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Description

In the 1990s, John H. Conway developed a visual approach to the study of integer-valued binary quadratic forms. His creation, the "topograph," sheds light on classical reduction theory, the solution of Pell-type equations, and allows tedious algebraic estimates to be simplified with straightforward geometric arguments. The geometry of the topograph arises from a Coxeter group of type (3, infinity) and its close relation to the group PGL(2,Z). From this perspective, the insights of Conway arise from an arithmetic hyperbolic Coxeter group. In this talk, I will survey Conway's approach to integer binary quadratic forms, and show how similar techniques can yield number theoretic results from other arithmetic hyperbolic Coxeter groups. These results are joint work with Chris D. Shelley and Suzana Milea.