Arithmetic Applications of Definable and Hyperbolic Geometry

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It is a frequent phenomenon in mathematics that it can be useful to forget certain structure. For example, when one is studying a polynomial function, it can be counterproductive to use the algebraic structure, and instead one should merely remember that one is dealing with, say, a continuous function. This gives one the freedom to perform operations that are impossibly in the algebraic world (such as cutting-and-pasting) but comes at the expense of certain nice properties (such as having finitely many solutions to equations). More generally,  mathematicians work quite hard to find just the right setting to work in: sufficiently general so as to be flexible in what one is 'allowed' to do, but sufficiently  concrete so as to have many enjoyable properties. A frequent example of this is the world of algebraic functions, versus the world of holomorphic functions. A large part of my proposal deals with developing an intermediate category that can be loosely described as `o-minimal holomorphic functions'. It turns out that many of the functions we are interested in - such as the exponential function, and automorphic functions that come up in the study of algebraic varieties - are not quite algebraic, but are much more well-behaved then general holomorphic functions. This theory was initiated by Peterzil and Starchenko and has already found much use in functional transcendence and number theory. Together with my coworkers, we are pushing this theory further to allow for studying much more nuanced algebraic phenomena (nilpotent  thickenings, deformation theory, coherent sheaves, etc...) This has already had enormous applications to Hodge Theory - a particularly powerful tool for understanding algebraic varieties via their cohomology. We have shown that the natural setting for hodge theory is in fact 'o-minimal holomorphic functions', and using this proven long-standing conjectures in the field. More importantly, many of the existing results become much more streamlined, making the whole subject more accessible. The holy grail of hodge theory (and one of the central questions of algebraic geometry) is the hodge conjecture. This allows one to derive information about algebraic subvarieties (solutions to polynomial equations) from their hodge structures (much simpler linear algebraic information). It is one of the goals of this proposal to attempt to make progress on the hodge conjecture using this technology. Specifically, we hope that an important piece called the "absolute hodge conjecture" can be resolved using these methods.