Michael Hutchings

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He published, and he published, and he still perished. -Epitaph.

I love writing, but I hate the paperwork. -Unknown.

Please feel free to email me with questions, comments, bug reports, or reprint requests.

[1] The shortest enclosure of three connected areas in R^2 (with C. Cox, L. Harrison, S. Kim, J. Light, A. Mauer, and M. Tilton),
Real Analysis Exchange 20 (1994-95), 313-35.
Given three prescribed areas A_1, A_2, A_3, we determine the shortest union of curves in R^2 whose complement has four connected components, three of which are bounded and have areas A_1,A_2,A_3. This is a generalization to multiple regions of the classical isoperimetric theorem. We introduce the ``overlapping bubbles'' trick, with which we can rule out some pathological competitors by temporarily allowing the curves to cross each other.

[2] The structure of area-minimizing double bubbles,
Journal of Geometric Analysis 7 (1997), 285-304. Review.
This paper investigates how to enclose and separate two (possibly disconnected) regions of prescribed volumes in R^n, S^n, or H^n with the least possible surface area. We prove that the exterior region must be connected, and we develop techniques to bound the number of components of the enclosed regions in an area-minimizer. The key is a topological argument which proves that the least area required to enclose two volumes is a concave function of the two volumes. We also prove a general symmetry theorem for minimal enclosures of m volumes in R^n for m < n, based on an idea due to Brian White.

[3] The double bubble conjecture (with J. Hass and R. Schlafly),
Electronic Research Announcements of the AMS 1 (1995), 98-102. Postscript PDF
The topological complexity bounds of [2], together with known symmetry results, reduce the isoperimetric problem for two volumes in R^n to a finite-dimensional minimization problem. Using rigorous numerical methods, Hass and Schlafly solved this problem for two equal volumes in R^3, showing that the unique minimizer is the ``standard double bubble''. Article [3] is a joint announcement of our results.

[4] Integration of singular braid invariants and graph cohomology,
Transactions of the AMS 350 (1998), 1791-1809. Postscript PDF
This paper attempts to explain some of the mysteries underlying the existence of Vassiliev knot invariants from a topological point of view, in the easier case of braids. In particular we prove necessary and sufficient conditions for an arbitrary invariant of singular braids with m double points to be ``integrable'' to a braid invariant. This gives a slight generalization of the existence theorem for Vassiliev invariants of braids.

[5] Circle-valued Morse theory, Reidemeister torsion, and Seiberg-Witten invariants of 3-manifolds (with Y-J. Lee),
Topology 38 (1999), 861-888. Postscript PDF
We show that by suitably counting closed orbits and flow lines between critical points of the gradient of a circle-valued Morse function on a manifold, one recovers a form of topological Reidemeister torsion. In the three-dimensional case, we conjecture that a finer version of the Morse theory invariant is equal to the Seiberg-Witten invariant, by analogy with Taubes' ``Seiberg-Witten = Gromov'' theorem in four dimensions. Combining our theorem with this conjecture, we recover the Meng-Taubes formula relating part of the Seiberg-Witten invariant to Milnor torsion.

[6] The isoperimetric problem on surfaces (with H. Howards and F. Morgan),
American Mathematical Monthly 106 (1999), 430-439.
This article is an elementary survey of isoperimetric theorems on various surfaces, with a few new proofs.

[7] The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature (with H. Howards and F. Morgan),
Transactions of the AMS 352 (2000), 4889-4909. Postscript PDF
We solve the problem described by the title. This paper expands on part of [6].

[8] An introduction to the Seiberg-Witten equations on symplectic four-manifolds (with C. H. Taubes),
Symplectic geometry and topology (Park City, UT, 1997), 103-142, AMS, 1999.
This expository article is based on Taubes's lectures at Park City, Utah in July 1997. Lecture 1: Background from differential geometry. Lecture 2: Spin and the Seiberg-Witten equations. Lecture 3: The Seiberg-Witten invariants. Lecture 4: The symplectic case, part I. Lecture 5: The symplectic case, part II.

[9] Circle-valued Morse theory and Reidemeister torsion (with Y-J. Lee),
Geometry and Topology 3 (1999), 369-396. Postscript PDF.
This paper proves a refinement of the main theorem of [5], using a different method. In the three dimensional case, combining this result with the conjecture of [5], we obtain a formula for the the full Seiberg-Witten invariant, which was conjectured by Turaev. (Update: the latter formula has since been proved by Turaev, refining the work of Meng-Taubes. Turaev's result and [9] indirectly prove the conjecture of [5].)

[10] Reidemeister torsion in generalized Morse theory,
Forum Mathematicum 14 (2002), 209-244. pdf
This paper re-proves the results of [5] and [9], and extends them from circle-valued functions to closed 1-forms. The strategy is to first give an a priori proof, by bifurcation analysis, that our Morse-theoretic analogue of Reidemeister torsion is a topological invariant. We then use invariance to reduce to the easier case of real-valued functions. It is hoped that the a priori proof of invariance will provide a model for the possible construction of torsion invariants in Floer theory.

[11] Proof of the double bubble conjecture (with F. Morgan, M. Ritore, and A. Ros),
Electronic Research Announcements of the AMS 6 (2000), 45-49. Postscript PDF Publicity Pictures.
This article is an announcement of the results in [12] below.

[12] Proof of the double bubble conjecture (with F. Morgan, M. Ritore, and A. Ros),
Annals of Mathematics 155 (2002), no. 2, 459-489. pdf
This paper proves that the standard double bubble in R^3 is uniquely area-minimizing for arbitrary volumes, without the help of a computer. (Compare [3].) The key is a new instability argument, which rules out the nonstandard possibilities that were not already ruled out by [2].

[13] An index inequality for embedded pseudoholomorphic curves in symplectizations
Journal of the European Math. Soc. 4 (2002), 313-361. PDF PS
Abstract: Let S be a surface with a symplectic form, let f be an automorphism of S, and let Y be the mapping torus of f. We show that the dimensions of moduli spaces of embedded pseudoholomorphic curves in R x Y, with cylindrical ends asymptotic to periodic orbits of f or multiple covers thereof, are bounded from above by an additive relative index. We deduce some compactness results for these moduli spaces.
This paper establishes some of the foundations for a program with Michael Thaddeus, to understand the Seiberg-Witten Floer homology of Y in terms of such pseudoholomorphic curves. Analogues of our results should also hold in three dimensional contact topology.

[14] The periodic Floer homology of a Dehn twist (with M. Sullivan),
June 2002, 48 pages, 5 pictures. Rated 'R' for spectral sequences. Postscript.
Abstract: The periodic Floer homology of a surface symplectomorphism, defined by the first author and M. Thaddeus, is the homology of a chain complex which is generated by certain unions of periodic orbits, and whose differential counts embedded pseudoholomorphic curves in R cross the mapping torus. It is conjectured to recover the Seiberg-Witten Floer homology of the mapping torus for most spin-c structures, and is related to contact homology. In this paper we compute the periodic Floer homology of some Dehn twists.

[15] Floer homology of families I [new]
August 2003, 33 pages. pdf postscript.
Abstract: For any version of Floer theory, suppose we are given a family of equivalent objects parametrized by a manifold B. Then there exists, modulo the usual analytical issues, a natural spectral sequence whose E^2 term is the homology of B with twisted coefficients in the Floer homology of the fibers. The higher terms and differentials in this spectral sequence give homotopy invariants of the family. This construction formally gives invariants of families of Hamiltonian isotopic symplectomorphisms, families of 3-manifolds, families of Legendrian knots, etc. This paper explains the spectral sequence in detail for the model case of finite dimensional Morse theory, and shows that it agrees here with the Leray-Serre spectral sequence. The spectral sequence for families of symplectomorphisms will be discussed in a sequel.

Coming soon: Periodic Floer homology (or something like that), with M. Thaddeus.
I'm sorry that this isn't finished already, but this paper will define periodic Floer homology. In the meantime, there is a summary of this material in paper [14] above.

Coming soon: Floer homology of families II (or something like that).
This paper will discuss Floer-theoretic invariants of families of symplectomorphisms, following [15].

Coming soon: Rounding corners of polygons and the embedded contact homology of T^3 (or something like that), with M. Sullivan.
This paper will generalize [14] to calculate the embedded contact homology of T^3, in terms of some amusing combinatorics involving polygons in the plane whose vertices are at lattice points.

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