Related papers: Explicit formulas for algebraic connections on ell…

It is well-known that del Pezzo surfaces of degree $9-n$ one-to-one correspond to flat $E_n$ bundles over an elliptic curve. In this paper, we construct $ADE$ bundles over a broader class of rational surfaces which we call $ADE$ surfaces,…

We show that isomorphism classes $[\mathcal{A}]$ of flat $q\times q$ matrix bundles $\mathcal{A}$ (or projectively flat rank-$q$ complex vector bundles $\mathcal{E}$) on a pro-torus $\mathbb{T}$ are in bijective correspondence with the…

We define and study the theory of derivation-based connections on a recently introduced class of bimodules over an algebra which reduces to the category of modules whenever the algebra is commutative. This theory contains, in particular, a…

We present an explicit and computationally actionable blueprint for constructing vector-valued Siegel modular forms associated to real multiplication (RM) abelian surfaces, leveraging the theta correspondence for the unitary dual pair…

For any additive functor from modules (or, more generally, from an abelian category with enough projectives or injectives), we construct long sequences tying up together the derived functors, the satellites, and the stabilizations of the…

For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs' result for a commutative noetherian ring. As a…

This paper explores a construction of the elliptic classes of the Springer resolution using the periodic Hecke module. The module is established by employing the Poincar\'e line bundle over the product of the abelian variety of elliptic…

In the paper I introduce a new characteristic class $c(E)$ for a finite rank vector bundle $E$ on an affine scheme $S:=Spec(A)$ - the fundamental class of $E$. The class $c(E)$ is not a characteristic class in the classical sense in the…

Associated to a symmetric space there is a canonical connection with zero torsion and parallel curvature. This connection acts as a binary operator on the vector space of smooth sections of the tangent bundle, and it is linear with respect…

We introduce graded $\mathbb{E}_{\infty}$-rings and graded modules over them, and study their properties. We construct projective schemes associated to connective $\mathbb{N}$-graded $\mathbb{E}_{\infty}$-rings in spectral algebraic…

We describe new families of the Knizhnik-Zamolodchikov-Bernard (KZB) equations related to the WZW-theory corresponding to the adjoint $G$-bundles of different topological types over complex curves $\Sigma_{g,n}$ of genus $g$ with $n$ marked…

Let $(G,H,\sigma)$ be a symmetric pair and $\mathfrak{g}=\mathfrak{m}\oplus\mathfrak{h}$ the canonical decomposition of the Lie algebra $\mathfrak{g}$ of $G$. We denote by ${\nabla}^0$ the canonical affine connection on the symmetric space…

In this paper, we construct new characteristic classes of fiber bundles via flat connections with values in infinite-dimensional Lie algberas of derivations. In fact, choosing a fiberwise metric, we construct a chain map to the de Rham…

We consider the notion of a connection on a module over a commutative ring, and recall the obstruction calculus for such connections. The obstruction calculus is defined using Hochschild cohomology. However, in order to compute with Grobner…

Equipping a non-equivariant topological $E_\infty$-operad with the trivial $G$-action gives an operad in $G$-spaces. For a $G$-spectrum, being an algebra over this operad does not provide any multiplicative norm maps on homotopy groups.…

If $\mathcal{M}=(M,\nabla)$ is an affine surface, let $\mathcal{Q}(\mathcal{M}):=\ker(\mathcal{H}+\frac1{m-1}\rho_s)$ be the space of solutions to the quasi-Einstein equation for the crucial eigenvalue. Let…

In two-dimensional loop models, the scaling properties of critical random curves are encoded in the correlators of connectivity operators. In the dense O($n$) loop model, any such operator is naturally associated to a standard module of the…

We define a congruence module $\Psi_A(M)$ associated to a surjective $\mathcal O$-algebra morphism $\lambda\colon A \to \mathcal{O}$, with $\mathcal{O}$ a discrete valuation ring, $A$ a complete noetherian local $\mathcal{O}$-algebra…

We study the moduli space $\mathbf{M}_X(\Lambda, n)$ of semistable $\Lambda$-modules of vanishing Chern classes over an abelian variety $X$, where $\Lambda$ belongs to a certain subclass of $D$-algebras. In particular, for $\Lambda =…

Connectivity is a homotopy invariant property of separable C*-algebras which has three notable consequences: absence of nontrivial projections, quasidiagonality and a more geometric realization of KK-theory for nuclear C*-algebras using…