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What examples are there of striking applications of the ideas of microlocal analysis? Ideally I'm looking for specific results in any of the relevant fields (PDE, algebraic/differential geometry/topol …

Let $P$ be an analytic linear differential operator defined on some open interval $X=(a,b)$ and $\mathcal{M}=\mathcal{D}_X / \mathcal{D}_X \bullet P$ the corresponding $\mathcal{D}$-module. I'm trying …

Disclaimer: I know very little about both of the fields in question. My question is pretty simple: What's the relation between differential Galois theory and D-modules over algebraic curves? …

Let $R$ be a regular algebra over a field $k$ of char 0. Let $D$ be its corresponding algebra of differential operators. As in the general setting of non-commutative algebra we can tensor right $D$- …

Let $R$ be a regular ring over a field of char 0. Let $X=Spec R$ and $D=\mathcal{D}_X$ the algebra of differential operators over it. The overall vague question is what kind of algebraic object is $ …

Let $k$ be an algebraically closed field of characteristic 0. Consider the field of fractions of formal power series $K:= Frac(k[[T_1,...,T_n]])$. We have the corresponding algebra of differential ope …

While reading these notes by Victor Ginzburg on $D$-modules I found a certain construction of Microlocailzation in the algebraic setting which unfortunately doesn't seem to be elaborated on a lot in t …

Let $X$ be a smooth algebraic (/projective if it simplifies things considerably) variety over $\mathbb{C}$ and consider the derived category $\mathcal{C}=D_h^b(\mathcal{D}_X)$ of bounded complexes of …

I intentionally phrased the title to match a different question which is almost identical to the one i'm asking. However similar, the answer there, which uses commutative algebraic geometry, is not di …

Let $X$ be a smooth affine variety over $\mathbb{C}$ and let $\mathcal{D}_X$ be its algebra of differential operators. Consider $\mathcal{C}=\mathcal{D}_X$-$\text{mod}$, the stable $\infty$ category …

Let $X$ be a smooth curve over an algebraically closed field of characteristic 0. The category of quasi-coherent $D$-modules $\mathcal{D}_X$-$Mod$ is a symmetric monoidal abelian category. We can ther …

(All functors and categories are derived, all $\mathcal{D}$-modules are holonomic and $f: X \to Y$ is proper for simplicity). We may use the adjunction $\int_f \dashv f^!$ to get some interpretation …

Let $C$ be a smooth projective curve over an algebraically closed field $k$. The tangent lie algeborid $\mathcal{T}_C$ of $C$ is just sheaf of vector fields on $C$ equipped with the usual lie bracke …