Motivated by a 1993 conjecture of Stanley and Stembridge, Shareshian and Wachs conjectured that the characteristic map takes the character of the dot action of the symmetric group on the cohomology of a regular semisimple Hessenberg variety to ωXG(t), where XG(t) is the chromatic quasisymmetric function of the incomparability graph G of the corresponding natural unit interval order, and ω is the usual involution on symmetric functions. We prove the Shareshian-Wachs conjecture.Our proof uses the local invariant cycle theorem of Beilinson-Bernstein-Deligne to obtain a surjection, which we call the local invariant cycle map, from the cohomology of a regular Hessenberg variety of Jordan type λ to a space of local invariant cycles. As λ ranges over all partitions, the local invariant cycles collectively contain all the information about the dot action on a regular semisimple Hessenberg variety. We then prove a result showing that, under suitable hypotheses, the local invariant cycle map is an isomorphism if and only if the special fiber has palindromic cohomology. (This is a general theorem, which independent of the Hessenberg variety context.) Applying this result to the universal family of Hessenberg varieties, we show that, in our case, the surjections are actually isomorphisms, thus reducing the Shareshian-Wachs conjecture to computing the cohomology of a regular Hessenberg variety. But this cohomology has already been described combinatorially by Tymoczko, and, using a new reciprocity theorem for certain quasisymmetric functions, we show that Tymoczko's description coincides with the combinatorics of the chromatic quasisymmetric function.

Conventional optical networks are based on SONET rings, but since rings are known to use bandwidth inefficiently, there has been much research into shared mesh protection, which promises significant bandwidth savings. Unfortunately, most shared mesh protection schemes cannot guarantee that failed traffic will be restored within the 50 ms timeframe that SONET standards specify. A notable exception is the p -cycle scheme of Grover and Stamatelakis. We argue, however, that p -cycles have certain limitations, e.g., there is no easy way to adapt p -cycles to a path-based protection scheme, and p -cycles seem more suited to static traffic than to dynamic traffic. In this paper we show that the key to fast restoration times is not a ring-like topology per se, but rather the ability to pre -cross-connect protection paths. This leads to the concept of a pre -cross-connected trail or PXT, which is a structure that is more flexible than rings and that adapts readily to both path-based and link-based schemes and to both static and dynamic traffic. The PXT protection scheme achieves fast restoration speeds, and our simulations, which have been carefully chosen using ideas from experimental design theory, show that the bandwidth efficiency of the PXT protection scheme is comparable to that of conventional shared mesh protection schemes.

Abstract-In this paper, we study the problem of traffic grooming to reduce the number of transceivers in optical networks. We show that this problem is equivalent to a certain traffic maximization problem. We give an intuitive interpretation of this equivalence and use this interpretation to derive a greedy algorithm for transceiver minimization. We discuss implementation issues and present computational results comparing the heuristic solutions with the optimal solutions for several small example networks. For larger networks, the heuristic solutions are compared with known bounds on the optimal solution obtained using integer programming tools.

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