研究生研讨班 2022-2024

Let $1<p<\infty$. We show the boundedness of operator-valued commutators $[\pi_a,M_b]$ on the noncommutative $L_p(L_\infty(\mathbb{R})\otimes \mathcal{M})$ for any von Neumann algebra $\mathcal{M}$, where $\pi_a$ is the $d$-adic martingale paraproduct with symbol $a\in BMO^d(\mathbb{R})$ and $M_b$ is the noncommutative left multiplication operator with $b\in BMO^d_\mathcal{M}(\mathbb{R})$. Besides, we consider the extrapolation property of semicommutative $d$-adic martingale paraproducts in terms of the $BMO^d_\mathcal{M}(\mathbb{R})$ space.

In this talk, we present a range of solutions and effective strategies for addressing the inverse problems, incorporating constraints on the Hilbert space and employing various optimization techniques. Specifically, the Kaczmarz-gradient, two-point gradient and homotopy perturbation methods are introduced. Additionally, leveraging deep learning technique as a tool, we derive an approach for adjusting convex penalties based on iterative adjustments. The rationality and efficiency of these methods are validated through numerical simulations involving multiple equations.

Collective behavior is ubiquitous in natural phenomena, such as the flocking of birds and the swarming of fish. In 2007, Cucker and Smale introduced the well-known Cucker-Smale (CS) model to describe the collective behavior of bird flocks. This study focuses on the clustering behavior of the CS model with an infinite number of particles. Research in this area is divided into two domains: one involves deriving kinetic models corresponding to finite-particle models through mean-field limits, and the other investigates dynamical systems on infinite graphs.

Recent work by Xin Li has shown how to naturally associate C*-algebras to Garside categories and to present these as groupoid algebras for appropriately chosen groupoids, obtaining a unifying theory encompassing many important special cases. This talk will be a quick tour of Garside theory on left cancellative small categories, as well as groupoids and C*-algebras arising from them, with an application to higher-rank graphs as a typical example.

In this talk, we further study the operator-valued Hardy and BMO spaces introduced by Tao Mei, and establish the wavelet characterizations of Hrady spaces and harmonic extension of non-commutative BMO functions. In addition, we introduced a kind of general Hardy spaces, that is, operator-valued Hardy spaces associated with anisotropic dilations, and we also establish the classical Fefferman's duality theorem between Hardy and BMO spaces in our setting. As applications, we also obtain the real and complex interpolations theory on these spaces. This is joint work with Dr. Cheng Chen, Prof. Guixiang Hong and Prof. Xinfeng Wu.

Comon's Conjecture asserts that for a symmetric tensor, the rank is equal to the symmetric rank. However, the symmetric rank does not always exist. We give a necessary and sufficient condition for symmetric tensors to have symmetric rank, and give some sufficient conditions for this conjecture to be true. Moreover, we propose an algebraic method to compute the symmetric rank and symmetric rank decomposition for symmetric tensors over the binary field. Finally, we completely characterize the maximum rank of m×n×2 tensors over an arbitrary field.

As is well known, the Dirac operator plays a crucial role in Alain Connes’ noncommutative geometry. In this talk, we will revisit the construction of Dirac operators on Riemannian spin manifolds. Some basic knowledge of classical geometry is required.
Working on a spectral triple (A,H,D), i.e. the noncommutative generalization of a Riemannian spin manifold, places you into the operator framework. So, in order to get some non-trivial results on it, both summability and regularity concerning the abstract Dirac operator D (self-adjoint, possibly unbounded) are usually assumed to be good enough. However, life is not smooth, especially when your algebra A is not ‘smooth’ either. Suppose now we have a nice algebra acting on Hilbert space H, then how to construct a proper Dirac operator D to guarantee the summability and regularity? There is no routine method in the noncommutative setting. This motivates us to look for some inspirations from the starting point: Riemannian geometry! And we will provide a few examples.

We have considered the primitive index for random quantum channels, which means the minimal natural number n such that the Choi state/matrix of the n-fold composition of channels is full rank. In the previous work, several upper bounds for the index have been obtained, which can be used to construct the so-called (quantum) Wielandt inequality. We note that the optimal upper bound is still an open problem. In our work, we have shown a generic lower bound for the index when the channels are randomly chosen. Our main method is the graphical Weingarten calculus, introduced by Collins and Nechita. Moreover, our result is closely related to the injectivity of the representation of matrix product states. Perez-Garcia et al. claimed that a similar lower bound for the injectivity could be numerically verified, and our result provides a rigorous proof for their argument.