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It is well known that the number of conjugacy classes of a finite group equals the number of irreducible representations. This can be regarded as a simple example of cocenter-representation duality. In this talk, I will explain how this idea can be used to study the affine Hecke algebras. It is based on joint works with Sian Nie, and with Dan Ciubotaru.

Mathematical concepts are the corner stones in the mathematical world. It is not easy to summarize and clearly state the inherent laws hidden in the concept's formation, variation, and the structure of the concept net. This talk is only a superficial try. We begin with the motivation and the formation of the notion of group, proceeding to the variations and alienations of groups. Certain generalizations of groups, such as affine group schemes and quantum groups, are also introduced, though they have a completely different look from the original face of a group.

In the second lecture, I will show how this idea is generalized to the quantum case and, in particular, to q-Schur superalgebras S_q(m|n,r). Thus, we obtain a classification theorem of simple modules of q-Schur superalgebras with m+n>=r at roots of unity. We believe that this work will be extended to all q-Schur superalgebras via the Schur functor. This is joint work of Gu Haixia and Wang Jianpan.

In this talk we establish weak factorizations for a weighted Bergman space A^p_a, with 1 < p < 1, into two weighted Bergman spaces on the unit ball of C^n. To obtain this result, we characterize bounded Hankel forms on weighted Bergman spaces on the unit ball of C^n. This is a joint work with Jordi Pau.

This talk reports on our recent development of high-resolution, nonionizing biophotonic imaging technologies, including optical coherence tomography (OCT) and multiphoton endomicros- copy. These fiber-optically based technologies have shown significant translational potential for imaging tissue microanatomiesin vivoat a resolution approaching or at that of standard histopathology but without the need for tissue removal or staining. The physics principles, engineering challenges and (big) data and image processing challenges, and potential solutions will be briefly discussed, including the development of advanced lasers, MEMS technology, custom optical fiber, ultracompact and high-performance micro optics, speckle noise reduction, image segmentation and optical staining techniques. Representative applications of these high-resolution technologies will be presented,

In the first lecture, I will discuss an extension of the Alperin weight conjecture to permutation modules M of a finite group. By introducing relative projectivity and the theory of vertices and sources, we may partition the set of primitive idempotents of End(M) in terms of their defect groups. In this way, we may provide a classification of simple modules of End(M).

In this talk, I will report our recent research on a reaction-advection- diffusion competition model in a periodic habitat. We first investigate the global attractivity of a semi-trivial steady state (i.e., the competitive exclusion) for the periodic initial value problem. Then we establish the existence of the rightward spreading speed and its coincidence with the minimal wave speed for spatially periodic rightward traveling waves. Further, we obtain a set of sufficient conditions for the rightward spreading speed to be linearly determinate. Finally, we apply the obtained results to a prototypical reaction-diffusion model. This is a joint work with Dr. Xiaoqiang Zhao.

I will talk about the application of algebraic and number theoretical methods in the study of two combinatorial structures: difference sets in design theory and Cameron-Libler line class in finite geometry. The first concerns the multiplier conjecture, while the other has its origin in the study of permutation groups. What they have in common is that results from algebraic number theory plays important role in the proof. The second part is a joint work with Koji Momihara and Qing Xiang.