It is not necessary that the tensor product of two Demazure crystals can be decomposed as a disjoint union of Demazure crystals. In this talk, I will talk about the necessary and sufficient condition of the decomposition of two Demazure crystals as a disjoint union of Demazure crystals for symmetrizable Kac-Moody Lie algebras.

In this talk, we will discuss the integrable structure of a conformal field theory with an extended W_3 symmetry algebra. These systems are expected to have infinitely many conserved local integrals of motion, known as quantum Boussinesq charges, which are in involution with each other. We will propose a prescription to systematically construct the conserved currents of such a system by combining two approaches. First, we will determine the eigenvalues of quantum Boussinesq charges on the highest-weight state, the first excited state and the second excited state using the ODE/IM correspondence. Second, we will compute thermal correlators of these charges using the Zhu recursion relation, evaluating traces of composite operators composed of the energy-momentum tensor, spin-3 fields W, and their derivatives on the higher-spin module of a torus. By combining these results, we will derive new currents of the quantum Boussinesq hierarchy

In this talk, we will focus on two classical partial covering problems under budget constraints: Max k-weight SAT and Maximum Coverage. We will examine these problems through two complementary optimization objectives — optimizing the number of resources required to satisfy given constraints, and optimizing the number of constraints satisfied given a bounded budget. Since both problems are computationally intractable in general, we will motivate the parameterized approximation framework under structural restrictions. We will present efficient parameterized approximation schemes (EPAS) for biclique-free instances and establish an equivalence between the two problems in this setting when parameterized by solution size. This equivalence extends to variants with additional constraints, allowing us to focus on algorithmic results for Maximum Coverage, from which we derive randomized EPAS for a broad family of satisfiability and covering problems. Our results both unify and significantly generalize prior work. The talk will also cover results parameterized by the number of satisfied constraints, as well as a novel extension of the lossy kernelization framework to multi-criteria optimization, developed in the course of designing lossy kernels under various constraints. Finally, we will outline how the techniques developed here extend to related problems such as Max k-weight NAE-SAT, underscoring their versatility. (Pre Synopsis Talk.)