Prominent among his contributions to algebraic geometry are the fundamental ideas he introduced to shape the modern theory of moduli spaces. Examples of moduli spaces include parameter spaces for families of algebraic varieties and algebraic objects over them. The problem of construction and geometry of moduli spaces has a long history going back to Riemann. Mumford achieved his goal by developing the theory of Geometric Invariant Theory where he introduced the concept of stability. The concept of stability has found important applications in many directions including the modern theory of geometric analysis. 

He applied his theory to construct moduli spaces of algebraic curves and abelian varieties. In collaboration with Harris, he proved that the moduli space of curves of high odd genus is of general type. His works paved the way to understand moduli spaces of more general algebraic varieties and sheaves over them. 

Via a different approach, he constructed moduli spaces of abelian varieties using Theta functions and therefore gave explicit algebraic equations of these moduli spaces. The theory led to deep understanding of Theta functions. Mumford proved that all abelian varieties over positive characteristics can be lifted to characteristic zero. With Bombieri, he extended the classification theory of algebraic surfaces to positive characteristics. 

In the 1980's, Mumford shifted from algebraic geometry to theoretical foundations of intelligence, both natural and artificial, where he made important contributions. 

There is no doubt that Mumford is one of the greatest living mathematicians."

Shamir is a co-inventor of the RSA public-key cryptography algorithm. In their 1978 paper “A method for obtaining digital signatures and public-key cryptosystems”, Rivest, Shamir and Adleman presented the first realistic implementation of public-key cryptosystem (a concept invented by Diffie and Hellman) which is ideal for internet applications. RSA is practical and theoretically elegant, and has been the most widely used encoding-decoding scheme in internet commercial applications.
Shamir’s 1979 paper “How to share a secret” showed how information about a master-key can be divided up among a group of users, so that the master-key cannot be reconstructed unless a quorum of the users pool together their shares of the information. Shamir’s secret-sharing scheme stands as among the most essential tools used in cryptography today. Other outstanding contributions initiated by Shamir include the identity-based cryptography, differential cryptanalysis (with Biham), and visual cryptography (with Naor) among others. Shamir also made seminal contributions to computational complexity theory. His 1992 paper “IP = PSPACE” resolved a major open problem regarding the power of interaction. 
Shamir is a formidable leader in computer science. His illustrious work will continue to influence the world for years to come. "