Related papers: $q$-Hypergeometric solutions of quantum differenti…

We consider an equisingularity problem for polynomial families of affine hypersurfaces $X_\tau \subset \mathbb C^n$ with (at worst) isolated singularities. We show that the constancy of the global polar invariants $\gamma^* (X_\tau)$ is…

We consider the Grassmannian X of (n-k)-dimensional subspaces of an n-dimensional complex vector space. We describe a `mirror dual' Landau-Ginzburg model for X consisting of the complement of a particular anti-canonical divisor in a…

In terms of several summation and transformation formulas for basic hypergeometric series, two forms of the Chinese remainder theorem for coprime polynomials, the creative microscoping method introduced by Guo and Zudilin, Guo and Li's…

We first give a pedagogical introduction to the differential calculus on q-groups and analize the relation between differential calculus and q-Lie algebra. Equivalent definitions of bicovariant differential calculus are studied and their…

Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parametrizing isomorphism classes of geometric objects (vector bundles, polarized…

We formulated a mirror-free approach to the mirror conjecture, namely, quantum hyperplane section conjecture, and proved it in the case of nonnegative complete intersections in homogeneous manifolds. For the proof we followed the scheme of…

We study projective unitary (co)representations of compact quantum groups and the associated second cohomology theory. We introduce left/right/bi/strongly projective corepresentations and study them in details. In particular, we prove that…

S. Weinberg pointed out a way to introduce a cosmological term by modifying the theory of gravity. This modification would be justified if the Einstein equations with the cosmological term could be obtained in the classical limit of some…

In recent years enormous progress has been made in perturbative quantum field theory by applying methods of algebraic geometry to parametric Feynman integrals for scalar theories. The transition to gauge theories is complicated not only by…

We prove a version of Quillen's stratification theorem in equivariant homotopy theory for a finite group $G$, generalizing the classical theorem in two directions. Firstly, we work with arbitrary commutative equivariant ring spectra as…

The super Macdonald polynomials indexed by the super partitions form a basis of the level zero super Fock module (combinatorial representation) of the quantum toroidal algebra $\mathcal{U}_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_{1|1})$.…

We prove a twisting theorem for nodal classes in permutation-equivariant quantum $K$-theory, and combine it with existing theorems of Givental to obtain a twisting result for general characteristic classes of the virtual tangent bundle.…

We prove in full generality that the $T$-equivariant quantum cohomology of any flag variety $G/P$ is isomorphic to the coordinate ring of a stratum of the Peterson scheme associated to the Langlands dual group scheme $G^{\vee}$. This result…

We compute the genus-0 permutation-equivariant quantum K-theory of Fermat singularities, in parallel with the Givental-Lee theory for projective varieties. We extend Givental-Tonita's formalism of adelic Lagrangian cones to the singularity…

We describe a bilinear identity satisfied by certain multidimensional q-hypergeometric integrals. The identity can be considered as a deformation of the Riemann bilinear relation for the twisted de Rham (co)homologies. The identity also…

We consider the joint system of equivariant quantum differential equations (qDE) and qKZ difference equations for the Grassmannian $G(k,n)$, which parametrizes $k$-dimensional subspaces of $\mathbb{C}^n$. First, we establish a connection…

It is shown that, given any finite dimensional, split basic algebra $\Lambda = K\Gamma/I$ (where $\Gamma$ is a quiver and $I$ an admissible ideal in the path algebra $K \Gamma$), there is a finite list of affine algebraic varieties, the…

This paper is devoted to study of differential calculi over quadratic algebras, which arise in the theory of quantum bounded symmetric domains. We prove that in the quantum case dimensions of the homogeneous components of the graded vector…

We study representations of $GL_{n}(\mathbb{F}_{q})$ that are distinguished with respect to a symmetric subgroup $H=GL_{n}(\mathbb{F}_{q})^{\sigma}$, where $\sigma$ is an involution. We prove that those representations satisfy $\pi \cong…

We prove the Landau-Ginzburg Mirror Symmetry Conjecture at the level of (orbifolded) Frobenius algebras for a large class of invertible singularities, including arbitrary sums of loops and Fermats with arbitrary symmetry groups.…