Symmetry plays an important role in quantum systems: it can constrain the dynamics, give rise to selection rules, and provide computation methods in quantum computers. In recent years there are also new types of symmetries called generalized symmetries discovered in many quantum systems, including non-invertible symmetry and higher group symmetry. These lectures will be about symmetries in various quantum systems and their applications such as constraints on the low energy dynamics. Examples will be discussed in the lectures include quantum mechanics systems, gauge theories, lattice models, and the symmetry includes ordinary and higher form symmetry as well non-invertible symmetry.

Symmetry plays an important role in quantum systems: it can constrain the dynamics, give rise to selection rules, and provide computation methods in quantum computers. In recent years there are also new types of symmetries called generalized symmetries discovered in many quantum systems, including non-invertible symmetry and higher group symmetry. These lectures will be about symmetries in various quantum systems and their applications such as constraints on the low energy dynamics. Examples will be discussed in the lectures include quantum mechanics systems, gauge theories, lattice models, and the symmetry includes ordinary and higher form symmetry as well non-invertible symmetry.

Symmetry plays an important role in quantum systems: it can constrain the dynamics, give rise to selection rules, and provide computation methods in quantum computers. In recent years there are also new types of symmetries called generalized symmetries discovered in many quantum systems, including non-invertible symmetry and higher group symmetry. These lectures will be about symmetries in various quantum systems and their applications such as constraints on the low energy dynamics. Examples will be discussed in the lectures include quantum mechanics systems, gauge theories, lattice models, and the symmetry includes ordinary and higher form symmetry as well non-invertible symmetry.

For almost four decades, PROMYS has been synonymous with deep exploratory mathematical learning for talented secondary school students and their teachers. In this presentation we will discuss the history of PROMYS and its underlying principles as well as strategies for developing mathematical habits of mind that encourage creativity and innovation. We will also share ideas for new outreach efforts, currently in development, designed to serve local students from underserved populations in the Boston area.

Thermalization is the process by which a physical system evolves toward a state of maximal entropy, as permitted by conservation laws. I will begin by outlining the framework used to understand this phenomenon in quantum systems with unitary evolution (Eigenstate Thermalization Hypothesis). Next, I will discuss factors that can hinder or slow down thermalization. One example is long-lived prethermalization, where certain effective (or pseudo-conserved) quantities significantly delay thermalization depending on specific model parameters. This theory is particularly relevant for periodically driven systems, which can exhibit remarkable resistance to heating over extended timescales. I will then explore the possibility of systems that robustly fail to thermalize. Here, robustness refers to the fact that no fine-tuning is required, in contrast with integrable models. Many-body localization (MBL) is the most well-known, and possibly the only example of systems that fail to thermalize on their own. I will examine MBL from both theoretical and numerical perspectives, covering its description in terms of local integrals of motion, the destabilizing effect of quantum avalanches, and recent mathematical advancements. These later developments are welcome given the challenges in properly interpreting numerical results in this field.

Symmetry plays an important role in quantum systems: it can constrain the dynamics, give rise to selection rules, and provide computation methods in quantum computers. In recent years there are also new types of symmetries called generalized symmetries discovered in many quantum systems, including non-invertible symmetry and higher group symmetry. These lectures will be about symmetries in various quantum systems and their applications such as constraints on the low energy dynamics. Examples will be discussed in the lectures include quantum mechanics systems, gauge theories, lattice models, and the symmetry includes ordinary and higher form symmetry as well non-invertible symmetry.

Thermalization is the process by which a physical system evolves toward a state of maximal entropy, as permitted by conservation laws. I will begin by outlining the framework used to understand this phenomenon in quantum systems with unitary evolution (Eigenstate Thermalization Hypothesis). Next, I will discuss factors that can hinder or slow down thermalization. One example is long-lived prethermalization, where certain effective (or pseudo-conserved) quantities significantly delay thermalization depending on specific model parameters. This theory is particularly relevant for periodically driven systems, which can exhibit remarkable resistance to heating over extended timescales. I will then explore the possibility of systems that robustly fail to thermalize. Here, robustness refers to the fact that no fine-tuning is required, in contrast with integrable models. Many-body localization (MBL) is the most well-known, and possibly the only example of systems that fail to thermalize on their own. I will examine MBL from both theoretical and numerical perspectives, covering its description in terms of local integrals of motion, the destabilizing effect of quantum avalanches, and recent mathematical advancements. These later developments are welcome given the challenges in properly interpreting numerical results in this field.