The combinatorial formula for open gravitational descendents

In recent works, [20],[21], descendent integrals on the moduli space of Riemann surfaces with boundary were defined. It was conjectured in [20] that the generating function of these integrals satisfies the open KdV equations. In this paper we develop the notions of symmetric Strebel-Jenkins differentials and of Kasteleyn orientations for graphs embedded in open surfaces. In addition we write an explicit expression for the angular form of the sum of line bundles. Using these tools we prove a formula for the descendent integrals in terms of sums over weighted graphs. Based on this formula, the conjecture of [20] was proved in [5]

Open rr-spin theory II: The analogue of Witten's conjecture for rr-spin disks

We conclude the construction of $r$-spin theory in genus zero for Riemann surfaces with boundary. In particular, we define open $r$-spin intersection numbers, and we prove that their generating function is closely related to the wave function of the $r$th Gelfand--Dickey integrable hierarchy. This provides an analogue of Witten's $r$-spin conjecture in the open setting and a first step toward the construction of an open version of Fan--Jarvis--Ruan--Witten theory. As an unexpected consequence, we establish a mysterious relationship between open $r$-spin theory and an extension of Witten's closed theory.Comment: The more foundational parts of the previous version, v3, were moved to the article arXiv:2003.01082. These include the description of objects, constructions of moduli spaces and bundles and proofs of orientations theorems. The name of the paper and the abstract were changed accordingl

Majority Dynamics and the Retention of Information

We consider a group of agents connected by a social network who participate in majority dynamics: each agent starts with an opinion in {−1, +1} and repeatedly updates it to match the opinion of the majority of its neighbors. We assume that one of {−1, +1} is the “correct” opinion S, and consider a setting in which the initial opinions are independent conditioned on S, and biased towards it. They hence contain enough information to reconstruct S with high probability. We ask whether it is still possible to reconstruct S from the agents’ opinions after many rounds of updates. While this is not the case in general, we show that indeed, for a large family of bounded degree graphs, information on S is retained by the process of majority dynamics. Our proof technique yields novel combinatorial results on majority dynamics on both finite and infinite graphs, with applications to zero temperature Ising models

The point insertion technique and open rr-spin theories II: intersection theories in genus-zero

The papers [5, 3, 6, 19, 20] initiated the study of open $r$-spin and open FJRW intersection theories, and related them to integrable hierarchies and mirror symmetry. This paper uses a new technique, the point insertion technique, developed in the prequel [36], to define new open r-spin and open FJRW intersection theories. These new constructions provide potential candidates for theories whose existence was conjectured before: $\bullet$ K. Hori [23] predicted the existence of open $r$-spin theory with $\lfloor\frac{r}{2}\rfloor$ types of boundary states. The one constructed in [5, 3] has only one type of boundary state. In this work we describe $\lfloor\frac{r}{2}\rfloor$ open $r$-spin theories, labelled by $\mathfrak{h}\in\{0,\ldots,\lfloor\frac{r}{2}\rfloor-1\},$ where the $\mathfrak{h}$-th one has $\mathfrak{h}+1$ boundary states. We prove that the $\mathfrak{h}=0$ theory is equivalent to the [5, 3] construction, and calculate all intersection numbers for all these theories. $\bullet$ In [1] K. Aleshkin and C.C.M. Liu conjectured the existence of a quintic Fermat FJRW theory. We construct such an FJRW theory, and provide evidence that this is the conjectured theory. We also explain how the point insertion technique can be used for constructing other open enumerative theories, satisfying the same universal recursions

Harmonic Labeling of Graphs

Which graphs admit an integer value harmonic function which is injective and surjective onto $\Z$? Such a function, which we call harmonic labeling, is constructed when the graph is the $\Z^2$ square grid. It is shown that for any finite graph $G$ containing at least one edge, there is no harmonic labeling of $G \times \Z$