Refined Razumov–Stroganov conjectures for open boundaries

Refined Razumov-Stroganov conjectures for open boundaries

Recently it has been conjectured that the ground-state of a Markovian Hamiltonian, with one boundary operator, acting in a link pattern space is related to vertically and horizontally symmetric alternating-sign matrices (equivalently fully-packed loop configurations (FPL) on a grid with special boundaries). We extend this conjecture by introducing an arbitrary boundary parameter. We show that t...

Open Conjectures on Congruences

Abstract. We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while conjectures in Part B have been recently confirmed. We hope that this material will interest number theorists and stimulate further research. Number theoris...

Open Positivity Conjectures for Integer Partitions

The purpose of this report is to collect some open positivity conjectures for integer partitions. The conjectures in §2-§3 are simple to state and initially appear easy to do, but remain open. The conjectures in §4-§5 should be related to representation theory, since they generalize known facts in this area. The conjectures in §6 are speculative, and if true, would require new combinatorial ins...

Euler Systems and Refined Conjectures of Birch Swinnerton-Dyer Type

The relationship between arithmetic objects (such as global fields, or varieties over global fields) and the analytic properties of their Lfunctions poses many deep and difficult questions. The theme of this paper is the Birch and Swinnerton Dyer conjecture, and certain refinements that were proposed by Mazur and Tate. We will formulate analogues of these conjectures over imaginary quadratic fi...

Nonautonomous difference equations: Open problems and conjectures