Theory and applications of algebraic combinatorics

这项研究将两股力量结合在一起，一股理论力量和一股应用力量。在理论方面，本课程从代数组合学探讨对称函数的恒等式。这些恒等式，以及类似代数中的大量相关恒等式，在理论物理、代数几何、表示理论以及组合学中发挥着重要作用。因此，找到这样的身份对所有这些主体来说都是至关重要的。在应用方面，本课程探讨匿名网络的安全性。匿名网络——例如Tor——隐藏了发送者和接收者之间关系的信息。它们已经成为持不同政见者、记者和执法官员的重要交流手段，随着普通人对电子隐私的关注日益增加，它们将变得更加普遍。但它们有多安全呢？

This research weaves together two strands, one theoretical and one applied. On the theoretical side, this programme explores identities of symmetric functions from algebraic combinatorics. These identities, as well as a host of related identities in similar algebras, play a significant role in theoretical physics, algebraic geometry, and representation theory as well as combinatorics. Thus finding such identities is crucial for all these subjects. On the applied side this programme explores the security of anonymity networks. Anonymity networks--for example, Tor--conceal information about the relationship between senders and receivers. They are already a vital means of communication for political dissidents, journalists, and law enforcement officials, and as the common person's concern with electronic privacy grows, they will become more widespread. But how secure are they?