Time crystals are crystalline structures in the time domain. In contrast to the usual crystals which manifest in spatial dimensions, the formation of a time crystal or even a proposal of such, can be tricky if not controversial.

The reason is closely related to the time translation symmetry. Time translation symmetry not only underlies the invariance of laws of physics, but also in the standard dynamical framework, is directly related to the con- servation of energy. It was thought until very recently that time translation symmetry could not be spontaneously broken.

In 2012, Frank Wilczek attempted such a possibility with degenerate Hamiltonians for a conservative system and later a quantum equivalent in a quantum many-body system (cf.[2, 3]). Met with enormous amount of crit- icism, the initial proposal eventually led to the discovery of a Floque time crystal, which breaks the discrete time symmetry of a quantum many-body system and was for the first time realized in laboratories in 2017 (cf. [1]).

In this talk, I will give a brief account of this development and share my own thoughts on the mathematical implications of these time crystals. These include degenerate dynamics around points of degeneracy, their contribution to time crystals and possible topological resolution of these degeneracies.

**References:**

[1] S. Choi, J. Choi, R. Landig, G. Kucsko, H. Zhou, J. Isoya, F. Jelezko, S. Onoda, H. Sumiya, V. Khemani, C. von Keyserlingk, N. Y. Yao, E. Demler and M. D. Lukin , Observation of discrete time-crystalline order in a disordered dipolar many-body system, Nature 2017 Mar 8; 543 (7644), 221225, letter.

[2] A. Shapere and F. Wilczek, Classical time crystals, Phys. Rev. Lett. 109, 160402 (2012).

[3] F. Wilczek, Quantum time crystals, Phys. Rev. Lett. 109, 160401 (2012).