30 years ago I proved that any tight contact structure on the
3-sphere is diffeomorphic to the standard one. I also
optimistically claimed at the same paper that similar methods could
be used to prove a multi-parametric version: the space of
tight...
What does the spectrum of a random matrix look like when we make
no assumption whatsoever about the covariance pattern of its
entries? It may appear hopeless that anything useful can be
said at this level of generality. Nonetheless, a widely used...
We study CM cycles on Kuga-Sato varieties over X(N) via theta
lifting and relative trace formula. Our first result is the
modularity of CM cycles, in the sense that the Hecke modules they
generate are semisimple whose irreducible components are...
The query model is one of the most basic computational
models. In contrast to the Turing machine model, fundamental
questions regarding the power of randomness and quantum computing
are relatively well-understood. For example, it is
well-known...
We will discuss the first steps in an approach to proving
homological mirror symmetry for Looijenga pairs through tropical
Lagrangian sections. Namely, we will see how to construct these
Lagrangian sections from tropical data corresponding to line...
The honest answer to the question is that I actually do not
know. I will therefore rather talk about several famous examples
that are widely called "h-principle results" and try to explain
some of the ideas behind the ones I am most familiar with.
We consider the problem of computing a Gradient Descent solution
of a continuously differentiable function f on a bounded convex
domain, i.e., finding a "point where Gradient Descent terminates".
Equivalently, this corresponds to computing a so...
Knots associated to overtwisted manifolds are less explored.
There are two types of knots in an overtwisted manifold – loose and
non-loose. Non-loose knots are knots with tight complements whereas
loose knots have overtwisted complements. While we...
