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Missing from braid group was the precise geometric definition, so I put that in.
am starting something at logarithmic cohomology operation, but so far there are just some general statements and some references
Finally created thick subcategory theorem with a quick statement of the theorem and a quick pointer to how this determines the prime spectrum of a monoidal stable (∞,1)-category of the (∞,1)-category of spectra.
Cross-linked vigorously with related entries.
Created some minimum at Bousfield-Kuhn functor, for the moment just so as to record some references.
Created BICEP2, currently with the following text:
BICEP2 is the name of an astrophysical experiment which released its data in March 2014. The experiment claims to have detected a pattern called the “B-mode” in the polarization of the cosmic microwave background (CMB).
This data, if confirmed, is widely thought to be due to a gravitational wave mode created during the period of cosmic inflation by a quantum fluctuation in the field of gravity which then at the era of decoupling left the characteristic B-mode imprint on the CMB. This fact alone is regarded as further strong evidence for the already excellent experimental evidence for cosmic inflation as such (competing models did not predict such gravitational waves to be strong enough to be detectable in this way).
What singles out the BICEP2 result over previous confirmations of cosmic inflation is that the data also gives a quantitative value for the energy scale at which cosmic inflation happened (the mass of the hypothetical inflaton), namely at around GeV. This is ntoeworthy as being only two order of magnituded below the Planck scale, and hence 12 or so orders of magnitude above energies available in current accelerator experiments (the LHC). Also, it is at least a curious coincidence that this is precisely the hypothetical GUT scale.
It is thought that this value rules out a large number of variant models of cosmic inflation and favors the model known as chaotic inflation.
am creating a table modalities, closure and reflection - contents and adding it as a floating table of contents to relevant entries
just for completeness so that I don’t have gray links elsewhere, I have created some minimum (nothing exciting) at quantum fluctuation.
Started something at chaotic inflation.
Created a webpage for Florian Ivorra, to eliminate a grey link.
I did some editing at exponential function, to restore what I had believed to be a clear argument, which had been edited out by Colin Tan in favor of his own argument. His argument has been moved to a footnote.
added a category:reference entry for
Loop Groups, Characters and Elliptic Curves
talk at ASPECTS of Topology, 17-19 December 2012, Oxford,
on (geometric) representation theory reflected in and incarnated as low-dimensional extended gauge field theory and specifically on formalizations of geometric Langlands duality in terms of 2-dimensional QFT motivated from compactification of N=4 D=4 super Yang-Mills theory.
added to chiral differential operator a paragraph briefly summarizing how the Witten genus of a complex manifolds is constructed by Gorbounov, Malikov and Schechtman. Copied the same paragraph also into the Properties-section at Witten genus
created a minimum at Hirzebruch-Riemann-Roch theorem (I really only wanted to un-gray links at Todd genus)
As we discussed in another thread but maybe it should be announced separately: I have started an entry Dirac operator on smooth loop space with some bare minimum.
Wrote up more stuff at pi.
Incidentally, there are some statements at irrational number that look a little peculiar to me. For example:
In the early modern era, Latin mathematicians began work with imaginary numbers, which are necessarily irrational. They subsequently proved the irrationality of pi, (…)
I suppose Legendre could qualify as a “Latin mathematician of the early modern era” if we take a sufficiently broad view (e.g., he spoke a language in the Latin “clade”), but somehow I feel this is not what the author really had in mind; there were those Renaissance-era Italians who began work with imaginaries IIRC. :-) Probably it would be good to rephrase slightly.
Also this:
There is an easy nonconstructive proof that there exist irrational numbers and such that is rational; let be and let be either or , depending on whether the latter is rational or irrational. A constructive proof is much harder
Not that hard actually: take and , where if I did my arithmetic correctly. Pretty sure that can be made constructive. (Again, I think it’s probably just a case of several thoughts being smooshed together.)
started some minimum at KO-dimension
added a new Examples-section Integral versus real cohomology to fiber sequence
(and renamed the original fibration sequence and made it a redirect to that – but the cache bug is in the way,as usual).
created stub for spectral action
Back in the days I had made several web postings on the “FRS formalism” and how it may be understood as rigorously implementing “holography” in the form of CS/WZW-correspondence. Ever since the nLab came into being there was a stub entry FFRS-formalism which collected some (not all) of these links.
Now I got a question on how it works. (As a student one cannot imagine yet that communication in academia/maths often has latency periods of several years….) While I have absolutely no time for this now, this afternoon I went and expanded that stub entry a bit more (and maybe it’s at least good for my own sanity in these days). Also renamed it to something more suggestive, now it is titled
There is still plenty and plenty of room to expand further (urgent would be to mention the tensor produc of the MTC with its dual, which currently the entry is glossing over), but I am out of time now.
I have added to SimpSet a list of a few properties of the internal logic of the 1-topos of simplicial sets.
Thomas Nikolaus recently gave an impressive talk in which he announced a number of impressive results in topological T-duality. I have already been referring to this from the page T-duality 2-group and I just felt I want to refer to it from elsewhere, too. So therefore I now gave it a category:reference entry:
started some minimum at geometric engineering of quantum field theory
Started some minimum at Stolz conjecture.
I gave the category:people entry Daniel Freed a bit of actual text. Please feel invited to edit further. Currently it reads as follows:
Daniel Freed is a mathematician at University of Texas, Austin.
Freed’s work revolves around the mathematical ingredients and foundations of modern quantum field theory and of string theory, notably in its more subtle aspects related to quantum anomaly cancellation (which he was maybe the first to write a clean mathematical account of). In the article Higher Algebraic Structures and Quantization (1992) he envisioned much of the use of higher category theory and higher algebra in quantum field theory and specifically in the problem of quantization, which has – and still is – becoming more widely recognized only much later. He recognized and emphasized the role of differential cohomology in physics for the description of higher gauge fields and their anomaly cancellation. Much of his work focuses on the nature of the Freed-Witten anomaly in the quantization of the superstring and the development of the relevant tools in supergeometry, and notably in K-theory and differential K-theory. More recently Freed aims to mathematically capture the 6d (2,0)-superconformal QFT.
In the category:people entry Vladimir Voevodsky I’d like to have a brief statement on the motivic work. I have now put in the following, but experts are please asked to fine tune this where necessary:
Владимир Воеводский (who publishes in English as Vladimir Voevodsky) (web site) is a famous mathematician.
he received a Fields medal in 2002 for a proof of the Milnor conjecture. The proof crucially uses A1-homotopy theory and motivic cohomology developed by Voevodsky for this purpose. In further development of this in 2009 Voevodsky announced a proof of the Bloch-Kato conjecture.
After this work in algebraic geometry, cohomology, homotopy theory Voevodsky turned to the foundations of mathematics and is now working on homotopy type theory which he is advertizing as a new “”univalent foundations” for modern mathematics.
I'm trying to keep some results where I can get at them at Taylor's theorem.
created computable function (analysis) with the definition of “continuously realizable functions”.
I ended up giving Baire space (computability) its (minimal) stand-alone entry after all
I had created a stub for effective topological space and cross-linked with equilogical space. Added some pointers to the literature, but otherwise no real content yet.
An entry which defines both the local category and the local Grothendieck category, two notions which generalize the notion of a category of modules over a commutative local ring.
stub for clutching construction
New entry for Samuel compactification of uniform spaces, and some references at uniform space.
started something at Church-Turing thesis, please see the comments that go with this in the thread on ’computable physics’.
This is clearly just a first step, to be expanded. For the moment my main goal was to record the results about physical processes which are not type-I computable but are type-II computable.
Old discussion at star-autonomous category, which I think was addressed in the entry, and which I’m now moving here:
+–{: .query} Mike: Can someone fill in some examples of -autonomous categories that are not compact closed?
Finn: Blute and Scott in ’Category theory for linear logicians’ (from here) give an example: reflexive topological vector spaces where the topologies are ’linear’, i.e. Hausdorff and with 0 having a neighbourhood basis of open linear subspaces; ’reflexive’ meaning that the map as above is an isomorphism. It seems this category is -autonomous but not compact. I don’t know enough topology to make much sense of it, though.
Todd: Finn, I expect that example is in Barr’s book, which would then probably have a lot of details. But I must admit I have not studied that book carefully. Also, the Chu construction was first given as an appendix to that book.
John: I get the impression that a lot of really important examples of -autonomous categories — important for logicians, anyway — are of a more ’syntactical’ nature, i.e., defined by generators and relations. =–
I am working on entries related to the (oo,1)-Grothendieck construction
started adding a bit of structure to (oo,1)-Grothendieck construction itself, but not much so far
added various technical details to model structure on marked simplicial over-sets
created stub for model structure for left fibrations to go in parallel with that
I gather the following is true and is shown in Battenfield-Schröder-Simpson (pdf), but I haven’t really fully absorbed yet how is actually embedded in .
The subcategory on the effectively computable morphisms of the function realizability topos is the Kleene-Vesley topos . The category of “admissible representations” (whose morphisms are computable functions (analysis), see there) is a reflective subcategory of (BSS) and the restriction of that to is
This is currently stated this way in the entry function ralizability and computable function (analysis), but please criticize/handle with care, I’ll try to further fine-tune as need be.
added to Kleene’s second algebra under “Properties” the sentence:
Kleeen’s second algebra is an abstraction of function realizability introduced for the purpose of extracting computational content from proofs in intuitionistic analysis. (e.g. Streicher 07, p. 17)
Am starting an entry computable physics. For the moment this is essentialy a glorified lead-in for
I had had the feeling that most previous literature on computability in physics is suffering from being not well informed of the relevant mathematical concepts, but then I found
which seems to be sober, well-informed and sensible. The main drawback seems to be, to me, that the author looks only at type-I computability and not really seriously at quantum physics. Both of this is what Streicher’s note above aims to do!
If anyone has more pointers to decent literature on this topic, please drop me a note.
Here is what it currently has in the entry text computable physics:
The following idea or observation or sentiment has been expressed independently by many authors. We quote from Szudzik 10, section 2:
The central problem is that physical models use real numbers to represent the values of observable quantities, Careful consideration of this problem, however, reveals that the real numbers are not actually necessary in physical models. Non-negative integers suffice for the representation of observable quantities because numbers measured in laboratory experiments necessarily have only finitely many digits of precision.
Diverse conclusions have been drawn from this. One which seems useful and well-informed by the theory of computability in mathematics is the following (further quoting from Szudzik 10, section 2)
So, we suffer no loss of generality by restricting the values of all observable quantities to be expressed as non-negative integers — the restriction only forces us to make the methods of error analysis, which were tacitly assumed when dealing with real numbers, an explicit part of each model.
In type-I computability the computable functions are partial recursive functions and in view of this some authors conclude (and we still quote Szudzik 10, section 2) for this:
To show that a model of physics is computable, the model must somehow be expressed using recursive functions.
However, in computability theory there is also the concept type-II computable functions used in the field of “constructive analysis”, “computable analysis”. This is based on the idea that for instance for specifying computable real numbers as used in physics, an algorithm may work not just on single natural numbers, but indefinitely on sequences of them, producing output that is in each step a finite, but in each next step a more accurate approximation.
!include computable mathematics – table
This concept of type-II computability is arguably closer to actual practice in physics.
Of course there is a wide-spread (but of course controversial) vague speculation (often justified by alluding to expected implications of quantum gravity on the true microscopic nature of spacetime and sometimes formalized in terms of cellular automata, e.g. Zuse 67) that in some sense the observable universe is fundamentally “finite”, so that in the end computability is a non-issue in physics as one is really operating on a large but finite set of states.
However, since fundamental physics is quantum physics and since quantum mechanics with its wave functions, Hilbert spaces and probability amplitudes invokes (functional) analysis and hence non-finite mathematics even when describing the minimum of a physical system with only two possible configurations (a “qbit”) a strict finitism perspective on fundamental physics runs into severe problems and concepts of computable analysis would seem to be necessary for discussing computability in physics.
This issue of computable quantum physics has only more recently been considered in (Streicher 12), where it is shown that at least a fair bit of the Hilbert space technology of quantum mechanics/quantum logic sits inside the function realizability topos .
I have started something at computability.
Mainly I was after putting some terms in organized context. That has now become
which I have included under “Related concepts” in the relevant entries.
at decidable proposition I found the simple basic idea a bit too deeply hidden in the text. In an attempt to improve on this I have added right before the subsections of the Idea-section this quick preview:
External decidability: either or may be deduced in the metalanguage;
Internal decidability: may be deduced, hence “ or not ” holds in the object language.
Okay?
created a minimum at computable real number, for the moment just so as to record the references with section numbers as given there.
happened to need Type Two Theory of Effectivity
just in case you are watching the logs and are wondering:
I think we should have another “floating table of contents” for collecting the topic cluster
so I am starting one at constructivism - contents and am including it into relevant entries.
But right now there is nothing much there yet. This is going to be expanded.
it seems that orthomodular lattice had been missing, so I created a bare minimum
stub tertiary radical with redirects third radical, tertiary decomposition theory
Created a brief entry transfer context in order to record an observation by Haugseng.
He defines a transfer context to be a linear homotopy-type theory aka Wirthmüller context in which not only but also satisfies its projection formula. Then he observes that a natural Umkehr map that may be built with this projection formula is (the abstract generalization of) the Becker-Gottlieb transfer.
(Have briefly cross-linked with these related entries.)
Thanks to Thomas Nikolaus for being reminded of Haugseng’s work when Joost Nuiten and me talked about something closely related as ESI yesterday.
uniform module (related also to essential submodule)
I created a link to axiom of pairing from constructible universe, then satisfied it.
I added some basic definitions to stability in model theory. No attempt yet to motivate them.
Some of the logic entries seem to be in a slight state of neglect, e.g., theory. I might want to get in there sometime soon, but anyone should please feel free to precede me.
I am adding some bits and pieces to geometric stability theory (which I am trying to learn more about these days).
created a stub for Chern-Simons gravity. But nothing much there yet.
started Hecke category with some bare minimum
started a bare minimum at horocycle correspondence
for some reason it seems we never had an entry compactly generated (infinity,1)-category (and out of all sections listed at HTT just 5.5.7 had been missing for some reason which is a mystery to me now).
I gave it a minimum of content. But this alerts one that there is a distinction being made here which we don’t have in the corresponding 1-categorical entries.
I’ve been adding material to Polish space, and plan on adding more (mostly in view of model-theoretic considerations).