we prove an effective equidistribution theorem for semisimple closed orbits on compact adelic quotients. the obtained error depends polynomially on the minimal complexity of intermediate orbits and the complexity of the ambient space. as an application, we establish a local-global principle for representations of quadratic forms, improving the codimension assumptions and providing effective bounds in a theorem of ellenberg and venkatesh. we will discuss these theorems not assuming any prior knowledge of any of the above concepts. this is based on joint work with manfred einsiedler, elon lindenstrauss, and amir mohammadi.

aiming to understand the solvability of semilinear elliptic boundary values problems in bounded domains, we will review the best known techniques for establishing the existence of critical points of functionals whose critical points are solutions to such problems. the mountain pass lemma, the nehari manifold,
the morse index, and bifurcation analysis will be discussed to conclude the existence of seven solutions for an asymptotically linear elliptic problem.

as we know, kaehler-ricci flow can be reduced to solve a class of  parabolic   complex monge-amp\`ere equations for kaehler potentials and  the solutions usually depend on the kaehler class of initial metric.   thus there  gives a  degeneration of kaehler metrics arising from the kaehler-ricci flow.  for a class of $g$-spherical manifolds,   we can  use  the local estimate  of  monge-amp\`ere equations as well as  the h-invariant for $c^*$-degeneration  to determine the limit of  kaehler-ricci flow after resales.  in particular,  on such manifolds,  the flow will develop the singularities of  type ii.  

we discuss an analogue of the goldbach conjecture for polynomials with coefficients in semidomains (i.e., subsemirings of an integral domain).