We propose a non-commutative generalization of Beilinson's Conjecture on the regulator map from algebraic K-theory to Deligne cohomology of algebraic varieties over Q. We also check a baby case of the generalized conjecture, namely, the case of finite-dimensional associative algebras.
We study the isoperimetric problem for the radially symmetric measures. Applying the spherical symmetrization procedure and variational arguments we reduce this problem to a one-dimensional ODE of the second order. Solving numerically this ODE we get an empirical description of isoperimetric regions of the planar radially symmetric exponential power laws. We also prove some isoperimetric inequalities for the log-convex measures. It is shown, in particular, that the symmetric balls of large size are isoperimetric sets for strictly log-convex and radially symmetric measures. In addition, we establish some comparison results for general log-convex measures.
We propose an algebraic model of the conjectural triply graded homology of S. Gukov, N. Dunfield and J. Rasmussen for some torus knots. It turns out to be related to the q,t-Catalan numbers of A. Garsia and M. Haiman.
We describe how certain simple p-adic techniques can be applied to get information about iterated orbits of algebraic points under a rational self-map of an algebraic variety defined over a number field.
We prove under GRH that zeros of $L$-functions of modular forms of level $N$ and weight $k$ become uniformly distributed on the critical line when $N+k\to\infty.$