We prove that the probability that a curve of the form $y^2 = f(x)$
over $\mathbb Q$ with $\deg f = 2g + 1$ has no rational point other
than the point at infinity tends to 1 as $g$ tends to infinity.
This is joint work with Michael Stoll.
This is joint work with G .Garkusha. Using the machinery of framed
sheaves developed by Voevodsky, a triangulated category of framed
motives is introduced and studied. To any smooth algebraic variety
$X$, the framed motive $M_{fr}(X)$ is associated...
Starting from an example in which the Hitchin correspondence can be
written down explicitly, we look at what might be said relating the
incidence complex of the boundary of the character variety, and the
Hitchin map.
Decoupling inequalities in harmonic analysis permit to bound the
Fourier transform of measures carried by hyper surfaces by certain
square functions defined using the geometry of the hyper surface.
The original motivation has to do with issues in...
At the heart of every local search algorithm is a directed graph on
candidate solutions (states) such that every unsatisfactory state
has at least one outgoing arc. In stochastic local search the hope
is that a random walk will reach a satisfactory...
In this talk, I will present the recent joint work with Yi Zhu on
$A^1$-connectedness for quasi-projective varieties. The theory of
$A^1$-connectedness for quasi-projective varieties is an analogue
of rationally connectedness for projective...
We consider compactifications of the Betti, de Rham and Dolbeault
realizations of the character variety. Starting from an example, we
look at what can be said, mostly conjecturally, about the
relationship between these spaces.
We prove that for any $\epsilon > 0$ it is NP-hard to
approximate the non-commutative Grothendieck problem to within a
factor $1/2 + \epsilon$, which matches the approximation ratio of
the algorithm of Naor, Regev, and Vidick (STOC'13). Our proof...