A vector partition of a vector v is an (unordered) decomposition of v as a sum of non-zero vectors with non-negative integer entries. We will discuss the relevance of these functions in the problem of plethysm, which lies at the heart of combinatorial representation theory. We will mention some open problems in this area. Number theorists may find the connections to classical partition theory of particular interest.

How much information can a physical system carry? For a standard d-level quantum system, the fundamental limit is log d bits of classical information. Conventional notion, together with the recent intuition, suggests that adding noise shouldn't change the big picture: a noisy quantum channel should perform much like an equally noisy classical channel. But does the same analogy still hold when we introduce additional resources over noisy quantum communication? In this talk, we explore two scenarios where the intuition drastically fails. We will show that when assisted by additional resources, the true classical communication capacity of a noisy quantum system can dramatically outperform its classical counterpart. Beyond revealing a fascinating, unexplored caveat in the foundations of information theory, these results suggest new design principles for the architecture of future quantum networks. Join Zoom Meeting https://zoom.us/j/91431488789 Meeting ID: 914 3148 8789 Passcode: 009545

In this talk, we discuss the rational dilation problem on domains in $\mathbb{C}^n$. Among the various approaches to this problem, we focus on techniques from operator space theory, highlighting the connection between rational dilation and complete contractivity. En route, we discuss the case when the domain is a unit ball. We determine the cotangent operator space structure at the origin of the tetrablock and use it to construct a commuting triple of operators for which the tetrablock is a spectral set but not a complete spectral set, thereby establishing the failure of rational dilation on the tetrablock.