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In the study of strongly-correlated insulators, a long-standing puzzle remained open for over 40 years. Some Kondo insulators (or mixed-valent insulators) display strange electrical transport that cannot be understood if one assumes that it is governed by the three-dimensional bulk. In this talk, I show that some 3D Kondo insulators have the right ingredients to be topological insulators and topological crystalline insulators, where the topologically protected surface states offer a natural explanation for the long-standing puzzle mentioned above. In addition, I will review recent experimental effort to test these ideas in SmB$_6$, and compare quantitatively results from various experiments, including electronic transport, quantum oscillations, STM, APRES and heat transport. Host: Shizeng Lin |