Related papers: Tautological equations in genus 2 via invariance c…

We study groups, exponential groups and ordered groups equipped with valuations. We investigate algebraic and topological features of such valued structures, and apply our findings in order to solve regular equations over groups using…

We demonstrate that the necessary condition for $SO(N) \times SO(N)$ duality invariance manifests as a partial differential equation in two-dimensional scalar theories. This condition, expressed as a partial differential equation,…

Here we study the abstract nonlinear differential equation of second order that in special case is the equation of the type of equation of traffic flow. We prove the solvability theorem for the posed problem under the appropriate conditions…

We formulate two conjectures about etale cohomology and fundamental groups motivated by categoricity conjectures in model theory. One conjecture says that there is a unique Z-form of the etale cohomology of complex algebraic varieties, up…

Quantum theory of the gauge models in the causal approach leads to some cohomology problems. We investigate these problems in detail.

The known facts about solvability of equations over groups are considered from a more general point of view. A generalized version of the theorem about solvability of unimodular equations over torsion-free groups is proved. In a special…

We determine all genus 2 curves, defined over $\mathbb C$, which have simultaneously degree 2 and 3 elliptic subcovers. The locus of such curves has three irreducible 1-dimensional genus zero components in $\mathcal M_2$. For each component…

We derive explicit universal equations for primary Gromov-Witten invariants by applying Getzler's genus one relation to quantum powers of Euler vector field. As an application, we provide some evidences for the genus-1 Virasoro conjecture.

An easy generalization of Beukers' integrals allows us to conjecture a double integral formula involving the zeta and the gamma functions. A special case of this formula is Sondow's double integral formula for Euler's constant gamma.

We prove the invariance of plurigenera under smooth projective deformations in full generality. The proof is done by several estimates of singular hermitian metrics in terms of $L^{2}$-extension theorem of holomorphic sections.

We formulate the necessary conditions for the integrability of a certain family of Hamiltonian systems defined in the constant curvature two-dimensional spaces. Proposed form of potential can be considered as a counterpart of a homogeneous…

Covariance is a useful property for handling supergravity theories. In this paper, we prove a covariance property of supergravity field equations: under reasonable conditions, field equations of supergravity are covariant modulo other field…

We compute the Euler characteristics of tautological vector bundles and their exterior powers over the Quot schemes of curves. We give closed-form expressions over punctual Quot schemes in all genera. For higher rank quotients of a trivial…

We describe mean value estimates for exponential sums of degree exceeding 2 that approach those conjectured to be best possible. The vehicle for this recent progress is the efficient congruencing method, which iteratively exploits the…

Some class of sums which naturally include the sums of powers of integers is considered. A number of conjectures concerning a representation of these sums is made.

We give canonical resolutions of singularities of several cone varieties arising from invariant theory. We establish a connection between our resolutions and resolutions of singularities of closure of conjugacy classes in classical Lie…

Justification theory is an abstract unifying formalism that captures semantics of various non-monotonic logics. One intriguing problem that has received significant attention is the consistency problem: under which conditions are…

We study the minimal Weierstrass equations for genus 2 curves defined over a ring of integers $\mathcal O_{\mathbb F}$. This is done via reduction theory and Julia invariant of binary sextics. We show that when the binary sextics has extra…

New invariants for 2-dimensional cell complexes are defined, which can be interpreted as curvature bounds. These invariants are proved to be rational and computable in a companion article. This document is a survey that collects theorems…

We study the symplectic Howe duality using two new and independent combinatorial methods: via determinantal formulae on the one hand, and via (bi)crystals on the other hand. The first approach allows us to establish a generalised version…