arXiv subjects
Publications and source records attributed to Cong Xue.
Let $G$ be a generically reductive group over a smooth projective curve $X$ over a finite field. For any finite set $I$, we show that nearby cycles commute with proper direct image from stacks of shtukas to $X^I$. This generalizes some results of Salmon and the authors.
The process of opinion expression and exchange is a critical component of democratic societies. As people interact with large language models (LLMs) in the opinion shaping process different from traditional media, the impacts of LLMs are increasingly recognized and being concerned. However, the knowledge about how LLMs affect the process of opinion expression and exchange of social opinion networks is very limited. Here, we create an opinion network dynamics model to encode the opinions of LLMs, cognitive acceptability and usage strategies of individuals, and simulate the impact of LLMs on opinion dynamics in a variety of scenarios. The outcomes of the simulations inform about effective demand-oriented opinion network interventions. The results from this study suggested that the output opinion of LLMs has a unique and positive effect on the collective opinion difference. The marginal effect of cognitive acceptability on collective opinion formation is nonlinear and shows a decreasing trend. When people partially rely on LLMs, the exchange process of opinion becomes more intense and the diversity of opinion becomes more favorable. In fact, there is 38.6% more opinion diversity when people all partially rely on LLMs, compared to prohibiting the use of LLMs entirely. The optimal diversity of opinion was found when the fractions of people who do not use, partially rely on, and fully rely on LLMs reached roughly 4:12:1. Our experiments also find that introducing extra agents with opposite/neutral/random opinions, we can effectively mitigate the impact of biased/toxic output from LLMs. Our findings provide valuable insights into opinion dynamics in the age of LLMs, highlighting the need for customized interventions tailored to specific scenarios to address the drawbacks of improper output and use of LLMs.
Sequential recommendation systems alleviate the problem of information overload, and have attracted increasing attention in the literature. Most prior works usually obtain an overall representation based on the user's behavior sequence, which can not sufficiently reflect the multiple interests of the user. To this end, we propose a novel method called PIMI to mitigate this issue. PIMI can model the user's multi-interest representation effectively by considering both the periodicity and interactivity in the item sequence. Specifically, we design a periodicity-aware module to utilize the time interval information between user's behaviors. Meanwhile, an ingenious graph is proposed to enhance the interactivity between items in user's behavior sequence, which can capture both global and local item features. Finally, a multi-interest extraction module is applied to describe user's multiple interests based on the obtained item representation. Extensive experiments on two real-world datasets Amazon and Taobao show that PIMI outperforms state-of-the-art methods consistently.
We prove, for all reductive groups, that the cohomology sheaves with compact support of stacks of shtukas are ind-smooth over $(X - N)^I$ and that their geometric generic fibers are equiped with an action of $\text{Weil}(X - N, \overline{\eta})^I$. Our method does not use any compactification of stacks of shtukas.
We construct the cohomology groups with compact support of stacks of shtukas with $\mathbb Z_{\ell}$-coefficients. We construct the cuspidal cohomology groups and prove that they are $\mathbb Z_{\ell}$-modules of finite type. We prove that the cohomology groups are modules of finite type over a Hecke algebra with $\mathbb Z_{\ell}$-coefficients. As an application, we prove that the cuspidal cohomology groups with $\mathbb Q_{\ell}$-coefficients are equal to the Hecke-finite cohomology groups with $\mathbb Q_{\ell}$-coefficients defined by V. Lafforgue. We also state the Eichler-Shimura relations for cohomology groups with $\mathbb Z_{\ell}$-coefficients and prove the compatibility of the excursion operators and the constant term morphisms.
In this paper we prove that the cohomology groups with compact support of stacks of shtukas are modules of finite type over a Hecke algebra. As an application, we extend the construction of excursion operators, defined by V. Lafforgue on the space of cuspidal automorphic forms, to the space of automorphic forms with compact support. This gives the Langlands parametrization for some quotient spaces of the latter, which is compatible with the constant term morphism.
Let $G$ be a connected split reductive group over a finite field ${\mathbb F}_q$ and $X$ a smooth projective geometrically connected curve over ${\mathbb F}_q$. The $\ell$-adic cohomology of stacks of $G$-shtukas is a generalization of the space of automorphic forms with compact support over the function field of $X$. In this paper, we construct a constant term morphism on the cohomology of stacks of shtukas which is a generalization of the constant term morphism for automorphic forms. We also define the cuspidal cohomology which generalizes the space of cuspidal automorphic forms. Then we show that the cuspidal cohomology has finite dimension and that it is equal to the (rationally) Hecke-finite cohomology defined by V. Lafforgue.