic resource for category-theory lovers
• David Ben-Zvi's list of Langlands resources, including lecture videos and many course notes
• Lecture videos from the Isaac Newton Institute for Mathematical Sciences, University of Cambridge
• Lecture videos from the IHES
• Lecture videos from the Institute for Advanced Study, School of Mathematics
• VMath video lectures from the MSRI
• OpenCourseWare (OCW) at the MIT
• A MathOverflow thread with lots of links to math videos
• Wolfram MathWorld, an encyclopedia of mathematics.
• Encyclopedia of Mathematics, a wiki-format encyclopedia of mathematics, in collaboration of Springer with the EMS. [UPDATE 12-05-09]

Recommend this to the interested non-mathematician

If you encounter people who seem to like math but aren't (yet) ready for research, send them to these places to play around:

• TED Talks: Numbers at play
• Khan Academy, the online math education website, with videos and exams
• The Art of Problem Solving, intriguing mathematical problems and solutions
• AMS Recommendations for Undergrads, a linklist (US-centric)

Innovative Projects & Experiments

A collection of partly well established (MathOverflow), and partly more experimental projects which might change the way you do mathematics, collaboratively, on the web.

• MathOverflow, the question&answer site for research math | customize RSS Feeds like this one.
• math.SE, the StackExchange question&answer site for undergrads.
• Polymath, massively collaborative mathematics.
• Tricki, a repository of mathematical know-how.
• Open Problem Garden, a collection of open math problems, ranging from easy to millenium-prize class.
• Wolfram Alpha, a computational knowledge engine.
• nCatLab Wiki, the n-Category Café open lab notebook.
• MathBlogging.org, your one-stop shop for mathematical blogs.
• The Stacks Project, an open source textbook and reference work on algebraic stacks and the algebraic geometry needed to define
  them.
• MFO (Oberwolfach) Photo Collection
• Mathematics Genealogy Project
• Math Online, a math books library.
• Boole's Rings Researchers. Connecting.
• Subject Wikis | most developed: Group Properties
• Mathematicians on Google+ | another such list.
• The On-Line Encyclopedia of Integer Sequences (OEIS)
• LMFDB, the database of L-functions, modular forms, and related objects.
• The Mathematical Atlas, a gateway to modern mathematics.

Notable Math Blogs

I warmly recommend clicking through this list and then through the blogrolls of each blog, until you found your niche in the blogosphere. There are vast amounts of content, and there is always some quality content hidden, deep inside. It is really worthwile to search once and to grab the RSS feeds of the 2-3 most interesting blogs. I also recommend to browse through the archives of interesting blogs, sometimes an article from some years ago might inspire you!

• MathBlogging.org Weekly Picks
• Timothy Gowers
• Terence Tao - What's New
• Burt Totaro - Geometry Bulletin Board
• Gil Kalai - Combinatorics and more
• Joel David Hamkins - Mathematics and Philosophy of the Infinite
• Andreas Holmstrom - Motivic Stuff
• Akhil Mathew - Climbing Mount Bourbaki
• Qiaochu Yuan - Annoying Precision
• Charles Siegel - Rigorous Trivialities
• Tanya Khovanova - Math Blog
• Martin Orr's Blog
• Adriana Salerno - PhD+Epsilon, an early-career mathematician blogs about her experiences and challenges.
• Low Dimensional Topology, a group blog
• n-Category Café, a group blog on math, physics and philosophy.
• Azimuth, John Carlos Baez' blog on what scientists can do to help save a planet in crisis.
• Secret Blogging Seminar, a group blog
• ResearchBlogging.org Mathematics section
• Researcher's blog, career advice.
• AMS Graduate Student Blog
• Cathy O’Neil - mathbabe
• Automorphic Forum

The Cost of Knowledge - The Academic Spring

In January 2012, Fields medallist Timothy Gowers wrote a blog post about the academic publisher Elsevier, its bad business practices and that he decided to boycott them. This lead to the creation of a boycott movement website "The Cost of Knowledge" and a discussion forum to change the world of mathematical publishing. The Academic Spring it was called.

• T.Gowers: Elsevier — my part in its downfall.
• the cost of knowledge (dot com)
• Math2.0 Forum
• Rehmann's Math Journal Price Survey.
• J.C.Baez: What We Can Do About Science Journals.
• M.Nielsen's comprehensive linklist on the academic spring, really a good starting point to get involved.

Recommended Reading for Young Researchers

This is mostly a list of "how to give a good talk" notes. Beware that there are many kinds of mathematics talks: seminar talks (audience should understand), workshop talks (experts should understand), colloquium talks (mathematicians should understand most), general public talks (non-mathematicians should understand most) - and all mixtures of these. Most talks take about 45 up to 90 minutes, although some plan only 20 to 30 minutes. It is crucial to know what kind of talk is expected and who the audience is. Besides that ... there are some tips & tricks:

• Bob L. Sturm: 25 points for PhD students
• Matilde Marcolli: The (Martial) Art of Giving Talks, (PDF).
• William T. Ross: How to give a good 20-minute talk
• John E. McCarthy: How to Give a Good Colloquium (PDF)
• P.R. Halmos: How to talk Mathematics, a classic
• Joseph A. Gallian: How to Give a Good Talk, comes with a handy preparation check-list (PDF).
• David Tong: How to Make Sure Your Talk Doesn’t Suck (PDF)
• Read the AMS Notices, in particular the "What is ..." series. It is very good at explaining concepts to the general mathematical public.

How to write mathematics texnically

If you have to write math, you either do it on paper, on a blackboard or on a computer.
On any computer, you'll likely use LaTeX, since the days of trying to use ASCII are long over.
LaTeXing is an art to be learned, but the basics are pretty easy and it's no longer limited to creating PS or PDF files, but you can literally TeX the web.

• TeX StackExchange, the question-answer forum.
• TeXample.net, ample resources for TeX users, especially for TikZ graphics.
• MathJax, an open source JavaScript display engine for mathematics that works in all modern browsers.
• HTML escape codes for math | HTML escape codes for greek letters
• LaTeX for WordPress Plug-in, the one I currently use (implemented with MathJax).
• Instiki, a wiki that supports LaTeX+Markdown syntax (implemented in Ruby, with MathJax)
• Detexify - LaTeX symbol classifier, you draw a math symbol with your mouse pointer, then you get LaTeX code.
• MathJax Bookmarklet, to enable LaTeX rendering via MathJax on any website.

Some conference & summer school listings

You should choose which one is most useful to you, then check that one more often and maybe ignore the rest.

• Conference List at the AMS Graduate Student Blog [UPDATE 12-05-09]
• Andreas Holmstrom's list of lists
• K-Theory Calendar
• AlgTop-Conf
• Conferences in Arithmetic Geometry
• Conferences and meetings on Topology and related topics
• Upcoming conferences in algebraic geometry

Mailing lists

They still exist. And they're mostly used for job offers and conference announcements.

• ALGTOP-L, the algebraic topology mailing list | Archives

For fun

• arXiv vs. snarXiv, guess real from fake scientific publication titles.
• Hypercube rotation training

The four functor formalism arises as part of the six functor formalism (add Hom and Tensor to make it six) in certain (co)homological set-ups. Where I encountered it first was in a paper I tried to read, about the stable motivic homotopy category, but most likely you'll see this stuff in papers dealing with perverse sheaves or motives and their realisations.

Disclaimer: We'll stay in the topological category for this post, i.e. the objects are topological spaces and the morphisms continuous maps. Sheaves are ordinary sheaves of abelian groups (no fancy Grothendieck topology necessary here), not $\mathcal{O}$-modules of some sort. However, the discussion doesn't change too much if you translate into the algebraic category, so this should be a good exercise for the bored reader.

Pushforward

Pushforward of sheaves is straightforward: given a space $X$, a sheaf $F$ on $X$ and a continuous map $f : X \to Y$, the sheaf $f_\ast F$ on $Y$ should be a sheaf that does on open subsets $Y$ what $F$ had done on the corresponding open subsets of $X$, i.e. $(f_\ast F)(U) := F(f^{-1}U)$. Check that this definition gives again a sheaf. Observe that the constant map $c : X \to pt$ yields $(c_\ast F)(pt) = F(X)$, so $c_\ast$ is almost the global section functor and we should think of any $f_\ast$ as some kind of generalized global section functor.

Pullback

I want to define the pullback functor $f^\ast : Sh(Y) \to Sh(X)$ as the left adjoint to $f_\ast$. Of course, I have to show existence.
If $f$ would be an open embedding, we would have $f(U)$ open in $Y$ for all open subsets $U$ of $X$, and it would be natural to define $(f^\ast G)(f(U)) := G(U)$. To see that we indeed have a left adjoint by this definition is up to you, but it fails for a general $f$, since $f(U)$ needn't be open in general.
So, given a sheaf $G$ on $Y$ I define a new presheaf on $X$ by $U \mapsto \lim_{\rightarrow} G(V)$, where the limit ranges over all open subsets $V$ such that $V$ contains $f(U)$. By this "trick" we circumvent the given problem (and introduce new behaviour) and it turns out that this is a correct definition, in the technical sense that we really have found a left adjoint to $f_\ast$.

Proof of the adjunction $Hom(f^\ast G, F) = Hom(G,f_\ast F)$:
for an open subset $U$ of $X$, a homomorphism from $(f^\ast G)(U)$ to $F(U)$ is just a homomorphism from $\lim_{\rightarrow} G(V)$ to $F(U)$ and a homomorphism from $G(V)$ to $(f_\ast F)(V)$ is just a homomorphism from $G(V)$ to $F(f(V))$. So you see, if we have homomorphisms $G(V) \to F(f(V))$ for all $V$, this gives in the limit homomorphisms $\lim_{\rightarrow} G(V) \to \lim_{\rightarrow} F(f(V)) = F(U)$.
For the other direction, observe that if we have homomorphisms $\lim_{\rightarrow} G(V) \to F(U)$ for all $U$, we certainly have this for all $U=f(V)$, where the limit is just $G(V)$, i.e. where we have just $G(V) \to F(f(V))$.

Pushforward with compact support

We have already seen how pushforward generalizes global sections. As global sections give (as derived functor) cohomology of sheaves, there is a global section with compact support functor, which gives cohomology with compact support. For the locally constant sheaf $\mathbb{Z}$ this gives back "singular" cohomology with compact support, as it appears in Poincaré duality. I will explain this in some more detail now, although I won't explain how to move from global sections to cohomology.

Poincaré duality states, for a smooth compact complex n-dimensional manifold X

$$H^k(X;\mathbb{R}) \simeq H_{2n-k}(X;\mathbb{R})$$

$$H^k_c(X;\mathbb{R}) \simeq H_{2n-k}(X;\mathbb{R})$$

The functor of global sections with compact support $\Gamma_c$ is defined as

$$\Gamma_c(F,U) := \{ s \in F(U) | supp(s) \text{ compact} \} \subset F(U) = \Gamma(F,U).$$

By analogy, we define the pushforward with compact support $f_{!}$ as a subfunctor of $f_\ast$ (which just means that $f_{!} F$ will be a subsheaf of $f_\ast F$ for every $F$, which in turn just means that $(f_{!}F) U$ is a subset of $(f_\ast F) U$ for every open set $U$).

$$(f_{!}F)(U) := \{ s \in F(f^{-1}U) | f|supp(s) : supp(s) \to U \text{ proper}\}.$$

An example:
Let $f$ be an open embedding $f : U \to X$, then $f_{!} F$ is just the "extension by zero", i.e. the stalks at all points of $U$ are just the same as those of $F$, and all other/new stalks (over $X \setminus U$) are plain $0$.

Another example:
Let $f$ be a proper map $f : Y \to X$, then $f_{!} = f_\ast$, as you can see from the definition.

A comprehensive example:
If $f$ can be factored into $f = p \circ j$ with $j$ an open embedding and $p$ proper, we have $f_{!} = p_\ast \circ j_{!}$, which gives a very explicit description of $f_{!}$.

Exceptional inverse image

We define a functor $f^{!}$ called exceptional inverse image, as the right adjoint to $f_{!}$, if it exists. We should say straightforward, that it doesn't exists, in general, on the level of sheaves and this is one of the things that makes working with complexes of sheaves necessary (in fact, the derived category).

However, for innocent maps $f$, we can actually define a functor that is right adjoint to $f_{!}$ and thus deserves to be called $f^{!}$.

For f an open embedding $f : U \to X$, we have just $f^{!} = f^\ast$, i.e. the functor $f^\ast$ is the left adjoint to $f_\ast$ and also the right adjoint to $f_{!}$.
The proof is similar to the proof of the adjointness of $f^\ast$ with $f_\ast$, so I leave it out.

Now I want to make clear why a right adjoint to $f_{!}$ doesn't exist (on the level of sheaves) in general, for categorical reasons.

Every left adjoint functor preserves colimits, since an adjunction like

$$Hom(f_{!} F, G) \simeq Hom(F, f^{!} G)$$

Now being right-exact is a special case of preserving colimits, since it means to preserve cokernels (which are special colimits). Clearly, $f_{!}$ is not right-exact, since it has cohomology: let $X$ be a compact space and $f$ the constant map to a point. Then for $f_{!} = f_\ast = c_\ast \simeq \Gamma$ to be right-exact, the cohomology on $X$ must vanish.

The salvation consists of enlarging the category of sheaves to the category of chain complexes of sheaves, only to make it smaller again by introducing the appropriate definition of morphisms, which in the end gives what is called the derived category of abelian sheaves. There, a general $f^{!}$ exists.

Concrete examples for four functors

Let us look at the embedding $j : \mathbb{C}^\times \to \mathbb{C}$ and its closed complement $i : \{0\} \to \mathbb{C}$.

First we will look at a skyscraper sheaf on $\{0\}$ with stalk some abelian group $A$ over $0$. We denote the skyscraper sheaf by $F$. By definition, we have $i_{!} F = i_\ast F$ a skyscraper sheaf with stalk $A$ over $0$. Now $j^{!} i_\ast F = j^\ast i_\ast F = 0$, since $j^\ast$ throws away all information from the stalk over $0$.

Okay, let's look at a local system on $\mathbb{C}$, i.e. a locally constant sheaf $F$.
This is the same data (an equivalent category) as the monodromy representation of the fundamental group, in this case $\pi_1(\mathbb{C}^\times,1) \simeq \mathbb{Z}$.
We have as $j_{!} F$ a sheaf with stalks just $F_x$ where $x \neq 0$, and $(j_{!} F)_0 = 0$, since every section with compact support is away from an arbitrarily small ball around the origin.
The sheaf $j_\ast F$ has the same stalks $F_x$ where $x \neq 0$ but it has a new one at the origin, given by the usual stalk-limit-formula you would write down - and in general, this is non-zero.

Cleary $i^\ast j_{!} F$ vanishes, since $i^\ast$ picks the stalk at the origin and throws away everything else. Of course, $i^\ast j_\ast F$ contains exactly the "new" stalk which might be interesting.
Thinking about it, the sheaves $i^{!} j_{!} F$ and $i^{!} j_\ast F$ are both zero, by the same argument we had for $(j_{!} F)_0$. Here you can also use the adjunction for reasoning!

Last words

The nice thing about this setting is that it generalizes to give the following:

Take $V$ a closed subspace in $X$ and $U$ its open complement, then you have an open embedding $j$ and a closed embedding $i$ which behave very much like our $j$ and $i$ from the last examples. It presents the category of sheaves on $X$ as an extension of the sheaves on $V$ by the sheaves on $U$. The same happens for the derived category. The magic word for this situation is "Recollement".

Da ich nun schon mehrmals darauf angesprochen wurde, möchte ich gerne den entsprechenden Fragebogen online stellen. Man kann so eine Evaluation relativ leicht selbst durchführen. Wenn man als Tutor die Antworten auszählt, so erkennt man noch an der Handschrift, wer was geschrieben hat, also sollte man das den Tutanden überlassen. Bei einer Gruppengröße von 10-20 Leuten geht das aber ganz fix und sollte kein Problem sein. Eine Computer-Auszählung ist eigentlich gar nicht so dringend nötig.

Hier gibt es das fertige PDF zum Ausdrucken

und hier gibt es den Quelltext als LaTeX-Dokument.

Das ist natürlich nur ein Vorschlag; ich bitte darum, das LaTeX-Dokument nach Gutdünken umzugestalten und so zu verwenden, wie man es für sinnvoll hält.

Wenn es jemanden interessiert, kann ich auch noch ein bisschen Arbeit darein stecken, unsere Erfassungsformulare (in HTML+CGI-Skript) sowie die Auswertungssoftware (Python+R) online zu stellen, aber das CGI-Skript sowie das R-Skript sind nicht von mir, daher dauert das ein bisschen. Man kann prinzipiell auch eine Erfassung am Rechner durchführen, dazu muss man im LaTeX-Dokument nur einen "Absenden"-Button hinzufügen und ein passendes CGI-Skript verwenden.
Eine sehr einfache Online-Lösung kann man sich auch mit Google Documents zusammenklicken (und sicherlich gibt es andere Anbieter mit ähnlichen Lösungen).

Zum Inhaltlichen: Wir hatten damals ein bisschen über den Fragebogen diskutiert und es gab verschiedene Meinungen. Im Großen und Ganzen wurden Freitext-Felder nicht so intensiv genutzt, wie man es sich vielleicht wünschen würde. Der Rest ist aber eher positiv angekommen. Um ehrliches Feedback zu bekommen, ist die Anonymität der ausfüllenden sehr wichtig. Es hilft auch, wenn die Teilnehmer wissen, dass sie am Ende profitieren, wenn man z.B. eine Evaluation nach 5 Wochen Tutorat durchführt, und noch einige Zeit vor sich hat. Man sollte nicht den Fehler machen, den Studenten zu viele Fragen zu stellen, denn dann bekommt man statt wohlüberlegte Antworten nur "passt schon" zurück (weniger ist mehr).

An manchen Universitäten gibt es bereits spezielle Evaluations-Büros (in Karlsruhe habe ich das gesehen), die sich generalstabsmäßig darum kümmern. Aber auch hierbei geht es oft nicht so sehr um einzelne Übungsgruppen, denn um ganze Massenveranstaltungen. Deshalb kann ich eine Evaluation solcher Kleingruppen-Übungen sehr empfehlen. Es hilft gerade Neulingen, die noch nicht so genau wissen, ob die Studenten wegen ihnen nichts kapieren, oder ob es an etwas anderem liegt :-)

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".

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).

There are some aspects which require an explanation:

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!).

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.

The solution is to save each website of "your purchases" (or other such lists of books) as HTML file and let a smart script do the extraction work. This way, you're not violating Amazon's terms of service (which most likely don't allow any robots scraping the website) and on the positive side, it works.

Here is my python script, which you can also download here (in a better version):
import sys, re
asinRegExString = "<tr valign=middle id=\"iyrListItem([A-Z0-9]{10})\">"
asinRegEx = re.compile(asinRegExString)
filename = sys.argv[1]
f = open(filename,'r')
asinlist = []
for line in f.readlines():
    match = asinRegEx.match(line)
    if match != None:
        asinlist+=[match.group(1)]
f.close()
print "\n".join(asinlist)

To run this script, you need a Python interpreter. On most common GNU/Linux systems, those are installed or easily installable, for example by "apt-get install python" on Debian-based systems.

I have tested it with Amazon.de and the "purchased books" website but I guess it would work equally well with Amazon.co.uk and Amazon.com. As always, leave a comment if it worked for you or not. If it doesn't work or if you have different needs (like, extracting ISBN and name and price) this will be easily possible by altering the regular expressions in the script (easy for a programmer, not that easy for anyone else).

I used this to import all books I bought via Amazon into my Google Books library which I use to maintain a list of all books I own. The nice thing about Google Books, on the other hand, is their XML export feature, which I commented on earlier.

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.