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The talk will consists of two parts in different natures. In the first 45 minutes we will introduce the concept of Farahat-Higman rings. This will be done through reviewing the original work of Farahat and Higman in which they studied the center of the group rings Z[Sn] over the symmetric group of n letters. We will present their proof of the existence of a universal filtered Z-algebra Z that governs the center of the group algebras Z[Sn]. We will also present a conceptual explanation of the work of Farahat and Higman. The conceptual explanation enables one to generalization the work of Farahat and Higman to groups such as wreath products G o Sn, where G is a finite group, the general linear group GLn(Fq) over finite fields or symplectic groups Spn(Fq) over finite fields. In the second lecture we will briefly apply the method to the center of the group rings Z[Spn(Fq)] and present the proof of the existence of a universal filtered Z-algebra that governs the center of the algebras Z[Spn(Fq)]. If time permits, we will present the semi-group method of Ivanov and Kerov. The semi-group method demystifies the work of Farahat and Higman.