An (r, s)-formation is a concatenation of s permutations of r letters. If u is a sequence with r distinct letters, then let Ex (u, n) be the maximum length of any r-sparse sequence with n distinct letters which has no subsequence isomorphic to u. For every sequence u define fw (u), the formation width of u, to be the minimum s for which there exists r such that there is a subsequence isomorphic to u in every (r, s)-formation. We use fw (u) to prove upper bounds on Ex (u, n) for sequences u such that u contains an alternation with the same formation width as u.We generalize Nivasch's bounds on Ex ((ab) t , n) by showing that fw ((12 . . . l) t ) = 2t−1 and Ex ((12 . . . l) t , n) = n2by the NSF Graduate Research Fellowship under Grant No. 1122374.obtained from v by only moving the first letter of v to another place in v, then we show that fw (u) = 4 and Ex (u, n) = Θ(nα(n)). Furthermore we prove that fw (abc(acb) t ) = 2t + 1 and Ex (abc(acb) t , n) = n2 1 (t−1)! α(n) t−1 ±O(α(n) t−2 ) for every t ≥ 2.

Spectral invariants arising from twisted periodic Floer homology have recently played a key role in resolving various open problems in two-dimensional dynamics. We resolve in great generality a conjecture of Hutchings regarding the relationship between the asymptotics of the twisted PFH spectral invariants and the Calabi homomorphism for area-preserving disk maps. Our result, by an argument similar to that of Irie in the setting of Reeb flows, allows us to prove the smooth closing lemma for area-preserving diffeomorphisms of a closed surface. DAN CRISTOFARO-GARDINER, ROHIL PRASAD, AND BOYU ZHANG 5. The relationship between twisted SWF and twisted PFH 72 5.1. Lee-Taubes' isomorphism in the untwisted setting 72 5.2. Lee-Taubes' isomorphism in the twisted setting 73 5.3. The twisted isomorphism and the relative Z-grading 76 5.4. Twisted SWF spectral invariants recover twisted PFH spectral invariants 85 6. Properties of twisted PFH spectral invariants 89 6.1. PFH cobordism maps 89 6.2. The pseudoholomorphic curve axiom 91 6.3. Spectrality 93 6.4. Hofer continuity 94 7. Exact triangles, vanishing and non-vanishing 98 8. Seiberg-Witten-Floer max-min families 8.1. Estimates on continuation instantons 8.2. Max-min for r ą ´2πρ 8.3. Max-min for r " ´2πρ 8.4. Differential equations 8.5. Recovering SWF spectral invariants 9. Proof of Hutchings' conjecture 10. Proof of the closing lemma References

We introduce a method for constructing invariant probability measures of a large class of non-singular volume-preserving flows on closed, oriented odd-dimensional smooth manifolds using pseudoholomorphic curve techniques from symplectic geometry. These flows include any non-singular volume preserving flow in dimension three, and autonomous Hamiltonian flows on closed, regular energy levels in symplectic manifolds of any dimension. As an application, we use our method to prove the existence of obstructions to unique ergodicity for this class of flows, generalizing results of Taubes and Ginzburg-Niche.

Edtmair-Hutchings have recently defined, using Periodic Floer homology, a "U-cycle property" for Hamiltonian isotopy classes of area-preserving diffeomorphisms of closed surfaces. They show that every Hamiltonian isotopy class satisfying the U-cycle property satisfies the smooth closing lemma and also satisfies a kind of Weyl law involving the actions of certain periodic points; they show that every rational isotopy class on the two-torus satisfies the U-cycle property. The purpose of this note is to explain why the U-cycle property holds for every rational Hamiltonian isotopy class.

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