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.

Distance set problems form a central theme in geometric measure theory and harmonic analysis, investigating the relationship between the geometric structure of a set and the distances determined by its points. A fundamental question in this area is the Falconer distance conjecture, which asks whether a set in Euclidean space with sufficiently large Hausdorff dimension must determine a set of distances of positive Lebesgue measure. In this talk, I will discuss some of the central questions and developments in distance set problems, with an emphasis on the role of harmonic analysis and geometric measure theory. Beginning with the Falconer distance problem, I will explore related questions concerning prescribed distance configurations, including finite chains and trees, and how the geometric structure and sparsity of sets influence the existence of such configurations. If time permits, I will also discuss some extensions of these problems to the Heisenberg group, where the underlying geometry leads to interesting variations of classical distance problems.