2020年12月24日（星期四）上午9：00—10：00 摘要: In this talk, we study a Boltzmann-type Mean Field Game model proposed in Achdou et al. (2014) for knowledge diffusion and economic growth, where knowledge diffusion results from imitation by searching and learning and from innovation subject to Brownian noises. Largely inspired by Dai et al. (2009), where the marginal value function has been used directly to study portfolio selection with transaction costs, we transform the original partial integro-differential equation system into an equivalent one by also studying a representative agent's marginal value function. We show a necessary condition to generate a sustained growth is that innovation cannot dominate imitation. In particular, when learning technology is sufficiently inefficient or discount rate is sufficiently low, either of which leads individuals to put no effort in imitation, sustained economic growth then disappears. Further

It is a classical problem to determine the algebra of automorphic forms on symmetric domains. The first breakthrough was due to Igusa, who showed in 1962 that the algebra of genus 2 Siegel modular forms of even weight is freely generated by four forms of weights 4, 6, 10, 12.In this talk, we introduce a new approach to classify and construct free algebras of modular forms on symmetric domains of type IV. We first establish a sufficient and necessary condition for the graded algebra of orthogonal modular forms being free. We then prove an explicit classification result and present more than 70 such

Data hiding or steganography in image is a scheme of hiding the secret message into the cover image. It can be used to transmit the secret data by the cover image or embed important information such as copyright information, authentication information or management information in the cover images. The quality of stego-image is the main object of the applications in copyright protection and the image authentication. The method to improve the quality is to enhance the embedding efficiency. In order to achieve a higher embedding capacity and embedding efficiency,

The projective characters of the symmetric group was studied by I. Schur in his seminal paper of 1911, where the famous Schur Q-functions were defined and a Frobenius type character table was established. I. Frenkel, W. Wang and I have generalized Schur's work and computed major part of spin characters of the wreath products in 2001. I will discuss how to compute the missing part of spin characters for the wreath products.

In this work, we introduce {\em topological representations of a quiver} as a system consisting of topological spaces and its relationships determined by the quiver. Such a setting gives a natural connection between topological representations of a quiver and diagrams of topological spaces. First, we investigate the relation between the category of topological representations and that of linear representations of a quiver via $P(\Gamma)$-$\mathcal{TOP}^o$ and $k\Gamma$-Mod, concerning (positively) graded or vertex (positively) graded modules. Second, we discuss the homological theory of topological representations of quivers via $\Gamma$-limit $Lim^{\Gamma}$ and using it, define the homology groups of topological representations of quivers via $H_n$

In this talk I shall report a joint work with Xiao Wei on the graded decomposition numbers and inverse graded decomposition numbers for parabolic Verma modules in any integral blocks of a parabolic category of a semisimple Lie algebra. A sum formula for the radical filtrations of parabolic Verma modules is also obtained, which generalizes the classical Jantzen sum formula for the usual Verma modules.

We investigate the contracting curvature flow of closed, strictly convex axially symmetric hypersurfaces in R^n+1 and S^n+1 by sigma_k^\alpha, where \sigma_k is the k-th elementary symmetric function of the principal curvatures and \alpha\ge 1/k. We prove that for any n\geq3 and any fixed k with 1\leq k\leq n, there exists a constant c(n,k)>0 such that if \alpha lies in the interval [1/k,1/k+c(n,k)], then we have a nice curvature pinching estimate involving the ratio of the biggest principal curvature to the smallest principal curvature at every point of the flow hypersurface, and we prove that the properly rescaled hypersurfaces converge exponentially to the unit sphere. Our results provide an evidence for the general convergence result without initial curvature pinching conditions.

In this talk we will study the limit of a sequence of harmonic maps from surfaces with uniformly bounded energy. We define the generalized neck domain as the union of ghost bubbles and necks that are directly connected and give a uniform upper bound of the energy density on the generalized neck domain. As an application, we prove an inequality of the index and nullity of the sequence.