To apply the technique of virtual fundamental cycle (chain) in
the study of pseudo-holomorphic curve, we need to construct certain
structure, which we call Kuranishi strucuture, on its moduli space.
In this talk I want to review certain points of...
We present some novel approaches to the instability problem of
Hamiltonian systems (in particular, the Arnold Diffusion problem).
We show that, under generic conditions, perturbations of geodesic
flows by recurrent dynamics yield trajectories whose...
In this general survey talk, we will describe an approach to
doing homotopy theory within Univalent Foundations. Whereas
classical homotopy theory may be described as "analytic", our
approach is synthetic in the sense that, in ``homotopy type
theory...
The classical Pell equation $X^2-DY^2=1$, to be solved in
integers $X,Y\neq 0$, has a variant for function fields (studied
already by Abel), where now $D=D(t)$ is a complex polynomial of
even degree and we seek solutions in nonzero complex...
Calibrated currents naturally appear when dealing with several
geometric questions, some aspects of which require a deep
understanding of regularity properties of calibrated currents. We
will review some of these issues, then focusing on the two...
We all know Shannon's entropy of a discrete probability
distribution. Physicists define entropy in thermodynamics and in
statistical mechanics (there are several competing schools), and
want to prove the Second Law, but they didn't succeed yet
(very...
he Weil height measures the “complexity” of an algebraic number.
It vanishes precisely at 0 and at the roots of unity. Moreover, a
finite field extension of the rationals contains no elements of
arbitrarily small, positive heights. Amoroso, Bombieri...
The theorem of the title is that if the L-function L(E,s) of an
elliptic curve E over the rationals vanishes to order r=0 or 1 at
s=1 then the rank of the group of rational rational points of E
equals r and the Tate-Shafarevich group of E is finite...
