Understanding the incompressible/compressible fluid is a fundamental, but
challenging, project not only in numerical analysis, but also in
theoretical analysis, especially when extra effects, such as the elastic
deformation or the magnetic field, interact with the flow. In this talk,
the incompressible fluid and its associated flow map will be reviewed first.
The main object of this talk devotes to a recent work in understanding
incompressible/compressible magnetohydrodynamic fluids with zero magnetic
diffusivity (which is equivalent to infinite conductivity). This is a
joint work with Fanghua Lin.

We complete the exposition on self-genericity axioms for ideals on P(Z) (Club Catch, Projective Catch and Stationary Catch). We have established some relations with forcing axioms and with the existence of certain regular forcing embeddings and projections, and also point out connections with Precipitousness. We give an rough overview of the method used for proving the existence of models with Woodin cardinals coming from these axioms, using the Core Model Theory. In this talk we finish explaining the mechanism of absorbing extenders in the core model, and lifting iterability from countable models to models of large cardinality.

I will explain several conjectures and results regarding the slope distribution of Up operator action on the space of modular forms.  Most notably, we prove that the slopes of modular forms with a highly p-divisible characters roughly form unions of arithmetic progressions.  This is a joint work with Daqing Wan and Jun Zhang.

The Sparse Time-Frequency Representation (STFR) method is based on the assumption that many physical signals inherently contain AM-FM representations. We propose a sparse optimization method to extract the AM-FM representations of such signals. We prove the convergence of the method for periodic signals under certain assumptions and provide practical algorithms specifically for the non-periodic STFR, which extends the method to tackle problems that former STFR methods could not handle, including stability to noise and non-periodic data analysis. This is a significant improvement since many adaptive and non-adaptive signal processing methods are not fully capable of handling non-periodic signals. Moreover, we propose a new STFR algorithm to study intrawave signals with strong frequency modulation and analyze the convergence of this new algorithm for periodic signals. Such signals have previously remained a bottleneck for all signal processing methods. Furthermore, we propose a modified version of STFR that facilitates the extraction of intrawaves that have overlaping frequency content. We show that the STFR methods can be applied to the realm of dynamical systems and cardiovascular signals. In particular, we present a simplified and modified version of the STFR algorithm that is potentially useful for the diagnosis of some cardiovascular diseases. We further explain some preliminary work on the nature of Intrinsic Mode Functions (IMFs) and how they can have different representations in different phase coordinates. This analysis shows that the uncertainty principle is fundamental to all oscillating signals.