## Department Colloquium

### Friday, September 11th, 2020

**Time:** 2:30 p.m. **Place:** Online (via Zoom)

**Speaker:** Serdar Yuksel (Queen's University)

**Title:** Geometry of Information Structures, Strategic Measures and Associated Control Topologies.

**Abstract: ** In many areas of applied mathematics (including control theory, information theory, game theory) decentralization of information among several decision makers is unavoidable. Information and correlation structures determine who knows what information and how the decisions may be dependent leading to various problems on the geometry of correlation structures among decisions/controls. We define information structures, place various topologies on them, and study closedness and compactness properties on the (strategic) measures induced by decentralized control/decision policies under varying degrees of relaxations with regard to access to private or common randomness. Ultimately, we present existence and approximation results for optimal decision/control policies. We then discuss various upper and lower bounding techniques, through realizable and classically non-realizable (such as quantum correlations and non-signaling) convex relaxations and quantization. For each of these, we review or establish closedness and convexity properties and present a hierarchy of correlation structures. As a second main theme, we review or introduce various topologies on decision/control strategies defined independently from information structures, but for which information structures determine whether the topologies entail utility in arriving at existence, compactness, convexification or approximation results. These approaches, which we will term as the strategic measures approach (where the induced joint measure is considered) and the control topology approach (where a product space of individual control policy spaces is considered), lead to complementary results and solution methods in optimal stochastic control. (Joint work primarily with Prof. Naci Saldi, other collaborators will also be acknowledged).