A priori estimate for a family of semi-linear elliptic equations with critical nonlinearity

We consider positive solutions of $\Delta u-\mu u+Ku^{\frac{n+2}{n-2}}=0$ on $B_1$ ($n\ge 5$) where $\mu$ and $K>0$ are smooth functions on $B_1$. If $K$ is very sub-harmonic at each critical point of $K$ in $B_{2/3}$ and the maximum of $u$ in $\bar B_{1/3}$ is comparable to its maximum over $\bar B_1$, then all positive solutions are uniformly bounded on $\bar B_{1/3}$. As an application, a priori estimate for solutions of equations defined on $\mathbb S^n$ is derived.Comment: 26 page

The cohomological support locus of pluricaonical sheaves and the Iitaka fibration

Let $alb_X: X \rightarrow A$ be the Albanese map of a smooth projective variety and $f: X \rightarrow Y$ the fibration from the Stein factorization of $alb_X$. For a positive integer $m$, if $f$ and $m$ satisfy the assumptions AS(1,2), then the translates through the origin of all components of cohomological locus $V^0(\omega_X^m, alb_X)$ generates $I^*Pic^0(S)$ where $I: X \rightarrow S$ denotes the Iitaka fibration. This result applies to studying pluricanonical maps. We also considered the problem about whether a fibration is isotrivial and isogenous to a product.Comment: 16 pages. Welcome comment

Video Synthesis from the StyleGAN Latent Space

Generative models have shown impressive results in generating synthetic images. However, video synthesis is still difficult to achieve, even for these generative models. The best videos that generative models can currently create are a few seconds long, distorted, and low resolution. For this project, I propose and implement a model to synthesize videos at 1024x1024x32 resolution that include human facial expressions by using static images generated from a Generative Adversarial Network trained on the human facial images. To the best of my knowledge, this is the first work that generates realistic videos that are larger than 256x256 resolution from single starting images. This model improves the video synthesis in both quantitative and qualitative ways compared to two state-of-the-art models: TGAN and MocoGAN. In a quantitative comparison, this project reaches a best Average Content Distance (ACD) score of 0.167, as compared to 0.305 and 0.201 of TGAN and MocoGAN, respectively