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I am a postdoc in maths with a degree in theoretical physics. I am interested in mathematical structures in quantum field theory and string theory, see here.
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Dec 18 
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What is torsion in differential geometry intuitively?
Further putting this all together, it seems we should say that given an $(H \to G)$Cartan connection ncatlab.org/nlab/show/Cartan+connection (which subsumes Gstructures and soldering forms) then torsion is the projection of its curvature under $\mathfrak{g}\to \mathfrak{g}/\mathfrak{h}$. 
Dec 17 
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What, precisely, does Klein's Erlangen Program state?
Riemannian geometry (and many other types of geometries) is subsumed by the globalization of Klein geometry known as "Cartan geometry". ncatlab.org/nlab/show/Cartan+geometry 
Dec 15 
awarded  Announcer 
Dec 11 
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Artin Lfunction and Zeta function of twisted Dirac operator
John Baez points out that discussion of more analogy along these lines is in D. Brown "Lifting properties of prime geodesics" ncatlab.org/nlab/show/Selberg+zeta+function#Brown09 . On p. 9 there is a comprehensive table of pertinent analogies. 
Dec 10 
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Questions about analogy between Spec Z and 3manifolds
I am wondering if it is not better to think of the prime fields as being not like chosen knots, but like all the prime geodesics in a hyperbolic 3manifold. Then everything falls into place: we want the Selberg zeta function given as a product over determinants of holonomies over all these prime geodesics (as in BunkeOlbrich 94) and so that's the product over all determinants of Frobenius, hence is the Artin Lfunction. 
Nov 28 
awarded  Announcer 
Nov 13 
awarded  Custodian 
Nov 13 
reviewed  Close Websites for Math Shopping 
Nov 12 
awarded  Announcer 
Nov 12 
comment 
Synthetic vs. classical differential geometry
Using modern developments in type theory, one may take this quite far by formalulating the axioms for SDG, incarnated in the guise of (differential) cohesion, in the new foundations given by homotopy type theory. Doing so leads to "cohesive homotopy type theory" ncatlab.org/nlab/show/cohesive+homotopy+type+theory . This connects to the question about applications of SDG in physics: in cohesive HoTT a large amount of cuttingedge modern physics may naturally be found and studied. That's the point of dcct ncatlab.org/schreiber/show/… 
Nov 12 
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Synthetic vs. classical differential geometry
+QiaochuYuan makes a nice point above about how synthetic reasoning is about passing from axioms for objects to axioms for their categories. Another perspective on that is foundational: Everyone is familiar with how the ZFC axioms lay foundations for sets. Similarly the axioms of "intuitionistic type theory" lay foundations for sets that may have geometric structure (may be interpreted as sheaves of sets). In "synthetic" reasoning we are adding axioms to this intuitionistic type theory that further specify the nature of this geometry. This way SDG connects to the foundations of mathematics. 
Nov 12 
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Synthetic vs. classical differential geometry
Consider the standard textbook on SDG by Anders Kock (ncatlab.org/nlab/show/synthetic+differential+geometry#KockBookA). The first chapter titled "The synthetic theory" develops differential geometry purely logically just by speaking intuitionistic logic+KLaxioms. Then chapter II "Categorical logic" recalls what it means to have a model for these axioms in a category, and finally chapter III "Models" discusses actual such models in certain categories. It may be useful to read these chapters in reverse order, because chapter III connects to classical theory, while I abstracts from that. 
Nov 12 
comment 
Synthetic vs. classical differential geometry
Yes, sorry, I should have used less jargon. As David Roberts and Qiaochu Yuan point out, the "models" that I was referring to are simply the categories (the toposes) which satisfy the axioms of SDG (or else those of differential cohesion). SDG in itself is the theory, in the formal sense of formal logic, obtained by starting with intuitionistic set theory and adding to it the KockLawvere axiom. The act of "speaking in this logic" is what people refer to when they say that "in SDG every function is smooth". This logic now has "models" in terms of categories built in ordinary set theory. 
Nov 12 
awarded  Nice Answer 
Nov 12 
revised 
Synthetic vs. classical differential geometry
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Nov 12 
revised 
Synthetic vs. classical differential geometry
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Nov 12 
answered  Synthetic vs. classical differential geometry 
Nov 11 
awarded  Revival 
Nov 4 
reviewed  Approve A technical question in Feix's construction of hyperkahler metric on cotangent bundles 