北京雁栖湖应用数学研究院

A Schubert class is called rigid if it can only be represented by Schubert varieties. We obtained a classification of rigid/multi-rigid Schubert classes for partial flag varieties. The rigidity problem is also related to the question that asks for a given (co)homology class, whether it can be represented by an irreducible subvariety. Besides Schubert classes, we investigate this problem for Chern-Schwartz-MacPherson classes which generalize the usual Chern classes to singular varieties.

Here, a boundary layer model is developed to analyze melting heat transfer in the flow of a tangent hyperbolic nanofluid past a permeable wedge-shaped surface.The governing equations for momentum, energy, and nanoparticle concentration are formulated, incorporating shear-dependent viscosity and phase change due to surface melting. Using similarity transformations, the system is reduced to ordinary differential equations and solved numerically with bvp4c function of MATLAB. The effects of the wedge angle, surface permeability, and melting parameter on velocity, temperature, and concentration profiles are examined, revealing significant impacts on boundary layer structure and nanofluid transport behavior.

In this study, we investigate the predictive capabilities of different news providers based on sentiment analysis, and propose a framework that endows different weights to different news providers for improving the prediction performance. In sentiment analysis, the prevalent Loughran-McDonald sentiment dictionary is utilized to calculate the sentiment scores of news articles, and the sentiment index of each news provider is obtained by integrating these sentiment scores. Based on the market data and sentiment indices of multiple news providers, we employ the recurrent neural network to build a number of base classifiers, and adopt the evidential reasoning rule to combine these base classifiers for predicting the stock market index movement. Additionally, the genetic algorithm is used to optimize the weights of base classifiers and important hyper-parameters of the recurrent neural network. In the experimental study, we apply the proposed approach to the daily movement prediction of the S&P 500 index, Dow Jones Industrial Average index and NASDAQ 100 index, and compare it with some state-of-the-art methods. The results show that our approach is effective for improving the prediction performance. Besides, the designed trading strategy based on the results of the proposed model achieves higher return rates than other trading strategies.

Significant success has been obtained in applying the homology of the directed flag complex to study digraphs arising as directed networks, including in traffic rout analysis and brain microcircuitry. In particular, this homology of directed cliques enjoys relative ease of computation when compared to other digraph homologies, making it preferable for use in applications concerning large network. However, computationally efficient homologies with good homotopical proprieties capable of distinguishing quivers have not previously been considered.

The classification of quantum phases of matter is a cornerstone of modern condensed matter physics. While the classifition of one-dimensional topological phases with onsite symmetries has been well-established for over a decade, their underlying mathematical structures--encoded in their macroscopic observables--has only recently been rigorously uncovered. We show that the observables, comprising the topological defects and symmetry actions, naturally form an enriched (fusion) category. The boundary-bulk relation, which plays the central role in the study of topological orders, also motivates new mathematical directions in enriched category theory. We address this by explicitly defining and computing the center of enriched (fusion) categories, and show that the results coincide with the physical intuitions.

Carrollian spacetime emerges from the ultra-relativistic limit ($c\to \infty$) of relativistic spacetime. It’s named after the pen name "Lewis Carroll" adopted by the mathematician Charles Dodgson. A defining characteristic of Carrollian geometry is its degenerate metric, for instance, the flat Carrollian metric is given by $ds^2=0dt^2+dx^i dx^i$. Recently, Carrollian structures have garnered significant attention due to their role in flat holography. The null infinity of asymptotically flat spacetimes shares the same degeneracy, motivating the framework of Carrollian holography. The central objective of my recent research is to formulate and analyze this holographic correspondence. The work I will present is the first paper of my series study on supersymmetric Carrollian holography. We systematically explore possible supersymmetric extensions of the Carrollian (conformal) algebra in dimensions $d=4$ and $d=3$. The Carrollian rotation algebra, consisting of spatial rotations and Carrollian boosts, is non-semisimple, leading to generally reducible yet indecomposable representations for the supercharges. Consequently, the Carrollian superalgebra admits multiple distinct structures. However, additional constraints arise in the conformal case. We found the nontrivial Carrollian superconformal algebras in $d=4$ and $d=3$ are isomorphic to the super-Poincar\'e algebras of $d=4$ and $d=3$ respectively. Notably, neither construction requires R-symmetry for algebraic closure. Furthermore, we identify two distinct classes of super-BMS4 algebras: a singlet super-BMS4 algebra and two multiplet chiral super-BMS4 algebras. The singlet case and one multiplet case originate from an extension of the 3D Carrollian superconformal algebras, whereas the other multiplet case do not admit this pathology due to their finite-dimensional subalgebra containing supercharges with conformal dimension $\Delta=\pm\frac{3}{2}$. This work lays the foundation for investigating supersymmetry within Carrollian holography. In particular, the discovery of multiplet chiral super-BMS symmetries introduces a novel framework for constructing super-BMS theories in holographic contexts. This work lead directly to: (i) constructing supersymmetric BMS theories, (ii) developing extrapolate dictionary and bulk reconstruction methods for supersymmetric fields, and (iii) analyzing superradiation in asymptotically flat spacetimes.

Let $(\Sigma,g_0)$ be a smooth closed Riemann surface, and $g=h\cdot g_0$ be a conical metric representing the divisor $\sum_{a\in\mathcal D}\alpha_{a}a$ with all $\alpha_a\in(-1,0)$. Then the singular Moser-Trudinger inequality holds $$\sup_{\substack{u\in H^1(\Sigma)\\ \|u\|_{h}^2=\beta}} \int_\Sigma he^{u^2}\mathrm d v_{g_0}<+\infty \quad\iff\quad \beta\le 4\pi\left(1+\min_{a\in\mathcal D}\alpha_a\right).$$ Building on this inequality, many works have investigated extremal functions, and more generally, the search for critical points of the constraint functional $$\int_\Sigma h(e^{u^2}-1)\mathrm d v_{g_0} \quad\text{on}~\Big\{u\in H^1(\Sigma) : \|u\|_{h}^2=\beta\Big\},$$ for $\beta\le 4\pi(1+\min_{a\in\mathcal D}\alpha_a)$. We prove that this functional also admits critical points for higher energies $\beta>4\pi(1+\min_{a\in\mathcal D}\alpha_a)$, using a min-max scheme and a subcritical approximation.

The wave equation is a second-order partial differential equation used in physics to describe various wave phenomena in nature. It is also an important model in the field of dispersive equations. This talk primarily studies equations such as the semilinear defocusing energy subcritical wave equation with an inverse-square potential. Our main goal is to understand the long time behavior of solutions. Following the proof of Dodson’s work, which uses Fourier truncation method and hyperbolic coordinates, we prove global wellposedness and scattering for radial initial data lying in the critical Sobolev space for higher dimension($d\ge 3$).

We propose a method called the $\tau$-optimal defining ($\tau$-OD) matrix-product (MP) construction to derive infinite families of quantum codes with good parameters. Through this scheme, we present 100 record-breaking quantum codes, which exceed the best-known lower bounds on the minimum distances of quantum codes listed in Grassl’s online database（Joint with Meng Cao).