

Awardees for Undergraduate ThesesNote: All prizes and awards are subjected to the awardee's presence at the Award Ceremony in 2010. The tentative date for the ceremony is December 17, 2010.Gold Awards(No order) Name: Lam Ka Kit Title: Achieving Capacity of Network Affiliation: The Chinese University of Hong Kong Abstract: In any communication network, we always want to find ways to transmit information to maximize the throughput and have no errors. However, if the network is huge and complex, with many sources and receivers, such a problem will become nontrivial. One way to approach it is to employ coding in the intermediate nodes inside the network. And in our project, the achievability of capacity was investigated with an emphasis on designing nonAbelian group codes. Matroidal networks and properties of nonAbelian group codes were studied. Moreover, some decidability network problems about achieving capacity were also studied. Name: Liang Zhao (赵亮) Title: Transcendence of hypergeometric functions for function fields Affiliation: Tsinghua University Abstract: In this article we study the transcendence of values of hypergeometric functions in the setting of function fields over finite fields. We have proved two main results: (i)If the number of parameters appearing at the denominator of the coecients is strictly large than the number of parameters appearing at the numerator, the values at nonzero rational arguments of hypergeometric function is transcendental. (ii)If the number of parameters that appearing at the denominator of the coecients is equal to the number of parameters appearing at the numerator, the values of hypergeometric function could be either transcendental or algebraic. This artical uses the criterion of transcendence developed by JY Yao. Silver Awards (No order) Name: Xiangyu Cao (曹翔宇) Title: Mobius transform, momentangle complexes and HalperinCarlsson conjecture Affiliation: Fudan University Abstract: In this paper, we give an algebracombinatorics formula of the M\"obius transform for an abstract simplicial complex $K$ on $[m]=\{1, ..., m\}$ in terms of the Betti numbers of the StanleyReisner face ring of $K$. Furthermore, we employ a way of compressing $K$ to estimate the lower bound of the sum of those Betti numbers by using this formula. As an application, associating with the momentangle complex $\mathcal{Z}_K$ (resp. real momentangle complex ${\Bbb R}\mathcal{Z}_K$) of $K$, we show that the HalperinCarlsson conjecture holds for $\mathcal{Z}_K$ (resp. ${\Bbb R}\mathcal{Z}_K$) under the restriction of the natural $T^m$action on $\mathcal{Z}_K$ (resp. $({\Bbb Z}_2)^m$action on ${\Bbb R}\mathcal{Z}_K$).
Name: Chao Li (李超) Title: 计算机代数系统的数学原理 Affiliation: Tsinghua University Abstract: 本文主要讨论计算机代数系统的数学原理，由十六个章节组成. 内容包含高精度运算, 数论, 数学常数, 精确线性代数, 多项式, 方程求解, 符号求和, 符号积分, 微分方程符号解等九大部分, 涵盖了构建计算机代数系统的最基础也是最重要的内容. 许多内容是第一次被系统地整理出现在中文文献中, 一些领域也追踪到了最新进展.
Name: Dai Shi (时代) Title: Upper bounds on the number of eigenvalues of a steadystate Schrodinger Equation Affiliation: Fudan University Abstract: Several upper bounds on the number of eigenvalues are provided, respectively, for Schrodinger equations of onedimensional case, central case and with generalized potential functions. Compared with Bargman’s and Calogero’s results, the new bounds can give finite estimation when the potential function approaches zero slowly as x approaches infinity. Besides, A generalization of Calogero bound is given, which can be applied to “bell shaped” potential functions rather than merely monotonous ones. Comparisons are presented with examples. With respect to Schrodinger operators containing generalized functions, some bounds are also established.
Name: Xu Chao Title: Hyperbolic KaehlerRicci flow Affiliation: Zhejiang University Abstract: In this paper, I consider the hyperbolic KaehlerRicci flow introduced by Kong and Liu, that is, the hyperbolic version of the famous KaehlerRicci floow.. I explain the derivation of the equation and calculate the evolutions of various quantities associated to the equation including the curvatures. Particularly on CalabiYau manifolds, the equation can be simplified to a scalar hyperbolic MongeAmpere equation which is just the hyperbolic version of the corresponding one in KaehlerRicciflow.. I briefly study its symmetries on Riemann surfaces and the symmetry theory of PDE is sketched in the appendix.
