Dr. Ernst Kani

Biography

Ernst Kani is a professor in the Department of Mathematics and Statistics at Queen's University. His main research area is Arithmetic Geometry, which synthesizes Number Theory and Algebraic Geometry, particularly focusing on Diophantine questions and Fermat's Theorem. He has developed a keen interest in the applications of Arithmetic Geometry to Public Key Cryptography, emphasizing his research on Galois representations, specifically those attached to modular elliptic curves. His contributions have played a crucial role in Wiles' proof of Fermat's Theorem. Kani's investigations lead to four inter-connected research lines: 1) Classifying isomorphisms of Modular Diagonal Quotient Surfaces; 2) Exploring curves of genus 2 with elliptic differentials through the fusion of elliptic curves with isomorphic Galois representations; 3) Studying Hurwitz schemes that arise from genus 2 covers of elliptic curves; and 4) Analyzing representations of fundamental groups in collaboration with G. Frey and H. Volklein. His topics, particularly those related to modular diagonal quotient surfaces, are closely tied to modular forms and Galois representations.

Research Interests