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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
comment 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 G-structures and soldering forms) then torsion is the projection of its curvature under $\mathfrak{g}\to \mathfrak{g}/\mathfrak{h}$.
Dec
17
comment 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
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Dec
11
comment Artin L-function 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
comment Questions about analogy between Spec Z and 3-manifolds
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 3-manifold. 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 Bunke-Olbrich 94) and so that's the product over all determinants of Frobenius, hence is the Artin L-function.
Nov
28
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Nov
13
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Nov
13
reviewed Close Websites for Math Shopping
Nov
12
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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 cutting-edge modern physics may naturally be found and studied. That's the point of dcct ncatlab.org/schreiber/show/…
Nov
12
comment 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
comment 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+KL-axioms. 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 Kock-Lawvere 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