Related papers: A 'Darboux theorem' for derived schemes with shift…

We develop an approach that allows to construct semiorthogonal decompositions of derived categories of surfaces with cyclic quotient singularities whose components are equivalent to derived categories of local finite dimensional algebras.…

Let $L$ be a $4$th order differential operator with coefficients in $\mathbb{K}(z)$, with $\mathbb{K}$ a computable algebraically closed field. The operator $L$ is called symplectic when up to rational gauge transformation, the fundamental…

This paper studies the derived category of the Quot scheme of rank $d$ locally free quotients of a sheaf $\mathscr{G}$ of homological dimension $\le 1$ over a scheme $X$. In particular, we propose a conjecture about the structure of its…

This work contains a brief and elementary exposition of the foundations of Poisson and symplectic geometries, with an emphasis on applications for Hamiltonian systems with second-class constraints. In particular, we clarify the geometric…

A stable generalized complex structure is one that is generically symplectic but degenerates along a real codimension two submanifold, where it defines a generalized Calabi-Yau structure. We introduce a Lie algebroid which allows us to view…

In this paper we derive new two-component integrable differential difference and partial difference systems by applying a Lax-Darboux scheme to an operator formed from an ${\mathfrak{sl}}_3({\mathbb{C}})$-based automorphic Lie algebra. The…

This paper studies the Cohomological Donaldson-Thomas theory of loop stacks of $0$-shifted symplectic stacks. In particular, we compare $(-1)$-shifted tangent stacks of these moduli problems, which we view as additive, to loop stacks, which…

In this paper we study derived equivalences for Symplectic reflection algebras. We establish a version of the derived localization theorem between categories of modules over Symplectic reflection algebras and categories of coherent sheaves…

We investigate the geometric and topological properties of the group of locally conformally symplectic (LCS) diffeomorphisms, utilizing the LCS flux homomorphism defined by S. Haller. By analyzing the flux map from the universal cover of…

We prove a conjecture of Hiraga-Ichino-Ikeda relating formal degrees of square-integrable representations to adjoint gamma factors for symplectic and even orthogonal groups over characteristic zero non-Archimedean local fields. The proof is…

In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves…

Let $G$ be a finite subgroup of $\mathrm{SU}(4)$ whose elements have age not larger than one. In the first part of this paper, we define $K$-theoretic stable pair invariants on the crepant resolution of the affine quotient $\mathbb{C}^4/G$,…

We solve the $K$-theoretically refined Donaldson-Thomas theory of local curves. Our results avoid degeneration techniques, but rather exploit direct localisation methods to reduce the refined Donaldson-Thomas partition function to the…

Cases of Deligne's companion conjecture for normal schemes over finite fields have been proven by L. Lafforgue, Drinfeld, and Zheng in recent years: L. Lafforgue proved the conjecture for curves, Drinfeld proved the conjecture for all…

We develop a framework for derived deformation theory, valid in all characteristics. This gives a model category reconciling local and global approaches to derived moduli theory. In characteristic 0, we use this to show that the homotopy…

We exhibit two three-parameter families of locally conformal symplectic forms on the solvmanifold $M_{n,k}$ considered in [1], and show, using the Hodge-de Rham theory for the Lichnerowicz cohomology that that they are not $d_{\omega}$…

We develop a characterisation of non-Archimedean derived analytic geometry based on dg enhancements of dagger algebras. This allows us to formulate derived analytic moduli functors for many types of pro-\'etale sheaves, and to construct…

In the generalized Hamiltonian formalism by Dirac, the method of constructing the generator of local-symmetry transformations for systems with first- and second-class constraints (without restrictions on the algebra of constraints) is…

Starting from an anti-symplectic involution on a K3 surface, one can consider a natural Lagrangian subvariety inside the moduli space of sheaves over the K3. One can also construct a Prymian integrable system following a construction of…

We give two proofs to the following theorem and its generalization: if a finite dimensional algebra $A$ is derived equivalent to a smooth projective scheme, then any derived equivalence between $A$ and another algebra $B$ is standard, that…