Mathematics Colloquia and Seminars

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One of the most important classes of special functions (originally: functions that show up a lot in applications) is the hypergeometric functions.  Among their many characterizations is as solutions of linear ODEs, and those equations are themselves quite special: they are "rigid", in the sense that they are uniquely determined by their local behavior near singular points.  If one makes the singularities slightly more complicated, one needs two parameters to specify the equation, and it turns out that if one changes the singularity structure in a smooth way, there is a natural flow, and the solutions (Painlev\'e transcendents) of the corresponding nonlinear ODEs began to appear in applications in the past few decades. In both cases, other characterizations of the functions have been used to produce generalizations, giving rise to hierarchies with "elliptic" special functions at the top.  It turns out that there is a corresponding hierarchy of equations, which are now discrete (difference equations), with the top level living on an elliptic curve.  To show this (and to get further generalizations) one must understand families of difference equations with specified singularities; I'll explain how this leads naturally to *noncommutative* algebraic geometry, why elliptic curves arise generically, and how derived equivalences of noncommutative surfaces give rise to new linear problems for Painlev\'e-type equations.