Slim cyclotomic q-Schur algebras were first introduced by Z. Lin and H. Rui, as centralizer subalgebras of Dipper-James-Mathas’ cyclotomic q-Schur algebras. Recent developments allow us to take the investigation to a new level. First, by using the matrix labelling of C. Mak for certain double cosets, we may construct an integral basis, simpler than the cellular basis, for these algebras. Second, the introduction of the Lusztig type form for quantum affine gl_n by Q. Fu and myself allows us to establish the cyclotomic Schur-Weyl duality at the integral level. Finally, when q is not a root of unity, we obtain a classification of irreducible representations. I will also mention some problems and conjectures. This is joint work with B. Deng and G. Yang.

In this paper, q-Gaussian distribution, q-analogy of Gaussian distribution, is introduced to characterize the prior information of unkown parameters for inverse problems. Based on Q-Hermite polynominals, we propase a spectral likelihood approximation algorithm of Bayesian inversion. Convergence results of the approximated posterior distribution in the sense of KullbackpLeibler divergence are obtained when the likelihood function is replaced with the SlA and the prior density function is truncated to its partial sum. In the end, two numerical examples are displayed, which verify our results.

With the development of the hardware, the numerical study on the all-electron Kohn-Sham equation has been attracting more and more attention, which potentially would play an important role in studying a variety of physical phenomena such as high harmonic generation. In this talk, towards the efficient numerical solver for the all-electron Kohn-Sham equation, an adaptive finite element framework

We address the following rainbow Ramsey problem: For posets P, Q what is the smallest number n such that any coloring of the elements of the Boolean lattice Bn either admits a monochromatic copy of PP or a rainbow copy of QQ. We consider both weak and strong (non-induced and induced) versions of this problem.

Vizing’s planar graph conjecture says that every class two planar has maximum degree at most 5. The conjecture is still open. We consider the problem for graphs embeddable on a surface Sigma and attempt to find the best possible upper bound for the maximum degree of class two graphs embeddable on Sigma. In this talk I will give a survey on this topic and present some new results.

The talk will describe various generalizations of the Hadamard jump condition, and how they can lead to information about polycrystal microstructure arising from martensitic phase transformations. (Joint work with Carsten Carstensen (Humboldt University, Berlin)

We propose a new and short proof for the Macdonald identities using only some easy facts from the theory of Jacobi forms of lattice index and classical root systems. (The needed basic features of these theories will be explained in the talk). We discuss applications and open questions related to the new proof, and we end the talk by deducing from the Macdonald identities, for 42 elliptic curves over the rationals and of rank 1 product identities for the (classical) Jacobi forms attached to elliptic curves by the theory of modular forms.