Math Day in Shenzhen

The Andrews-Curtis conjecture is an interesting puzzle in combinatorial group theory and low-dimensional topology. It asserts that any balanced presentation of the trivial group can be simplified to the trivial presentation by a finite sequence of elementary transformations. In this talk, I will give a brief survey about the conjecture and its stable version.

Poincaré emphasized the importance of periodic solutions in understanding the dynamics of the N-body problem. In this talk, we will present some of our results on the existence of periodic solutions in the following settings:

1. simple choreographies in the N-body problem with equal masses;
2. regularizable periodic solutions in the collinear N-body problem with arbitrary choice of masses;
3. relative periodic solutions in the spatial isosceles three body problem with two equal masses.

The interplay between tensor-network and machine learning models has attracted considerable attentions. Here we introduce the fundamental features of tensor-network machine learning models, with a focus on their trainability, generalization ability, vulnerability, and so on. Concretely, we first rigorously prove the presence and absence of barren plateau phenomena in different tensor network models. Then we rigorously prove the no-free-lunch theorems for both the 1D and 2D tensor network learning models, by introducing the combination method associated to “the puzzle of polyominoes”. Furthermore, we rigorously establish an adversarial theorem that emerges naturally as a practical consequence of our no-free-lunch theorem. Our findings reveal the intrinsic features of tensor-network machine learning models, and would open up an avenue for further analytical explorations on tensor-network models.

Quasi-periodic Schrödinger operators possess a rich background in quantum physics. From a mathematical perspective, they are deeply connected to diverse fields such as dynamical systems, fractal geometry, harmonic analysis, and number theory. In this talk, I will survey recent advancements in the study of the spectral properties of quasi-periodic Schrödinger operators, with a particular emphasis on methodologies rooted in dynamical systems. This work is primarily collaborative, undertaken with A. Avila, Y. Last, M. Shamis, and J. You.