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On lattice models and finite fields

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If you have a question about this talk, please contact Helen Innes.

I will discuss a surprising analogy between (a) lattice models of statistical physics (e.g. the Ising model) and (b) some generalizations of point-counting in algebraic varieties over finite fields. In particular, Hecke operators in the functional case of the Langlands correspondence give rise to an infinite tower of matrices of exponentially growing size, whose spectra behave similarly to the spectra of transfer matrices for Ising-type lattice models with periodic boundary conditions. I propose a conjecture that could explain this phenomenon. This conjecture could also give an explicit, “quantum computer”-type construction of a general finite field.

This talk is part of the Kuwait Foundation Lectures series.

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