School of Science Department of Mathematics 49 Quantum Groups Supervisor: IP Ivan Chi Ho / MATH Student: MAN Ryuichi / MATH-IRE Course: UROP4100, Spring We study the actions of the generalized Casimir operators of the quantum groups on the family of parabolic positive representations introduced in . Our main results (Theorems 3.1, 3.4 and 3.5) provide an explicit algorithm to compute the action of the Casimir operators on from their action on the positive representations , which can be computed using the formula in [4, Theorem 6.2]. In this report, we first provide the necessary background knowledge on the root systems and the quantum groups, and introduce the notations used. Followed by this is a summary of the key results obtained from our previous research. Then, we proceed to state and prove the algorithm to compute the Casimir action, and illustrate its use with an example.  I. Ip, Positive Casimir and central characters of split real quantum groups, Communication in Mathematical Physics, 344(3), 857-888, 2016. arXiv: 1503.00543.  I. Ip, Parabolic positive representations of , preprint, 2020. arXiv: 2008.08589. Integro-differential Equations: Theory and Applications Supervisor: JIN Tianling / MATH Student: LIU Jiaming / MATH-IRE Course: UROP1100, Spring We study the fundamental structure of the Sobolev space so as to have a more rigorous understanding of the weak solution of the boundary value problem of second-order partial differential equations as well as the variational principle for the principal eigenvalue of symmetric elliptic operators in partial differential equations. In this study report, the basic structure of Sobolev space is introduced, including the weak derivatives of locally integrable functions, in which we highlight the similarities and differences compared with ordinary derivatives. Besides, the elementary properties of Sobolev space are presented. Supplementary concepts, such as mollifier, are introduced. Moreover, after understanding the fundamental structure of Sobolev space, we consider the explanation of the weak solution of boundary value problems and the eigenvalues of symmetric elliptic operators, based on the properties of Sobolev space, linear algebra and real analysis.