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Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

34 votes
1 answer
3k views

Does the Pfaffian have a geometric meaning?

While reviewing the proof of Gauss-Bonnet in John Lee's book, I noticed the following paragraph: " ...In a certain sense, this might be considered a very satisfactory generalization of Gauss-Bonnet. …
Bombyx mori's user avatar
  • 6,249
19 votes

Atiyah's May 2018 paper on the 6-sphere

Since the question has been "hanging on" for a while, I think it makes sense to give an outline of Atiyah's argument in the paper. Note that the paper is short (page 1 introduction, page 5-6 reference …
Bombyx mori's user avatar
  • 6,249
18 votes
1 answer
2k views

Did differential geometry undergo a notation change?

As a graduate student, I found the old books of differential geometry used a different set of notation from modern textbooks. For example, Chern and Milnor defined the curvature 2-form by $d\omega-\om …
Bombyx mori's user avatar
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10 votes
Accepted

what is the universal cover of GL(2,R)?

For the sake of completeness here is an explicit proof. Let $m\in GL_{2}(\mathbb{R})^{+}$, then $m\rightarrow \frac{m}{\det(m)}$ maps it to an element in $SL_{2}(\mathbb{R})$. And we know $SL_{2}(\mat …
Bombyx mori's user avatar
  • 6,249
7 votes
0 answers
278 views

How to interpret heat kernel at unit time on a Riemann surface?

Let $M$ be a compact Riemann Surface. Let $P$ be a fixed point on $M$ and let $\delta_{P}$ be the Dirac point distribution at $M$. Consider the fundamental solution of the heat equation $$ (\partial_{ …
Bombyx mori's user avatar
  • 6,249
6 votes

Theta functions on an elliptic curve and Serre duality

Here is a 'low-brow' approach. One type of the result you are talking about has been written up implicitly in Lang's book Introduction to Arakelov theory. The case for cohomology of the elliptic curv …
Bombyx mori's user avatar
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5 votes

Mathematical uses of string theory

I recall that Richard Wentworth's first paper on precise constants in bosonization formula (which is part of his PhD thesis?) extensively used computational methods from bosonic string theory. I am no …
5 votes
1 answer
404 views

On Ray-Singer's proof of the metric invariance of analytical torsion

The Ray-Singer paper "R-torsion and the Laplacian on Riemannian manifolds" claimed that one may prove the metric invariance of analytical torsion by forming a homotopy between metric $\rho_{0},\rho_{1 …
Bombyx mori's user avatar
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4 votes

Constant term in Green's kernel expansion

This turned out to be a highly non-trivial topic. For a detailed analysis, see Jorgenson and Kramer's paper Bounds on canonical Green functions. Their result can be summarized as $$ g_{\textrm{can},X …
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3 votes

On determinants of Laplacians on Riemann surfaces

I do not know if this is really the end of the story. You may be interested in the following paper by Jay Jorgenson. Basically he extended the work by Ray-Singer by fixing all unknown invariants, bu …
Bombyx mori's user avatar
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2 votes
Accepted

Reference for Weyl's law for higher order operators on closed Riemannian manifolds

One possible reference is Seeley's paper on Complex powers of Elliptic Operators, where Seeley did it for the Laplacian (page 6). But the discussion carries over to all elliptic $\Psi DO$s without muc …
Bombyx mori's user avatar
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2 votes
Accepted

Atiyah-Patodi-Singer for manifolds with cusps

This type of questions has been investigated systematically by Melrose in the framework of 'c-calculus', where $c$ stands for the cusp. The basic idea, if I recall correctly is to blow up the boundary …
Bombyx mori's user avatar
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2 votes

Gaussian upper heat kernel bounds on closed Riemannian manifolds

This is related to Li-Yau gradient estimates. I think the usual set up is for a complete manifold with $\textrm{Ric}(M)>-k$. You probably need some Harnark type inequalities for parabolic equations. I …
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1 vote

K-homology classes of Dirac operators on Hermitian manifolds

I am not entirely familiar with $KK$-theory, so please correct me if there are mistakes. I think ultimately you are trying to show the topological $K$-theory class you get from taking the horizontal d …
Bombyx mori's user avatar
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1 vote

Accessible reference for (scattering) $\Psi DO$'s on manifolds

I take a very brief look at the paper and I did not see $\Psi DO$ on manifold with boundary being used heavily anywhere (no conormal distribution, multiple blow-ups, heavy handed symbol estimates, etc …

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