Related papers: Cusp Eigenforms and the Hall Algebra of an Ellipti…

From the structure of the category of representations of an affine cycle-free quiver, we determine an explicit linear form on the space of regular cuspidal functions over a finite field: its kernel is exactly the space of cuspidal…

In this survey article, we summarise the known results towards the conjecture: elliptic curves over totally real number fields are modular. For understanding these recent results in the literature, we present some necessary background along…

Let $A$ be the path algebra of a finite acyclic quiver $Q$ over a finite field. We realize the quantum cluster algebra with principal coefficients associated to $Q$ as a sub-quotient of a certain Hall algebra involving the category of…

We determine the Hall algebra, in the sense of Toen, of the algebraic triangulated category generated by a spherical object.

We obtain explicit formulas for the number of non-isomorphic elliptic curves with a given group structure (considered as an abstract abelian group). Moreover, we give explicit formulas for the number of distinct group structures of all…

Isomorphisms of separable Hilbert spaces are analogous to isomorphisms of n-dimensional vector spaces. However, while n-dimensional spaces in applications are always realized as the Euclidean space R^n, Hilbert spaces admit various useful…

In this paper we compute the convolution algebra in the equivariant K-theory of the Hilbert scheme of A^2. We show that it is isomorphic to the elliptic Hall algebra, and hence to the spherical DAHA of GL_\infty. We explain this coincidence…

We study an arithmetic analog of the Hall algebra of a curve, when the curve is replaced by the spectrum of the integers compactified at infinity. The role of vector bundles is played by lattices with quadratic forms. This algebra H…

It is argued that a nonsingular elliptic curve admits a natural or fundamental abelian heap structure uniquely determined by the curve itself. It is shown that the set of complex analytic or rational functions from a nonsingular elliptic…

We study polynomial deformations of the fuzzy sphere, specifically given by the cubic or the Higgs algebra. We derive the Higgs algebra by quantizing the Poisson structure on a surface in $\mathbb{R}^3$. We find that several surfaces,…

We study the derived Hall algebra of the partially wrapped Fukaya category of a surface. We give an explicit description of the Hall algebra for the disk with m marked intervals and we give a conjectural description of the Hall algebras of…

The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and a scroll of stationary bisecants. We…

We define the Hopf algebra structure on the Grothendieck group of finite-dimensional polynomial representations of $U_q \hat{gl}_N$ in the limit $N \to \infty$. The resulting Hopf algebra $Rep U_q \hat{gl}_\infty$ is a tensor product of its…

We generalize a theorem of Kapranov by showing that the Hall algebra of the category of coherent sheaves on a weighted projective line (over a finite field) provides a realization of the (quantized) enveloping algebra of a certain nilpotent…

Characteristic classes of Schubert varieties can be used to study the geometry and the combinatorics of homogeneous spaces. We prove a relation between elliptic classes of Schubert varieties on a generalized full flag variety and those on…

If $D$ is the definite quaternion algebra over $\qu$ of discriminant $p$, we compute, for any prime $p>3$, the number of infinite dimensional cusp forms on $D^*$ which are trivial at infinity, tamely ramified at $p$, and have given…

We consider two algebras of curves associated to an oriented surface of finite type - the cluster algebra from combinatorial algebra, and the skein algebra from quantum topology. We focus on generalizations of cluster algebras and…

We define elliptic generalization of W-algebras associated with arbitrary quiver using the formalism of arXiv:1512.08533 applied to six-dimensional quiver gauge theory compactified on elliptic curve.

We classify the Lagrangian orientable surfaces in complex space forms with the property that the ellipse of curvature is always a circle. As a consequence, we obtain new characterizations of the Clifford torus of the complex projective…

We show how the modular symmetries that have been found to be consistent with most available scaling data from quantum Hall systems, derive from a rigid family of algebraic curves of the elliptic type. The complicated special functions…