IMath Seminar (Feb 13): Dr. Eduardo R. Mendoza

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You are all invited to a seminar featuring:
Dr. Eduardo R. Mendoza (Mathematics and Statistics Department, De La Salle University)

Talk: Serre’s Tree and its Generalizations

This is an in-person event and it will be held on Thursday, February 13, 2025, 2:30 PM at MB 105. 

Abstract:
Jean-Pierre Serre (1926 – present), professor at the elite College de France in Paris, is considered by many colleagues as the greatest mathematician of the second half of the 20th century. Between the dual distinction of being the youngest Fields Medalist to date (at 27 in 1954) and the first Abel Prize laureate in 2003, the latter part of his stellar career encompassed connecting group theory, topology, arithmetic (or number theory) and geometry. An early highlight of this effort was the mid-1970s study of discrete groups acting on trees, with the primary example of the modular group PSL (2,Z) acting on a trivalent infinite tree in the hyperbolic plane, soon known as Serre´s tree. In 1980, the speaker introduced a generalization of Serre´s tree, the “minimal incidence set” in hyperbolic 3-space for any Bianchi group, i.e., PGL (2, O K ) where is the ring of integers in the imaginary-quadratic field K.
After a brief overview of Serre´s mathematical work, the talk will discuss the evolution of the construct to a widely used tool in topological arithmetic, particularly to study the arithmetic of hyperbolic 3-manifolds after K. Vogtmann (Cornell University) in 1985 used it to resolve the renowned “Cuspidal Cohomology Problem” (and renamed it the “Mendoza complex” of a Bianchi group). It also enabled a more geometric approach to integral binary Hermitian forms, both indefinite and positive definite. Two recent developments are significant: first, the construct was extended by J. Parkonnen and F. Poulin to hyperbolic 5-space and matrix groups of quaternions, thus completing the binary framework. Furthermore, M. Planat and collaborators have found a very surprising application of Bianchi groups to permutation gate-based quantum computing.
From a personal retrospective, these mathematical developments constitute an anecdote of luck, coincidence and “happy ends”. The talk concludes with an outlook on related research in topological arithmetic and potential connections to topological quantum computing, one of the contending technologies in the race between IBM, Google, Microsoft and others to build a quantum supercomputer.