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We present a construction of the explicit Hodge decomposition for $\bar\partial$-equation on Riemann surfaces.

We give an explicit form of Gross-Zagier formula on Shimura Curves and an explicit form of Waldspurger formula.

Indefinite Kaehler solutions of the Einstein equations are studied, and it is almost completely determined which compact complex surfaces admit such metrics.

Following an overview of the relevant theory, we construct several explicit examples of height-3 K3 spectra.

Some special solutions to the multidimensional Lam\'e and Bourlet type equations are constructed in an explicit form.

We pose the problem to determine explicit defining equations of various elliptic fibrations on a given $K3$ surface, and study the case of the Kummer surfaces of the product of two elliptic curves.

We derive the implicit equations for certain parametric surfaces in three-dimensional projective space termed tensor product surfaces. Our method computes the implicit equation for such a surface based on the knowledge of the syzygies of…

Any constant-scalar-curvature Kaehler (cscK) metric on a complex surface may be viewed as a solution of the Einstein-Maxwell equations, and this allows one to produce solutions of these equations on any 4-manifold that arises as a compact…

We present a covariant formulation of the Gauss-Weingarten equations and the Gauss-Mainardi-Codazzi equations for surfaces in 3-dimensional curved spaces. We derive a coordinate invariant condition on the first and second fundamental form…

We derive explicit, uniform, a priori interior Hessian and gradient estimates for special Lagrangian equations of all phases in dimension two.

We study the equations of universal torsors on rational surfaces.

Gersten-Witt cohomologies of Hirzebruch surfaces are computed.

We present new constructions of Kaehler metrics with constant scalar curvature on complex surfaces, in particular on certain del Pezzo surfaces. Some higher dimensional examples are provided as well.

In this paper we develop an abstract theory for the Codazzi equation on surfaces, and use it as an analytic tool to derive new global results for surfaces in the space forms ${\bb R}^3$, ${\bb S}^3$ and ${\bb H}^3$. We give essentially…

We survey some results on real rational surfaces focused on their topology and their birational geometry.

We construct singular solutions to special Lagrangian equa- tions with subcritical phases and minimal surface systems. A priori estimate breaking families of smooth solutions are also produced cor- respondingly. A priori estimates for…

Several exact expansions as well as lower and upperbounds of the Kermack and McKendrick SIR equations are presented.

We provide explicit equations of some smooth complex quartic surfaces with many lines, including all 10 quartics with more than 52 lines. We study the relation between linear automorphisms and some configurations of lines such as twin lines…

We give a simple and explicit constructions of various semi-discrete surfaces and discrete $K$-surfaces in terms of the Jacobi elliptic functions using $\tau$-functions. Their periodicities are also determined.

We give explicit formulas for the Betti numbers, both stable and unstable, of the unordered configuration spaces of an arbitrary surface of finite type.