Related papers: The GKZ hypergeometric $\mathcal D$-module

We study a class of spatial discretizations for the Vlasov-Poisson system written as an hyperbolic system using Hermite polynomials. In particular, we focus on spectral methods and discontinuous Galerkin approximations. To obtain L 2…

For all $k \ge 2$, we show that there exists a group $G$ and a non-free stably free $\mathbb{Z} G$-module of rank $k$. We use this to show that, for all $k \ge 2$, there exist homotopically distinct finite $2$-complexes with fundamental…

We study smooth higher symmetry groups and moduli $\infty$-stacks of generic higher geometric structures on manifolds. Symmetries are automorphisms which cover non-trivial diffeomorphisms of the base manifold. We construct the smooth higher…

Let X be a proper scheme and Z a prestack over X equipped with a flat connection. We give a local-to-global description of D-modules on the prestack S(Z) of flat sections of Z. Examples of S(Z) include the moduli stacks of principal…

Discontinuous Galerkin (DG) methods for hyperbolic partial differential equations (PDEs) with explicit time-stepping schemes, such as strong stability-preserving Runge-Kutta (SSP-RK), suffer from time-step restrictions that are…

We construct geometric models for classifying spaces of linear algebraic groups in G-equivariant motivic homotopy theory, where G is a tame group scheme. As a consequence, we show that the equivariant motivic spectrum representing the…

This paper is devoted to an elementary new construction of $1$-singular Gelfand-Tsetlin modules using complex geometry. We introduce a universal ring $\mathcal D_o$ together with the vector space $\mathcal S=\mathcal S(\mathcal D_o)$ with…

We describe the category of regular holonomic modules over the ring D[[h]] of linear differential operators with a formal parameter h. In particular, we establish the Riemann-Hilbert correspondence and discuss the additional t-structure…

In recent years, significant progress has been made in the study of integrable systems from a gauge theoretic perspective. This development originated with the introduction of $4$d Chern-Simons theory with defects, which provided a…

A modified $A$-hypergeometric system is a system of differential equations for the function $f(t^w \cdot x)$ where $f(y)$ is a solution of an $A$-hypergeometric system in $n$ variables and $w$ is an $n$ dimensional integer vector, which is…

Let $K$ be a field of characteristic zero, $R = K[X_1,\ldots,X_n]$. Let $A_n(K) = K<X_1,\ldots,X_n, \partial_1, \ldots, \partial_n>$ be the $n^{th}$ Weyl algebra over $K$. We consider the case when $R$ and $A_n(K)$ is graded by giving $\deg…

Homotopical algebraic $D$-geometry combines aspects of homotopical algebraic geometry of Toen and Vezzosi and $D$-geometry of Beilinson and Drinfeld. It was introduced by the paper's last two authors and di Brino as a suitable framework for…

Geometric integrators for nonholonomic systems were introduced by Cort\'es and Mart\'inez in [Nonholonomic integrators, Nonlinearity, 14, 2001] by proposing a discrete Lagrange-D'Alembert principle. Their approach is based on the definition…

For a connected reductive group $G$ and an affine smooth $G$-variety $X$ over the complex numbers, the localization functor takes $\mathfrak{g}$-modules to $D_X$-modules. We extend this construction to an equivariant and derived setting…

Let X be a smooth toric variety. David Cox introduced the homogeneous coordinate ring S of X and its irrelevant ideal B. Extending well-known results on projective space, Cox established the following: (1) the category of quasi-coherent…

We use weighted polynomial approximation to prove the existence of a compact set K with non-empty interior and a function f is dense in the space A(K) of all continuous functions on K that are holomorphic in the interior of K, endowed with…

Stable cohomology is a generalization of Tate cohomology to associative rings, first defined by Pierre Vogel. For a commutative local ring $R$ with residue field $k$, stable cohomology modules $\widehat{\mathrm{Ext}}{\vphantom…

We consider the quantized Knizhnik-Zamolodchikov difference equation (qKZ) with values in a tensor product of irreducible sl(2) modules, the equation defined in terms of rational R-matrices. We solve the equation in terms of…

In [DKY], it was conjectured that there is a uniform bound $B$, depending only on the degree $d$, so that any pair of holomorphic maps $f, g :\mathbb{P}^1\to\mathbb{P}^1$ with degree $d$ will either share all of their preperiodic points or…

We initiate a study of Hilbert modules over the polynomial algebra A=C[z_1,...,z_d] that are obtained by completing A with respect to an inner product having certain natural properties. A standard Hilbert module is a finite multiplicity…