5 research outputs found
Likelihood Geometry of Reflexive Polytopes
We study the problem of maximum likelihood (ML) estimation for statistical
models defined by reflexive polytopes. Our focus is on the maximum likelihood
degree of these models as an algebraic measure of complexity of the
corresponding optimization problem. We compute the ML degrees of all 4319
classes of three-dimensional reflexive polytopes, and observe some surprising
behavior in terms of the presence of gaps between ML degrees and degrees of the
associated toric varieties. We interpret these drops in the context of
discriminants and prove formulas for the ML degree for families of reflexive
polytopes, including the hypercube and its dual, the cross polytope, in
arbitrary dimension. In particular, we determine a family of embeddings for the
-cube that implies ML degree one. Finally, we discuss generalized
constructions of families of reflexive polytopes in terms of their ML degrees.Comment: 30 pages, 6 figures, 5 table
Supercongruences for rigid hypergeometric Calabi-Yau threefolds
We establish the supercongruences for the fourteen rigid hypergeometric Calabi--Yau threefolds over conjectured by Rodriguez-Villegas in 2003. Our first method is based on Dwork's theory of -adic unit roots and it allows us to establish the supercongruences between the truncated hypergeometric series and the corresponding unit roots for ordinary primes. The other method makes use of the theory of hypergeometric motives, in particular, adapts the techniques from the recent work of Beukers, Cohen and Mellit on finite hypergeometric sums over . Essential ingredients in executing the both approaches are the modularity of the underlying Calabi--Yau threefolds and a -adic perturbation method applied to hypergeometric functions
Mirror Symmetry and Polar Duality of Polytopes
This expository article explores the connection between the polar duality from polyhedral geometry and mirror symmetry from mathematical physics and algebraic geometry. Topics discussed include duality of polytopes and cones as well as the famous quintic threefold and the toric variety of a reflexive polytope