相关论文: On the dimension formula for the hyperfunction sol…
In this text, we illustrate the use of local methods in the theory of (irregular) holonomic D-modules. I. (The Euler characteristic of the de~Rham complex) We show the invariance of the global or local Euler characteristic of the de~Rham…
In an influential $L^2$ extension theorem due to Demailly, the finiteness of an $L^2$ norm called the Ohsawa norm determines whether a given holomorphic function can be extended. This result has been further generalized by Zhou and Zhu to…
We establish a dimension formula for certain union $X^G(\mu,b)_J$ of affine Deligne-Lusztig varieties associated to arbitrary parahoric level structures of split reductive groups, under certain genericity hypotheses.
We introduce a deformation of Cayley's second hyperdeterminant for even-dimensional hypermatrices. As an application, we formulate a generalization of the Jacobi-Trudi formula for Macdonald functions of rectangular shapes generalizing…
When ${\cal{D}}$ is a linear partial differential operator of any order, a direct problem is to look for an operator ${\cal{D}}_1$ generating the compatibility conditions (CC) ${\cal{D}}_1\eta=0$ of ${\cal{D}}\xi=\eta$. We may thus…
We give complete and exact descriptions of spaces of ultradifferentiable functions that are closed under composition with either holomorphic or ultradifferentiable functions -- which are two distinct cases. The proof works by considering…
It is well-known that any solution of the Laplace equation is a real or imaginary part of a complex holomorphic function. In this paper, in some sense, we extend this property into four order hyperbolic and elliptic type PDEs. To be more…
The codimension-three conjecture states that any regular holonomic module extends uniquely beyond an analytic subset with codimension equal to or larger than three. We give a proof of this conjecture.
Fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients were constructed recently. These fundamental solutions are directly connected with multiple Lauricella hypergeometric function and…
We study some size estimates for the solution of the equation d-bar u=f in one variable. The new ingredient is the use of holomorphic functions with precise growth restrictions in the construction of explicit solutions to the equation.
A new family of solutions of the Jacobi partial differential equations for finite-dimensional Poisson systems is investigated. This family is mathematically remarkable, as the functional dependences of the solutions appear to be associated…
We define for arbitrary modules over a finite von Neumann algebra $\cala$ a dimension taking values in $[0,\infty]$ which extends the classical notion of von Neumann dimension for finitely generated projective $\cala$-modules and inherits…
In this contribution we first summarize how contour integration methods can be used to derive closed formulae for functional determinants of ordinary differential operators. We then generalize our considerations to partial differential…
A solution to the effectiveness problem in Kohn's algorithm for generating subelliptic multipliers is provided for domains that include those given by sums of squares of holomorphic functions (also including infinite sums). These domains…
In this note we sketch a proof of a fundamental conjecture, the codimension-three conjecture, for microdifferential holonomic systems with regular singularities. It states that any regular holonomic E-module extends beyond a…
This paper considers a tiny modification of Justin Campbell's construction of the Kontsevich compactification in arXiv:1606.01518 [math.AG]. We construct a resolution of singularities of Drinfeld compactification with an Iwahori structure…
G\"{o}ttsche-Nakajima-Yoshioka K-theoretic blowup equations characterize the Nekrasov partition function of five dimensional $\mathcal{N}=1$ supersymmetric gauge theories compactified on a circle, which via geometric engineering correspond…
In his notebooks, Ramanujan presented without proof many remarkable formulae for the solutions to generalized modular equations. Much later, proofs of the formulae were provided by making use of highly nontrivial identities for theta series…
In the previous papers \cite{L1, L2} the author constructed Mabuchi and Aubin-Yau functionals over any complex surfaces and three-folds, respectively. Using the method in \cite{L2}, we construct those functionals over any complex manifolds…
A celebrated theorem of Harich-Chandra asserts that all invariant eigendistributions on a semisimple Lie group are locally integrable functions. We show that this result is a consequence of an algebraic property of a holonomic D-module…