A Landau--Ginzburg mirror theorem without concavity

We provide a mirror symmetry theorem in a range of cases where the state-of-the-art techniques relying on concavity or convexity do not apply. More specifically, we work on a family of FJRW potentials named after Fan, Jarvis, Ruan, and Witten's quantum singularity theory and viewed as the counterpart of a non-convex Gromov--Witten potential via the physical LG/CY correspondence. The main result provides an explicit formula for Polishchuk and Vaintrob's virtual cycle in genus zero. In the non-concave case of the so-called chain invertible polynomials, it yields a compatibility theorem with the FJRW virtual cycle and a proof of mirror symmetry for FJRW theory.Comment: 50 pages (accepted in Duke Mathematical Journal

A remarkable sequence of integers

A survey of properties of a sequence of coefficients appearing in the evaluation of a quartic definite integral is presented. These properties are of analytical, combinatorial and number-theoretical nature.Comment: 20 pages, 5 figure

Real Zeros and Partitions without singleton blocks

We prove that the generating polynomials of partitions of an $n$-element set into non-singleton blocks, counted by the number of blocks, have real roots only and we study the asymptotic behavior of the leftmost roots. We apply this information to find the most likely number of blocks.Comment: 16 page