I will discuss recent work on the global stability of the
Euler-Maxwell equations in 3D (joint work with Guo and Pausader),
and of the gravity water-wave system in 2D (joint work with
Pusateri).
We study conformal invariants that arise from nodal sets and
negative eigenvalues of conformally covariant operators, which
include the Yamabe and Paneitz operators. We give several
applications to curvature prescription problems. We establish
a...
Informally, uncertainty principle says that function and its
Fourier transform can not be both concentrated. Uncertainty
principle has a lot of applications in areas like compressed
sensing, error correcting codes, number theory and many others.
In...
We study some problems relating to polynomials evaluated either
at random Gaussian or random Bernoulli inputs. We present some new
work on a structure theorem for degree-d polynomials with Gaussian
inputs. In particular, if p is a given degree-d...
We propose an “analytical” framework for studying parallel
repetitions of one-round two-prover games. We define a new
relaxation of the value of a game, val+, and prove that it is both
multiplicative and a good approximation for the true value
of...
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...