Abstract

A graph can be viewed, in many respects, as a discrete analogue of an algebraic curve. We begin by formulating the theory of "divisors" on graphs and on tropical curves, and discuss the combinatorial analogues of the classical Riemann-Roch theorem. We also describe the analogues of Jacobian varieties in these settings, which are intimately related to the classical matrix-tree theorem. Connections with commutative algebra and algebraic geometry will also be discussed. We then turn our attention to non-archimedean geometry (in the sense of Berkovich) and the problem of "faithful tropicalization" of abelian varieties in terms of p-adic and tropical theta functions. .
(The non-expository parts will be based on joint works with M. Baker -- Y. An, M. Baker, G. Kuperberg -- F. Mohammadi -- T. Foster, J. Rabinoff, A. Soto).



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