We associate a bivariant theory to any suitable oriented Borel–Moore homology theory on the category of algebraic schemes or the category of algebraic $G$-schemes. Applying this to the theory of algebraic cobordism yields operational cobordism rings and operational $G$-equivariant cobordism rings associated to all schemes in these categories. In the case of toric varieties, the operational $T$-equivariant cobordism ring may be described as the ring of piecewise graded power series on the fan with coefficients in the Lazard ring.

Let $Q$ be a quiver, $M$ a representation of $Q$ with an ordered basis $\mathcal{ℬ}$ and $\underset{¯}{e}$ a dimension vector for $Q$. In this note we extend the methods of Lorscheid (2014) to establish Schubert decompositions of quiver Grassmannians Gr${}_{\underset{¯}{e}}\left(M\right)$ into affine spaces to the ramified case, i.e., the canonical morphism $F:T\to Q$ from the coefficient quiver $T$ of $M$ w.r.t. $\mathcal{ℬ}$ is not necessarily unramified.

In particular, we determine the Euler characteristic of Gr${}_{\underset{¯}{e}}\left(M\right)$ as the number of extremal successor closed subsets of $T_0$, which extends the results of Cerulli Irelli (2011) and Haupt (2012) (under certain additional assumptions on $\mathcal{ℬ}$).

We construct the elliptic Painlevé equation and its higher dimensional analogs as the action of line bundles on $1$-dimensional sheaves on noncommutative surfaces.

We introduce a new class of “electrical” Lie groups. These Lie groups, or more precisely their nonnegative parts, act on the space of planar electrical networks via combinatorial operations previously studied by Curtis, Ingerman and Morrow. The corresponding electrical Lie algebras are obtained by deforming the Serre relations of a semisimple Lie algebra in a way suggested by the star-triangle transformation of electrical networks. Rather surprisingly, we show that the type $A$ electrical Lie group is isomorphic to the symplectic group. The electrically nonnegative part ${\left({EL}_{2n}\right)}_{\ge 0}$ of the electrical Lie group is an analogue of the totally nonnegative subsemigroup ${\left(U_n\right)}_{\ge 0}$ of the unipotent subgroup of ${SL}_n$. We establish decomposition and parametrization results for ${\left({EL}_{2n}\right)}_{\ge 0}$, paralleling Lusztig’s work in total nonnegativity, and work of Curtis, Ingerman and Morrow and of Colin de Verdière, Gitler and Vertigan for networks. Finally, we suggest a generalization of electrical Lie algebras to all Dynkin types.

We prove the Kac–Wakimoto character formula for the general linear Lie superalgebra $\mathfrak{g}\mathfrak{l}\left(m|n\right)$, which was conjectured by Kac and Wakimoto in 1994. This formula specializes to the well-known Kac–Weyl character formula when the modules are typical and to the Weyl denominator identity when the module is trivial. We also prove a determinantal character formula for KW-modules.

In our proof, we demonstrate how to use odd reflections to move character formulas between the different sets of simple roots of a Lie superalgebra. As a consequence, we show that KW-modules are precisely Kostant modules, which were studied by Brundan and Stroppel, thus yielding a simple combinatorial defining condition for KW-modules and a classification of these modules.

We obtain an effective version of Matsusaka’s theorem for arbitrary smooth algebraic surfaces in positive characteristic, which provides an effective bound on the multiple that makes an ample line bundle $D$ very ample. The proof for pathological surfaces is based on a Reider-type theorem. As a consequence, a Kawamata–Viehweg-type vanishing theorem is proved for arbitrary smooth algebraic surfaces in positive characteristic.

We present a new method for determining the Galois module structure of the cohomology of coherent sheaves on varieties over the integers with a tame action of a finite group. This uses a novel Adams–Riemann–Roch-type theorem obtained by combining the Künneth formula with localization in equivariant $K$-theory and classical results about cyclotomic fields. As an application, we show two conjectures of Chinburg, Pappas, and Taylor in the case of curves.