Abstract:
In this paper we study the supremum of Perelman's \lambda-functional {\lambda }_M(g) on Riemannian 4-manifold M by using the Seiberg-Witten equations. We prove among others that, for a compact K\"{a}hler-Einstein complex surface (M, J, g_{0}) with negative scalar curvature, (i) If g_{1} is a Riemannian metric on M with \lambda_{M}(g_{1})= \lambda_{M}(g_{0}), then Vol_{g_{1}}(M)\geq Vol_{g_{0}}(M). Moreover, the equality holds if and only if g_{1} is also a K\"{a}hler-Einstein metric with negative scalar curvature. (ii) If g_{t}, t\in [-1,1], is a family of Einstein metrics on M with initial metric g_{0}, then g_{t} is a K\"{a}hler-Einstein metric with negative scalar curvature.

Abstract:
Let $g(t)$, $t\in [0, +\infty)$, be a solution of the normalized K\"ahler-Ricci flow on a compact K\"ahler $n$-manifold $M$ with $c_{1}(M)>0$ and initial metric $g (0)\in 2\pi c_{1}(M)$. If there is a constant $C$ independent of $t$ such that $$ \int_{M}|Rm(g(t))|^{n}dv_{t}\leq C,$$ then, for any $t_{k}\to \infty$, a subsequence of $(M, g(t_{k}))$ converges to a compact orbifold $(X, h)$ with only finite many singular points $\{q_{j}\}$ in the Gromov-Hausdorff sense, where $h$ is a K\"ahler metric on $X\backslash \{q_{j}\}$ satisfying the K\"ahler-Ricci soliton equation, i.e. there is a smooth function $f$ such that $$Ric(h)-h=\nabla\bar{\nabla}f, {\rm and}\it \nabla \nabla f=\bar{\nabla} \bar{\nabla} f=0. $$

Abstract:
Autophagy is a tightly controlled self-degradation process utilised by cells to sustain cellular homeostasis and to support cell survival in response to metabolic stress and starvation. Thus, autophagy plays a critical role in promoting cell integrity and maintaining proper function of cellular processes. Defects in autophagy, however, can have drastic implications in human health and diseases, including cancer. Described as a double-edged sword in the context of cancer, autophagy can act as both suppressor and facilitator of tumorigenesis. As such, defining the precise role of autophagy in a multistep event like cancer progression can be complex. Recent findings have implicated a role for components of the autophagy pathway in oncogene-mediated cell transformation, tumour growth, and survival. Notably, aggressive cancers driven by Ras oncoproteins rely on autophagy to sustain a reprogrammed mitochondrial metabolic signature and evade cell death. In this review, we summarize our current understanding of the role of oncogene-induced autophagy in cancer progression and discuss how modulators of autophagic responses can bring about therapeutic benefit and eradication of a subset of cancers that are addicted to this ancient recycling machinery. 1. Introduction Almost two decades ago, the Ohsumi laboratory first discovered and characterized the autophagy-related (ATG) genes in yeast [1, 2]. Since then, researchers around the world work relentlessly to unravel the biology of autophagy and its roles in a variety of human diseases. Autophagy can be broadly categorized into macroautophagy, microautophagy, and chaperone-mediated autophagy. Macroautophagy (hereafter autophagy) is a tightly regulated catabolic mechanism in the cell. It involves the sequestration of dysfunctional cytoplasmic constituents, ranging from misfolded proteins, proteoglycans and damaged organelles into double membrane vesicles, known as the autophagosomes. These autophagic vesicles eventually fuse with lysosomes, within which the dysfunctional cytoplasmic cargoes are degraded. This self-cannibalisation process in the cell appears to play a crucial role in supporting the bioenergetics and biosynthetic programs in response to nutrient deprivation and metabolic duress [3]. Under optimal growth conditions of a normal cell, the autophagic activity is kept in a minimal or basal state. Such basal autophagy is important for maintaining intracellular protein homeostasis and preservation of cellular integrity, through effective clearance of protein aggregates and damaged organelles. Under

Abstract:
In this paper we study non-singular solutions of Ricci flow on a closed manifold of dimension at least 4. Amongst others we prove that, if M is a closed 4-manifold on which the normalized Ricci flow exists for all time t>0 with uniformly bounded sectional curvature, then the Euler characteristic $\chi (M)\ge 0$. Moreover, the 4-manifold satisfies one of the following \noindent (i) M is a shrinking Ricci solition; \noindent (ii) M admits a positive rank F-structure; \noindent (iii) the Hitchin-Thorpe type inequality holds 2\chi (M)\ge 3|\tau(M)| where $\chi (M)$ (resp. $\tau(M)$) is the Euler characteristic (resp. signature) of M.

Abstract:
The main result of this paper shows that, if $g(t)$ is a complete non-singular solution of the normalized Ricci flow on a noncompact 4-manifold $M$ of finite volume, then the Euler characteristic number $\chi(M)\geq0$. Moreover, $\chi(M)\neq 0$, there exist a sequence times $t_k\to\infty$, a double sequence of points $\{p_{k,l}\}_{l=1}^{N}$ and domains $\{U_{k,l}\}_{l=1}^{N}$ with $p_{k,l}\in U_{k,l}$ satisfying the followings: [(i)] $\dist_{g(t_k)}(p_{k,l_1},p_{k,l_2})\to\infty$ as $k\to\infty$, for any fixed $l_1\neq l_2$; [(ii)] for each $l$, $(U_{k,l},g(t_k),p_{k,l})$ converges in the $C_{loc}^\infty$ sense to a complete negative Einstein manifold $(M_{\infty,l},g_{\infty,l},p_{\infty,l})$ when $k\to\infty$; [(iii)] $\Vol_{g(t_{k})}(M\backslash\bigcup_{l=1}^{N}U_{k,l})\to0$ as $k\to\infty$.

Abstract:
We consider maximum solution $g(t)$, $t\in [0, +\infty)$, to the normalized Ricci flow. Among other things, we prove that, if $(M, \omega) $ is a smooth compact symplectic 4-manifold such that $b_2^+(M)>1$ and let $g(t),t\in[0,\infty)$, be a solution to (1.3) on $M$ whose Ricci curvature satisfies that $|\text{Ric}(g(t))|\leq 3$ and additionally $\chi(M)=3 \tau (M)>0$, then there exists an $m\in \mathbb{N}$, and a sequence of points $\{x_{j,k}\in M\}$, $j=1, ..., m$, satisfying that, by passing to a subsequence, $$(M, g(t_{k}+t), x_{1,k},..., x_{m,k}) \stackrel{d_{GH}}\longrightarrow (\coprod_{j=1}^m N_j, g_{\infty}, x_{1,\infty}, ...,, x_{m,\infty}),$$ $t\in [0, \infty)$, in the $m$-pointed Gromov-Hausdorff sense for any sequence $t_{k}\longrightarrow \infty$, where $(N_{j}, g_{\infty})$, $j=1,..., m$, are complete complex hyperbolic orbifolds of complex dimension 2 with at most finitely many isolated orbifold points. Moreover, the convergence is $C^{\infty}$ in the non-singular part of $\coprod_1^m N_{j}$ and $\text{Vol}_{g_{0}}(M)=\sum_{j=1}^{m}\text{Vol}_{g_{\infty}}(N_{j})$, where $\chi(M)$ (resp. $\tau(M)$) is the Euler characteristic (resp. signature) of $M$.

Abstract:
In this paper, we prove two generalized versions of the Cheeger-Gromoll splitting theorem via the non-negativity of the Bakry-\'Emery Ricci curavture on complete Riemannian manifolds.

Abstract:
We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry-\'{E}mery Ricci tensor has a positive lower bound, and either of the following conditions: (i) the Ricci curvature is bounded from above; (ii) the Ricci curvature is bounded from below and injectivity radius is bounded away from zero. Moreover, a complete shrinking Ricci soliton has finite topological type if its scalar curvature is bounded.

Abstract:
We developed a method to measure the phase retardation and birefringence of muscovite mica plate in the temperature range of 223K to 358K within the spectrum of 300 to 700 nm. The phase retardation data is gained through the standard transmission ellipsometry using spectroscopic ellipsometer. With the phase retardation and thickness of the mica plate we can calculate its birefringence dispersion. Our results give abundant phase retardation and birefringence data of muscovite mica in the ultraviolet and visible spectrum from 223K to 358K. From the experimental data, the phase retardation and birefringence will drop down at the fixed wavelength when the temperature rises. The accuracy of the birefringence of mica plate is better than 3.5e-5.

Abstract:
A method for measuring the phase retardation and birefringence of crystalline quartz wave plate in the ultraviolet and visible spectrum is demonstrated using spectroscopic ellipsometer. After the calibration of the crystalline quartz plate, the experimental data are collected by the photodetector and sent to the computer. According to the outputted data, the retardation can be obtained in the range of 190 to 770 nm. With the retardation data, the birefringence for the quartz can be calculated in the same spectrum with an accuracy of better than . The birefringence results enrich the crystalline quartz birefringence data especially in the ultraviolet spectrum.