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

4 votes
0 answers
115 views

Delta distributions that are smooth on strata of a singular manifold

This is a mild reformulation of a previous question. Let $R = C^\infty(\mathbb{R}^N)$ and let $I$ be an ideal in $R$ which cuts out an $n$-dimensional "singular $C^\infty$ manifold $X$" in $\mathbb{R} …
4 votes

About Lie group $G$ has this escape property?

Let me rewrite a comment I made that has a typo, since it's too late to edit it. The property fails for any compact connected noncommutative Lie group. Note that if $G$ is a compact group then it has …
Dmitry Vaintrob's user avatar
6 votes
0 answers
94 views

forms on singular spaces that can be integrated on an LCI

I'd like to characterize rational $k$-forms on a singular scheme $X$ which can be integrated on any (real) bounded submanifold that is locally LCI in $X$ in a real sense (i.e., which is the real preim …
10 votes
0 answers
340 views

Hodge structure and rational coefficients

Suppose $X$ is a complex projective variety with a model $X_\mathbb{Q}$ defined over the rational numbers. Then there is a rational de Rham lattice $H^k_{dR}(X_\mathbb{Q}, \mathbb{Q})\subset H^k(X, \m …
6 votes
1 answer
496 views

Path integral as quantum mechanics on the tangent bundle

Let $X$ be a configuration space, a finite-dimensional manifold. By "quantum mechanics on $X$" I mean a linear evolution equation on complex-valued functions on $X$, determined by a Hamiltonian $H\in …
5 votes
0 answers
114 views

Flattening a connection on a Kähler manifold

Say $M$ is a closed Kähler manifold and $(V, \nabla)$ is a (say) constant Hermitian bundle on $V$ with (say) trivial flat connection. Now $M$ Kähler gives several distinguished classes of closed one-f …
2 votes
1 answer
185 views

effectively distinguishing knots

It was proven, I think by Mijatović EDIT: by Waldhausen, that there is an effective algorithm for distinguishing knot complements (the effective constants were found by Coward and Lackenby). The bound …
12 votes
3 answers
2k views

Intuition for Levi-Civita connection via Hamiltonian flows

A Riemannian metric on a manifold $X$ defines a function on the symplectic space $T^*X$ whose Hamiltonian flow gives geodesics. Is there a similar interpretation of the Levi-Civita connection?
5 votes

What kind of geometric object is the Pauli spin matrix vector $\vec{\sigma} = (\sigma_1, \si...

The Pauli spin vector encodes a two-complex-dimensional unitary representation of the Lie algebra $\bf{su}_2$. A basis of the Lie algebra maps to the matrix algebra $\bf{gl}_2(\mathbb{C}),$ giving a $ …
Dmitry Vaintrob's user avatar