The Littlewood conjecture states that $\liminf_{n\to\infty}||n\alpha|| ||n\beta|=0|$ holds for all real numbers $\alpha$ and $\alpha$, where $||\cdot||$ denotes the distance to the nearest integer. There are several other formulations of Littlewood conjecture, including the $p$-adic and mixed Littlewood conjecture. In this talk, I start with an introduction to the history of different versions of Littlewood conjecture.  Then I will present several refined results of mixed Littlewood conjecture for pseudo-absolute values.
Let $\mathcal{D}_1$,$\mathcal{D}_2,\cdots, \mathcal{D}_k$ be $k$ pseudo absolute sequences and define the  $\mathcal{D}$-adic norm $|\cdot|_{\mathcal{D}}:\N\to \{n_k^{-1}:k\ge 0\}$ by $|{n}|_\mathcal{D} = \min\{ n_k^{-1} : n\in  n_k\Z \}.$
Under some minor condition of $\mathcal{D}_1$,$\mathcal{D}_2,\cdots, \mathcal{D}_k$,  I set up the criteria of sequence $\psi(n)$ such that for almost every $\alpha$ the inequality
\begin{equation*}
    |n|_{\mathcal{D}_1}|n|_{\mathcal{D}_2}\cdots |n|_{\mathcal{D}_k}||n\alpha||\leq \psi(n)
\end{equation*}
has infinitely many solutions for $n\in\N$. Under some minor condition of the pseudo absolute sequence $\mathcal{D}$, I also show that
for any $\alpha\in\R$, $\liminf_{n\to \infty}n|n|_a|n|_\mathcal{D}\|n\alpha\|=0.$

Polynomial dynamical systems over finite fields or rings provide a useful tool for studying network dynamics, such as those of gene regulatory networks. In this talk, I will discuss linear dynamical systems over finite commutative rings. The limit cycles of a linear dynamical system over a finite field can be described by using the elementary divisors of the corresponding matrix. The extension of the study to a general finite commutative ring is natural and has applications. To address the difficulties in the commutative ring setting, we developed a computational approach. In an earlier work, we gave an efficient algorithm to determine whether such a system over a finite commutative ring is a fixed-point system or not. In a more recent work, we further analyzed the structure of such a system and provided a method to determine its limit cycles. 

Given a projective variety X over a field of characteristic 0, and a positive integer r, we study the rth secant variety of Veronese re-embeddings of X. In particular, I'll explain recent work which shows that the degrees of the minimal equations (and more generally, syzygies) defining these secant varieties can be bounded in terms of X and r independent of the Veronese embedding. This is based on http://arxiv.org/abs/1510.04904 and http://arxiv.org/abs/1608.01722.

We study the spectral transition line of the extended Harper's model
in the positive Lyapunov exponent regime. We show that both pure point
spectrum and purely singular continuous spectrum occur for dense subsets
of frequencies on the transition line.

In this talk we prove a discrete version of the Bethe-Sommerfeld conjecture. 

Namely, we show that the spectra of multi-dimensional discrete periodic Schr\"odinger operators on ℤd​ lattice with sufficiently small potentials contain at most two intervals. 

Moreover, the spectrum is a single interval, provided one of the periods is odd, and can have a gap whenever all periods are even. This is based on a joint work with Lana.