Rigidity and non-rigidity of commutative algebraic actions

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The proposed project lies in the area of dynamical systems and also involves number theory. The main objects are algebraic actions on tori and other homogeneous spaces by higher-rank commutative groups. The studies of such actions have contributed signi cantly to advances both in dynamical sytems and in Diophantine approximations. This research aims to investigate rigidity of these systems, a phenomenon characterized by the scarcity of invariant objects under the action. The main goal is to shed light on how higher-rank, hyperbolicity and irreducibility, the three typical conditions required in order for an action to enjoy rigidity, balance among themselves and quantitatively a ffect rigidity properties.The field of dynamical system studies long-term trajectories of points evolving over time according to certain given rules. Having its origin in Newton's mechanical laws, the subject has found numerous applications in physics, biology, economics and other sciences. The proposed research will focus on dynamical systems of arithmetic nature. In other words, the rules of evolution can be characterized by a set of integers and rational numbers. By studying dynamical behaviors of such systems, one expects to discover number-theoretical properties. This work also seeks to strengthen the connections between ergodic theory, number theory, Lie groups, Fourier analysis and additive combinatorics by combining methods from these mathematical branches. The PI intends to develop expository notes, as well as to teach graduate-level courses on relevant topics during the project.